Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration

Abstract

The paper describes essential reaction–diffusion models with delay arising in population theory, medicine, epidemiology, biology, chemistry, control theory, and the mathematical theory of artificial neural networks. A review of publications on the exact solutions and methods for their construction is carried out. Basic numerical methods for integrating nonlinear reaction–diffusion equations with delay are considered. The focus is on the method of lines. This method is based on the approximation of spatial derivatives by the corresponding finite differences, as a result of which the original delay PDE is replaced by an approximate system of delay ODEs. The resulting system is then solved by the implicit Runge–Kutta and BDF methods, built into Mathematica. Numerical solutions are compared with the exact solutions of the test problems.

1. Introduction

In natural science, there are numerous nonlinear systems that are hereditary when the quantities under consideration depend both on the present state and on the history of the process. In particular, the state of the system may be determined only by a certain moment in the past. These systems are referred to as systems with delay. To model such systems, one should use the differential equations containing not only the unknown function $u\left(t\right)$ but also the function $u(t-\tau )$, where t is the time and $\tau >0$ is the delay [1,2]. Typically, the delay time is assumed to be constant. The simplest ordinary differential equation (ODE) with delay has the form

$$u_t^{\prime }=F(u,w),w=u(t-\tau ),$$

(1)

where F is a function. Delay ODEs arise in population theory [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23], medicine [24,25,26,27,28,29,30,31,32,33,34,35], epidemiology [36,37,38,39,40,41,42,43,44], economy [45,46,47,48], climatology [49], mechanics [50], control theory [51], the mathematical theory of artificial neural networks [52,53,54,55,56,57], etc.

However, the processes occurring in such systems are often spatially inhomogeneous, which leads to the use of more complex reaction–diffusion equations (RDEs) with a constant delay (see, e.g, [58,59]):

$$u_t=a{u}_{xx}+F(u,w),w=u(x,t-\tau ),$$

(2)

where $u=u(x,t)$, $a>0$ is the diffusion coefficient, and F is the kinetic function. In the special case, when $F(u,w)=f\left(w\right)$ in (2), we have a simple physical interpretation: the substance transfer in a local non-equilibrium medium possesses inertial properties, i.e., the reaction of the system is not instantaneous but delayed for $\tau$.

In most cases, models described by nonlinear delay RDEs (2) are obtained by formally adding the diffusion term $a{u}_{xx}$ to the right side of the delay ODE (1). With this approach, random walks are modeled, in which the movement of each considered object (subject) is due to Fickian diffusion, when the flow is proportional to the concentration gradient and the proportionality constant is negative. For example, in population dynamics models, a phenomenon similar to diffusion arises from the tendency of any species to migrate to regions with a lower population density [60]. For simplicity, it is generally assumed that food is supplied continuously and uniformly in time and space. Thus, food will become scarce in regions with a higher population density, and species will tend to migrate to regions with a lower density in order to have a higher chance of survival. Papers [61,62,63] discuss the diffusion process from an ecological point of view.

Remark 1. 

The formal introduction of the diffusion term into the right side of a delay ODE can lead to some difficulties. The fact is that, although diffusion and delay are related to space and time, respectively, they are not independent of each other, because the individuals, cells, neurons, molecules, etc., are not at the same location at previous times. Possible ways to overcome this problem by introducing a distributed (non-local) delay are discussed in [64].

Nonlinear delay RDEs of form (2) and related, more complex equations and systems arise in various applications, such as population theory [65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80], medicine [81,82,83,84,85,86,87,88,89], epidemiology [90,91,92,93,94,95,96,97,98,99,100], biology [101,102,103,104,105,106], chemistry [58,107,108,109,110], control theory [59,111,112,113], the theory of artificial neural networks [114,115,116,117,118,119], and many others (see [59,120]). Some models described by nonlinear delay RDEs are considered below.

Delay greatly complicates the study of nonlinear equations of form (2). Such equations have solutions of a traveling-wave type $u=u\left(z\right)$, where $z=kx+\lambda t$ (see, for example, [59,121,122,123]), but do not have self-similar solutions $u=t^{\beta }\phi \left(x{t}^{\lambda }\right)$, which non-delay RDEs often admit. More complex exact solutions of nonlinear delay RDEs were obtained in [124,125,126,127,128,129,130,131,132,133,134]. Some exact solutions are given below.

The known methods of the numerical integration of non-delay RDEs, after certain modification, can also be used in solving initial-boundary value problems for delay RDEs. The method of lines and finite-difference methods are most widely used (see [120], Section 5.2). The method of lines allows one to reduce a partial differential equation (PDE) with delay to a system of delay ODEs. The resulting system can be solved using Mathematica [135], Maple [136], or MATLAB [137], which are not designed for solving delay PDEs directly (for other data on using Maple and Mathematica for scientific research, see [138]). Below, these numerical methods will be considered in more detail.

It is important to note that delay differential equations have a number of specific features [59,120,139,140] that are not inherent in non-delay equations and significantly complicate obtaining adequate numerical solutions. The fact is that even in the absence of delay, theoretical estimates of the accuracy of numerical solutions of nonlinear RDEs contain constants that depend on the smoothness of the solution and cannot be calculated a priori (this applies to non-smooth solutions, which are typical for delay equations). The practical convergence of numerical methods, based on the refinement of the grid, also cannot fully guarantee the reliability of the schemes used and the accuracy of calculations (especially near the values of the problem parameters corresponding to unstable solutions, or near those values of variables that correspond to singularities of the equation or large gradients of solutions).

In many cases, the most efficient and most illustrative way to assess the range of the applicability and accuracy of numerical methods is a direct comparison of numerical and exact solutions of test problems constructed using delay RDEs, allowing solutions in elementary functions. In the final part of the paper, some test problems containing free parameters are presented, and the numerical and exact solutions of these test problems are compared. The numerical solutions are obtained by the method of lines in combination with the implicit Runge–Kutta and BDF methods built into Mathematica.

2. Models Using Delay RDEs

2.1. Models of Population Theory

At present, much attention is paid to reaction–diffusion models with delay in population theory. The delay $\tau$ characterizes the average reproductive age, i.e., the average time from the moment of birth (sometimes from the moment the egg is laid) to the moment of becoming sexually mature. The diffusion logistic equation with delay has the form [66,141,142]:

$$\begin{array}{c}\hfill u_t=a{u}_{xx}+bu(1-w/k),w=u(x,t-\tau ),\end{array}$$

(3)

where $u=u(x,t)\ge 0$ is the population density, $b>0$ is the population growth rate, k is the carrying capacity of the environment, $\tau$ is the average reproductive age, $0<a\ll 1$ is the diffusion coefficient. The corresponding delay ODE (with $a=0$) is called the Hutchinson equation and is widely studied [16,17,19,20]. A more complex reaction–diffusion model, taking into account the limited food supply, is considered in [65,66].

The diffusive Nicholson’s blowflies equation with delay, describing the dynamics of the blowfly population, has the form

$$\begin{array}{c}\hfill u_t=a\mathsf{\Delta }u-\delta u+pw{e}^{-\kappa w},w=u(x,t-\tau ),\end{array}$$

(4)

where $p>0$ is the maximum possible per capita egg production rate (corrected for egg to adult survival), $1/\kappa >0$ is the population size at which the population achieves maximum reproductive success, $\delta >0$ is the per capita adult death rate, $\tau >0$ is the time it takes for an egg to develop into a sexually mature adult. Stability of solutions of the diffusive Hutchinson’s equation with delay (4) is studied in [72,73,74,75,76].

The reaction–diffusion Lotka–Volterra model with several delays is described by the system of equations:

$$\begin{array}{cc}\hfill & u_t=a_1{u}_{xx}+b_{1}u(1-c_1{\overline{u}}_1+d_1{\overline{v}}_2),\hfill \\ \hfill & v_t=a_2{v}_{xx}+b_{2}v(1+d_2{\overline{u}}_3-c_2{\overline{v}}_4),\hfill \end{array}$$

(5)

where $u=u(x,t)$ and $v=v(x,t)$ are the unknown functions; ${\overline{u}}_i=u(x,t-{\tau }_i)$, ${\overline{v}}_j=v(x,t-{\tau }_j)$ ($i=1,3$; $j=2,4$); ${\tau }_i\ge 0$ and ${\tau }_j\ge 0$ are the delays. System (5) describes the interaction of two species (in the case $d_1=d_2=0$, we have two independent diffusion logistic equations (3)). The unknown functions $u(x,t)$ and $v(x,t)$ and the coefficients $a_i$, $b_i$, $c_i$ ($i=1,2$) are non-negative and have a physical meaning similar to functions and coefficients of Equation (3). The delays ${\tau }_1$ and ${\tau }_4$, as in the diffusion logistic equation, characterize the average reproductive age of individuals, and the delays ${\tau }_2$ and ${\tau }_3$ are responsible for the time required for changes in the size of one population to lead to changes in another. All delays are non-negative and can be equal to zero in certain models. Terms with nonzero coefficients $d_1$ and $d_2$ distinguish the considered model from the diffusion logistic equation. These coefficients are responsible for the interaction between two populations. In the case of cooperative interaction, when one species persists in the absence of the other, and when the species mutually increase each other’s growth rate, both coefficients $d_1$ and $d_2$ are positive. In the case of competitive interaction, an increase in one population leads to a decrease in another (for example, an increase in the number of predators leads to a decrease in the population of prey), and the coefficients $d_1$ and $d_2$ are negative. Delay Lotka–Volterra cooperative models are discussed in [67,68], competitive models in [69,70,78], three-species model in [71].

In [68], a simpler than (5) system was studied in the form

$$\begin{array}{cccccc}\hfill & u_t=u_{xx}+bu(1-\overline{u}+d_1\overline{v}),\hfill & \hfill & 0<x<\pi ,\hfill & \hfill & t>0,\hfill \\ \hfill & v_t=v_{xx}+bv(1+d_2\overline{u}-\overline{v}),\hfill & \hfill & 0<x<\pi ,\hfill & \hfill & t>0,\hfill \end{array}$$

(6)

where $\overline{u}=u(x,t-\tau )$, $\overline{v}=v(x,t-\tau )$, with boundary and initial conditions

$$\begin{array}{cc}\hfill & u(0,t)=u(\pi ,t)=v(0,t)=v(\pi ,t)=0,t\ge 0,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & u(x,t)=v(x,t)=0.1(1+t/\tau )sinx,-\tau \le t\le 0,0\le x\le \pi .\hfill \end{array}$$

(8)

System (6) is obtained from (5), if we put $a_1=a_2=1$, $b_1=b_2=b$, $c_1=c_2=1$ and ${\tau }_1=\dots ={\tau }_4=\tau$. In [68], the following statements are proved.

$1^{\circ }$

For all $\tau >0$, the trivial solution $u=v=0$ of system (6) is stable for $b<1$ and unstable for $b>1$.

$2^{\circ }$

For $d_1{d}_2<1$ and $b=1+\epsilon$, where $0<\epsilon \ll 1$, there exists a positive stationary solution of system (6) satisfying the boundary conditions (6) such that $u,v\ne \mathrm{const}$.

$3^{\circ }$

Under the conditions from item $2^{\circ }$, there exists a value ${\tau }_b$ such that the stationary solution of system (6) is asymptotically stable for $0\le \tau <{\tau }_b$ and unstable for $\tau >{\tau }_b$.

Figure 1 shows graphs of the numerical solutions of problems (6)–(8) at the point $x=\pi /2$ obtained in Mathematica by a combination of the method of lines (for $N=200$) and the BDF method for $d_1=0.4$, $d_2=0.7$, and various b and $\tau$ (see [120], Section 6.2.4). Figure 1a corresponds to the case of a stable trivial equilibrium, Figure 1b to a stable positive stationary solution, Figure 1c to an unstable positive stationary solution. The dotted lines in Figure 1b correspond to the stationary solution, which, for $x=\pi /2$, has the values $u\approx 0.023$, $v\approx 0.028$.

Remark 2. 

Paper [68] provides graphs of solutions obtained for other parameter values by other methods: using MATLAB by combining the method of steps and an implicit numerical method of integrating PDEs.

2.2. Models Describing the Spread of Epidemics and the Development of Diseases

Models of the spread of epidemics and the development of diseases are related to models of population theory. Models of the spread of epidemics describe the dynamics of interactions between susceptible, infectious, and immune individuals (sometimes other groups of individuals are also considered). The best known cause of the delay in epidemiological models is the latent period of the disease in the infected organism: it takes some time before the infection in the infected host develops to a level sufficient for further transmission. Sometimes the latent period coincides with the incubation period, that is, the time from the moment of infection to the first signs of the manifestation of the disease. However, in general, these two periods do not coincide. Due to the high mobility of people within one country or even around the world, spatially homogeneous models do not adequately describe the spread of the disease. To make the model more realistic, it is necessary to take into account the heterogeneity of space. If the environment is spatially continuous, random diffusion is often used to describe population mobility, leading to models based on RDEs [61].

The main difference between models of disease development from models of the spread of epidemics is to explicitly take into account an intermediate agent (virus) in the process of transmission of infection from an infected cell to an uninfected one. Infection occurs not from a contact between an infected and susceptible individual, as is the case in epidemic models, but from a contact between a susceptible cell and a virus. Such models take into account the intracellular delay between the moment of cell infection and the start of cell production of new viruses.

The delay diffusion SIR model of the spread of epidemics of Kermack–Mckendrick type is described by the system of equations:

$$\begin{array}{cc}\hfill & {\left(u_1\right)}_t=a_1{\left(u_1\right)}_{xx}-\beta u_1{w}_2,\hfill \\ \hfill & {\left(u_2\right)}_t=a_2{\left(u_2\right)}_{xx}+\beta u_1{w}_2-\gamma u_2,\hfill \\ \hfill & {\left(u_3\right)}_t=a_3{\left(u_3\right)}_{xx}+\gamma u_2.\hfill \end{array}$$

(9)

Here, $u_1=u_1(x,t)$, $u_2=u_2(x,t)$, $u_3=u_3(x,t)$ are the population densities of susceptible, infected, and removed from the population (due to the acquisition of immunity or death) individuals; $w_2=u_2(x,t-\tau )$ is a function with delay; $\beta$ is the transmission coefficient (the average number of contacts of an infectious individual per unit of time); $\gamma$ is the recovery rate of infected individuals; $a_1$, $a_2$, $a_3$ are the diffusion coefficients. In system (9), the nonlinear term $\beta u_1{w}_2$ is responsible for mass incidence, and the removal from the population occurs in proportion to the share of infected individuals $\gamma u_2$; Model (9) is called the SIR model because individuals move from Susceptible to Infectious and then Removed.

Other epidemic models are discussed, for example, in [90,91,92,93,94,95,96,97,98,99]. Systems of six equations, including two nonlinear delay RDEs, describing the dynamics of the spread of the COVID-19 epidemic, are considered in [100]. Related to models of the spread of epidemics, models of the development of diseases inside the organism, for example, hepatitis B, are proposed in [81,82,83,84,85,86]. Models of interaction between immune and tumor cells are explored in [87,88,89].

2.3. Other Delay Reaction–Diffusion Models

To describe the dynamics of a homogeneous population of mature circulating blood cells of density $u=u\left(t\right)$, the following delay RDE can be used [101,102]:

$$u_t=a{u}_{xx}-\gamma u+{\beta }_0\frac{{\theta }^{n}w}{{\theta }^n+w^n},w=u(t-\tau ),$$

(10)

where ${\beta }_0$, $\theta$, n, and $\gamma$ are some parameters, the description of which can be found in [143]. Similar equations and systems were also studied in [103,104,105,106].

The Belousov–Zhabotinsky reaction is a class of chemical reactions occurring in an oscillatory mode in which some reaction parameters (color, concentration of components, temperature, etc.) change periodically, forming a complex spatiotemporal structure of the reaction medium (see [144,145,146]). To study circular spatiotemporal waves that arise in a solution prepared in a certain way, a model with delay is used [58,107,108,109,110]:

$$\begin{array}{cc}\hfill u_t& =u_{xx}+u(1-u-b\overline{v}),\overline{v}=v(x,t-\tau ),\hfill \\ \hfill v_t& =v_{xx}-cuv,\hfill \end{array}$$

(11)

where $u=u(x,t)$ and $v=v(x,t)$ are the dimensionless concentrations of bromous acid and bromide ions, respectively, and ($0\le u,v\le 1$), $b>0$, and $c>0$ are dimensionless parameters (see [146]).

Delay RDEs are used to describe the heat treatment of metal sheets [59,111,112,113]. The delay is responsible for the time between the moment the temperature values are taken, and the signal arrives at the controllers.

Delay RDEs and systems of such equations are widely used in the mathematical theory of artificial neural networks, the results of which are used for signal and image processing, in pattern recognition problems, in associative machine memory, in determining the speed of moving objects. The delay occurs in artificial neural networks due to the finite switching speed of amplifiers and the finite speed of signal propagation between neurons [114,115,116,117,118,119].

Remark 3. 

The models considered above and some other models based on ODEs, RDEs, and systems with delays are discussed in detail in [120], Section 6.

3. Exact Solutions of Nonlinear Delay RDEs

3.1. The Concept of ‘Exact Solution’

The general solution of nonlinear delay RDEs cannot be found even in the simplest cases. This means that we can only search for and study particular solutions, which are often referred to as exact solutions. In our study, we will use the concept of ‘exact solution’ for nonlinear delay RDEs in the same way as it was used in [126,128,147,148].

3.2. Exact Solutions of Traveling-Wave Type

RDEs of form (2) admit exact solutions of traveling-wave type:

$$\begin{array}{c}\hfill u=U\left(z\right),z=kx+\lambda t,\end{array}$$

(12)

where k and $\lambda$ are nonzero constants. Substituting (12) into (2), we obtain the nonlinear ODE with constant delay:

$$a{k}^2{U}_{zz}^{″}-\lambda U_z^{\prime }+F(U,W)=0,$$

(13)

where $W=U(z-\sigma )$, $\sigma =\lambda \tau$.

Representable in elementary functions, exact solutions of traveling-wave type to reduced nonlinear delay ODEs (13) and the corresponding original delay RDEs (2), whose kinetic function contains arbitrary functions, were obtained in [123].

Numerous studies have been devoted to the existence and stability of traveling-wave solutions of delay RDEs (see, for example, [58,67,76,79,80,149] and references therein).

3.3. More Complex Exact Solutions

Numerous papers contain more complex exact solutions expressed in elementary functions with a generalized or functional separation of variables for various classes of nonlinear delay PDEs. Solutions are constructed by the method of functional constraints [126] or its modifications [127,131,150]. The method of functional constraints is to seek for solutions with a generalized

$$u=\sum _{n=1}^N{\phi }_n\left(x\right){\psi }_n\left(t\right)$$

or functional

$$u=U\left(z\right),z=\sum _{n=1}^N{\phi }_n\left(x\right){\psi }_n\left(t\right),$$

separation of variables, which must satisfy one of two functional constraints:

$$\begin{array}{cc}\hfill z(u,w)=p\left(x\right),& w=u(x,t-\tau );\hfill \\ \hfill z(u,w)=q\left(t\right),& w=u(x,t-\tau ).\hfill \end{array}$$

Thus, an admissible form of the exact solution is found, which is then substituted into the original equation to determine the final form.

Papers [125,126,127,128,131] present exact solutions with generalized and functional separation of variables for nonlinear RDEs of form (2) in which the kinetic function F depends on one or more arbitrary functions of one/two arguments, or on free parameters, and for more complex delay equations of a higher order:

$$\begin{array}{c}L\left[u\right]=M\left[u\right]+F(u,w),\\ L\left[u\right]=\sum _{i=1}^n{b}_i\left(t\right)u_t^{\left(i\right)},M\left[u\right]=\sum _{i,j=1}^m{\left[k_{ij}\left(x\right)u_{x_j}\right]}_{x_i}+\sum _{i=1}^m{c}_i\left(x\right)u_{x_i};\\ u=u(x,t),w=u(x,t-\tau ),x=(x_1,\dots ,x_m).\end{array}$$

Paper [133] contains exact solutions with generalized separation of variables (generalized traveling wave) of the form $u=u\left(z\right),z=t+\theta \left(x\right)$ for nonlinear delay RDEs with variable coefficients

$$c\left(x\right)u_t={\left[a\left(x\right)u_x\right]}_x+b\left(x\right)F(u,w),w=u(x,t-\tau );$$

it also describes a method for constructing such solutions based on grouping the coefficients of the equation in order to reduce it to a delay ODE of a second order. Exact solutions of such and similar equations are also considered in [150,151,152].

Exact solutions of nonlinear RDEs with several delays are considered in [127]; with proportional delay (for $w=u(x,pt)$, where $p>0$ is some constant), in [147]; and with a time-varying delay $\tau =\tau \left(t\right)$ in [126,127]. Papers [127,131] give exact solutions to equations with a nonlinear transport coefficient:

$$u_t={\left[G\left(u\right)u_x\right]}_x+F(u,w),w=u(x,t-\tau ).$$

In [153], there are exact solutions of nonlinear delay Klein–Gordon equations:

$$u_{tt}=a{u}_{xx}+F(u,w),w=u(x,t-\tau );$$

(14)

and, in [139,154], of nonlinear delay equation of telegraph type:

$$u_{tt}+H\left(u\right)u_t={\left[G\left(u\right)u_x\right]}_x+F(u,w),w=u(x,t-\tau ).$$

Papers [130,132] contain a large number of exact solutions for nonlinear systems of delay RDEs. The ‘generating equations’ method for constructing such solutions is proposed in [132]. In [155], exact solutions for systems of Lotka–Volterra type with several delays are constructed.

Papers [156,157] describe a method for constructing exact solutions of nonlinear delay PDEs based on exact solutions of simpler PDEs without delay; nonlinear equations of higher orders, nonlinear equations with varying delay of a general form, and systems of nonlinear delay equations are considered.

A number of studies are devoted to the search for exact solutions of delay PDEs by methods of group analysis. In [124], nonlinear RDEs of form (2) were investigated; four equations that admit invariant solutions were obtained (for two of them, only degenerate solutions linear in x were found). In [158], eight nonlinear and two linear Klein–Gordon equations (14) admitting invariant solutions were established. One linear and seven nonlinear Klein–Gordon equations with varying delay were found in [159].

Remark 4. 

A large number of exact solutions of nonlinear delay PDEs and methods for their construction are considered in [120], Sections 3 and 4.

3.4. Some Exact Solutions with Separation of Variables

Equation 1. Consider the nonlinear RDE with a constant delay:

$$u_t=a{u}_{xx}+uf(w/u),w=u(x,t-\tau ).$$

(15)

$1^{\circ }$. Equation (15) has a solution with multiplicative separation of variables, periodic in space coordinate x, of the form:

$$u=[C_{1}cos\left(\beta x\right)+C_{2}sin\left(\beta x\right)]\psi \left(t\right),$$

where $C_1$, $C_2$, $\beta$ are arbitrary constants, and the function $\psi \left(t\right)$ satisfies the first-order ODE with a constant delay:

$${\psi }_t^{\prime }\left(t\right)=-a{\beta }^2\psi \left(t\right)+\psi \left(t\right)f\left(\psi (t-\tau )/\psi \left(t\right)\right).$$

(16)

$2^{\circ }$. Equation (15) has another solution with multiplicative separation of variables of the form:

$$u=[C_{1}exp(-\beta x)+C_{2}exp\left(\beta x\right)]\psi \left(t\right),$$

where $C_1$, $C_2$, $\beta$ are arbitrary constants, and the function $\psi \left(t\right)$ satisfies the first-order ODE with a constant delay:

$${\psi }_t^{\prime }\left(t\right)=a{\beta }^2\psi \left(t\right)+\psi \left(t\right)f\left(\psi (t-\tau )/\psi \left(t\right)\right).$$

(17)

$3^{\circ }$. Equation (15) has a degenerate solution with multiplicative separation of variables of the form:

$$u=(C_{1}x+C_2)\psi \left(t\right),$$

(18)

where $C_1$ and $C_2$ are arbitrary constants, and the function $\psi \left(t\right)$ satisfies the delay ODE (16) with $\beta =0$.

$4^{\circ }$. Equation (15) also has the solution with multiplicative separation of variables of mixed type:

$$u=e^{\alpha x+\beta t}\theta \left(z\right),z=\lambda x+\gamma t,$$

(19)

where $\alpha$, $\beta$, $\gamma$, $\lambda$ are arbitrary constants, and the function $\theta \left(z\right)$ satisfies the second order ODE with a constant delay:

$$\begin{array}{cc}\hfill a{\lambda }^2{\theta }_{zz}^{″}\left(z\right)& +(2a\alpha \lambda -\gamma ){\theta }_z^{\prime }\left(z\right)+(a{\alpha }^2-\beta )\theta \left(z\right)+\hfill \\ \hfill & +\theta \left(z\right)f\left(e^{-\beta \tau }\theta (z-\sigma )/\theta \left(z\right)\right)=0,\sigma =\gamma \tau .\hfill \end{array}$$

Solution (19) can be treated as a nonlinear superposition of two traveling-wave solutions.

Remark 5. 

Delay ODEs (16) and (17) admit partial solutions of exponential type:

$$\psi \left(t\right)=A{e}^{{\lambda }_{n}t},n=1,2,$$

where A is an arbitrary constant, and ${\lambda }_1$, ${\lambda }_2$ are roots of the transcendental equations:

$$\begin{array}{ccc}\hfill {\lambda }_1& =-a{\beta }^2+f\left(e^{-{\lambda }_1\tau }\right)\hfill & \hfill forEquation\left(16\right),\\ \hfill {\lambda }_2& =a{\beta }^2+f\left(e^{-{\lambda }_2\tau }\right)\hfill & \hfill forEquation\left(17\right).\end{array}$$

$5^{\circ }$. Equation (15) admits a solution of the form:

$$u=e^{ct}v(x,t),v(x,t)=v(x,t-\tau ),$$

(20)

where c is an arbitrary constant, and $v(x,t)$ is a $\tau$-periodic function to be determined.

Substituting (20) into (15), we obtain the linear problem for the function v:

$$v_t=a{v}_{xx}+bv,v(x,t)=v(x,t-\tau ),$$

(21)

where $b=f\left(e^{-c\tau }\right)-c$.

For convenience, we denote the general solution of problem (21) by $v=V_1(x,t;b)$. It has the form:

$$\begin{array}{cc}\hfill & V_1(x,t;b)=\sum _{n=0}^{\infty }exp(-{\lambda }_{n}x)\left[A_{n}cos({\beta }_{n}t-{\gamma }_{n}x)+B_{n}sin({\beta }_{n}t-{\gamma }_{n}x)\right]+\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \phantom{V_1(x,t;b)}+\sum _{n=1}^{\infty }exp\left({\lambda }_{n}x\right)\left[C_{n}cos({\beta }_{n}t+{\gamma }_{n}x)+D_{n}sin({\beta }_{n}t+{\gamma }_{n}x)\right],\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & {\beta }_n=\frac{2\pi n}{\tau },{\lambda }_n={\left(\frac{\sqrt{b^2+{\beta }_n^2}-b}{2a}\right)}^{1/2},{\gamma }_n={\left(\frac{\sqrt{b^2+{\beta }_n^2}+b}{2a}\right)}^{1/2},\hfill \end{array}$$

(23)

where $A_n$, $B_n$, $C_n$, $D_n$ are arbitrary constants, for which series (22) and (23) and its derivatives ${\left(V_1\right)}_t$, ${\left(V_1\right)}_{xx}$ converge (convergence, for example, can certainly be ensured if we set $A_n=B_n=C_n=D_n=0$ for $n>N$, where N is an arbitrary natural number).

We single out the following particular cases:

(i)

$\tau$-periodic in time solutions of problem (21) decaying as $x\to \infty$ are given by formulas (22) and (23) with $A_0=B_0=0$, $C_n=D_n=0$, $n=1,2,\dots$;

(ii)

$\tau$-periodic in time solutions of problem (21) bounded as $x\to \infty$ are given by formulas (22) and (23) with $C_n=D_n=0$, $n=1,2,\dots$;

(iii)

a stationary solution is given by Formulas (22) and (23) with $A_n=B_n=C_n=D_n=0$, $n=1,2,\dots$

Summing up, we have the exact solution of Equation (15):

$$u=e^{ct}{V}_1(x,t;b),b=f(e^{-c\tau })-c,$$

(24)

where c is an arbitrary constant, and the $\tau$-periodic function $V_1(x,t;b)$ is determined by formulas (22) and (23).

$6^{\circ }$. Equation (15) also admits a solution of the form:

$$u=e^{ct}v(x,t),v(x,t)=-v(x,t-\tau ),$$

(25)

where c is an arbitrary constant, $v(x,t)$ is $\tau$-antiperiodic function.

Substituting (25) into (15), we obtain the linear problem for the function v:

$$v_t=a{v}_{xx}+bv,v(x,t)=-v(x,t-\tau ),$$

(26)

where $b=f(-e^{-c\tau })-c$.

The general solution of problem (26), which for convenience we denote by $v=V_2(x,t;b)$, has the form:

$$\begin{array}{cc}\hfill & V_2(x,t;b)=\sum _{n=1}^{\infty }exp(-{\lambda }_{n}x)\left[A_{n}cos({\beta }_{n}t-{\gamma }_{n}x)+B_{n}sin({\beta }_{n}t-{\gamma }_{n}x)\right]+\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \phantom{V_2(x,t;b)}+\sum _{n=1}^{\infty }exp\left({\lambda }_{n}x\right)\left[C_{n}cos({\beta }_{n}t+{\gamma }_{n}x)+D_{n}sin({\beta }_{n}t+{\gamma }_{n}x)\right],\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & {\beta }_n=\frac{\pi (2n-1)}{\tau },{\lambda }_n={\left(\frac{\sqrt{b^2+{\beta }_n^2}-b}{2a}\right)}^{1/2},{\gamma }_n={\left(\frac{\sqrt{b^2+{\beta }_n^2}+b}{2a}\right)}^{1/2},\hfill \end{array}$$

(28)

where $A_n$, $B_n$, $C_n$, $D_n$ are arbitrary constants, for which series (27) and (28) and its derivatives ${\left(V_2\right)}_t$ and ${\left(V_2\right)}_{xx}$ converge. Decaying as $x\to \infty$, $\tau$-antiperiodic in time solutions of problem (26) are given by Formulas (27) and (28) with $C_n=D_n=0$, $n=1,2,\dots$

Summing up, we have the exact solution of Equation (15):

$$u=e^{ct}{V}_2(x,t;b),b=f(-e^{-c\tau })-c,$$

(29)

where c is an arbitrary constant, and the $\tau$-antiperiodic function $V_2(x,t;b)$ is determined by Formulas (27) and (28).

Remark 6. 

Solutions (22)–(23) and (27)–(28) are very similar in appearance. However, in the first solution the first sum starts with $n=0$, and in the second solution, with $n=1$, the values of ${\beta }_n$ also differ.

Equation 2. The nonlinear RDE with a constant delay

$$u_t=a{u}_{xx}+bulnu+uf(w/u)$$

admits a solution with multiplicative separation of variables of the form

$$u=\phi \left(x\right)\psi \left(t\right).$$

Here, the functions $\phi \left(x\right)$ and $\psi \left(t\right)$ are described, respectively, by the ODE and the ODE with constant delay:

$$\begin{array}{cc}\hfill & a{\phi }_{xx}^{″}=C_1\phi -b\phi ln\phi ,\hfill \\ \hfill & {\psi }_t^{\prime }\left(t\right)=C_1\psi \left(t\right)+\psi \left(t\right)f(\psi (t-\tau )/\psi \left(t\right))+b\psi \left(t\right)ln\psi \left(t\right),\hfill \end{array}$$

(30)

where $C_1$ is an arbitrary constant.

Remark 7. 

The second-order ODE (30) does not explicitly depend on x and its general solution can be expressed implicitly. This equation has the particular one-parameter solution:

$$\phi =exp\left[-\frac{b}{4a}{(x+C_2)}^2+\frac{C_1}{b}+\frac{1}{2}\right],$$

where $C_2$ is an arbitrary constant.

Equation 3. Consider the nonlinear RDE with a constant delay:

$$u_t=a{u}_{xx}+f(u-w).$$

(31)

$1^{\circ }$. Equation (31) has an exact solution with additive separation of variables, quadratic in x, of the form:

$$u=C_2{x}^2+C_{1}x+\psi \left(t\right),$$

(32)

where $C_1$ and $C_2$ are arbitrary constants, and the function $\psi \left(t\right)$ is described by the first-order ODE with a constant delay:

$${\psi }_t^{\prime }\left(t\right)=2C_{2}a+f\left(\psi \left(t\right)-\psi (t-\tau )\right).$$

(33)

$2^{\circ }$. Equation (31) also has a more general than (32) solution of the form:

$$u=C_1{x}^2+C_{2}x+C_{3}t+\theta \left(z\right),z=\beta x+\gamma t,$$

(34)

where $C_1$, $C_2$, $C_3$, $\beta$, $\gamma$ are arbitrary constants, and the function $\theta \left(z\right)$ is described by the second-order ODE with a constant delay:

$$a{\beta }^2{\theta }_{zz}^{″}\left(z\right)-\gamma {\theta }_z^{\prime }\left(z\right)+2C_{1}a-C_3+f\left(\theta \left(z\right)-\theta (z-\sigma )+C_3\tau \right)=0,\sigma =\gamma \tau .$$

The values $C_1=C_2=C_3=0$ in (34) correspond to a traveling-wave solution.

Remark 8. 

The delay ODE (33) has a particular solution, linear in t, of the form $\psi \left(t\right)=\lambda t+C_3$, where $C_3$ is an arbitrary constant, and λ is the root of the algebraic (transcendental) equation $2C_{2}a-\lambda +f\left(\tau \lambda \right)=0$.

Equation 4. Consider the nonlinear RDE with a constant delay:

$$u_t=a{u}_{xx}+bu+f(u-w),$$

(35)

which for $b=0$ turns into Equation (31).

$1^{\circ }$. Equation (35) for $ab>0$ has a solution with additive separation of variables, periodic in space coordinate x, of the form:

$$u=C_{1}cos\left(\lambda x\right)+C_{2}sin\left(\lambda x\right)+\psi \left(t\right),\lambda =\sqrt{b/a},$$

(36)

where $C_1$ and $C_2$ are arbitrary constants, and the function $\psi \left(t\right)$ satisfies the delay ODE:

$${\psi }_t^{\prime }\left(t\right)=b\psi \left(t\right)+f\left(\psi \left(t\right)-\psi (t-\tau )\right).$$

(37)

$2^{\circ }$. Equation (35) for $ab<0$ has another solution with additive separation of variables of the form:

$$u=C_{1}exp(-\lambda x)+C_{2}exp\left(\lambda x\right)+\psi \left(t\right),\lambda =\sqrt{-b/a},$$

(38)

where $C_1$ and $C_2$ are arbitrary constants, and the function $\psi \left(t\right)$ satisfies the delay ODE: (37).

$3^{\circ }$. Equation (35) with $b=0$ has a degenerate solution with additive separation of variables of the form:

$$u=C_{1}x+C_2+\psi \left(t\right),$$

where the function $\psi \left(t\right)$ satisfies the delay ODE (37) for $b=0$.

$4^{\circ }$. Equation (35) for $ab>0$ also has a more general than (36) solution of the form:

$$u=C_{1}cos\left(\lambda x\right)+C_{2}sin\left(\lambda x\right)+\theta \left(z\right),z=\beta x+\gamma t,\lambda =\sqrt{b/a},$$

(39)

where $C_1$, $C_2$, $\beta$, $\gamma$ are arbitrary constants, and the function $\theta \left(z\right)$ is described by the delay ODE:

$$\gamma {\theta }_z^{\prime }\left(z\right)=a{\beta }^2{\theta }_{zz}^{″}\left(z\right)+b\theta \left(z\right)+f(\theta \left(z\right)-\theta (z-\sigma )),\sigma =\gamma \tau .$$

(40)

Unlike (36), solution (39) is not periodic in x; it describes the nonlinear interaction of a periodic standing wave and a traveling wave.

$5^{\circ }$. Equation (35) for $ab<0$ also has a more general than (38) solution of the form:

$$\begin{array}{cc}\hfill u& =C_{1}exp(-\lambda x)+C_{2}exp\left(\lambda x\right)+\theta \left(z\right),\hfill \\ \hfill & z=\beta x+\gamma t,\lambda =\sqrt{-b/a},\hfill \end{array}$$

where $C_1$, $C_2$, $\beta$, $\gamma$ are arbitrary constants, and the function $\theta \left(z\right)$ satisfies the delay ODE (40).

More complex exact solutions of Equation (35) (containing any number of arbitrary parameters) can be obtained using the above solutions and the following theorem.

Theorem 1 

(on nonlinear superposition of solutions). Let $u_0(x,t)$ be some solution of the nonlinear delay Equation (35) and the function $v=V_1(x,t;b)$ be any τ-periodic solution of the linear diffusion equation with a source (21). Then, the sum

$$u=u_0(x,t)+V_1(x,t;b)$$

(41)

is also a solution of Equation (35). The general form of the function $V_1(x,t;b)$ is defined by Formulas (22) and (23).

This theorem is proved by direct substitution of Formula (41) into the original delay Equation (35) using the equation for the function v (21).

Remark 9. 

In Formula (41), as a particular solution $u_0(x,t)$ of the nonlinear Equation (35) one can use, for example, a spatially homogeneous solution $u_0\left(t\right)$, a stationary solution $u_0\left(x\right)$ or a traveling-wave solution $u_0=u_0(\alpha x+\beta t)$.

Equation 5. Consider the equation

$$u_t=a{u}_{xx}+bu+f(u-kw),k>0.$$

(42)

$1^{\circ }$. Equation (42) admits a solution with generalized separation of variables of the form

$$u=e^{ct}\phi \left(x\right)+\psi \left(x\right),c=\frac{1}{\tau }lnk,$$

(43)

where the functions $\phi =\phi \left(x\right)$ and $\psi =\psi \left(x\right)$ are determined from the ODEs:

$$\begin{array}{cc}\hfill a{\phi }_{xx}^{″}+(b-c)\phi & =0,\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill a{\psi }_{xx}^{″}+b\psi +f\left(\eta \right)& =0,\eta =(1-k)\psi .\hfill \end{array}$$

(45)

The linear ODE (44) is easy to integrate; the nonlinear ODE (45) is autonomous, and its general solution can be represented in an implicit form.

$2^{\circ }$. More complex exact solutions of Equation (42) (containing any number of arbitrary parameters) can be obtained using the above solutions (43)–(45) and the following theorem.

Theorem 2 

(generalizes Theorem 1). Let $u_0(x,t)$ be some solution of the nonlinear delay Equation (42), and the function $v=V_1(x,t;b)$ be any τ-periodic solution of the linear diffusion equation with a source (21). Then, the sum

$$u=u_0(x,t)+e^{ct}{V}_1(x,t;b-c),$$

(46)

where $c=(lnk)/\tau$, is also a solution of Equation (42). The general form of the function $V_1(x,t;b)$ is defined by Formulas (22) and (23).

This theorem is proved by direct substitution of Formula (46) into the original delay Equation (42) using the equation for the function v (21).

Formula (46) makes it possible to obtain a wide class of exact solutions of the nonlinear delay Equation (42). As the simplest particular solution of Equation (42), in (46), one can use the constant $u_0$, which satisfies the algebraic (or transcendental) equation $b{u}_0+f\left((1-k)u_0\right)=0$. As a function $u_0(x,t)$ in (46), one can also take simple particular solutions of the form $u_0=u_0\left(x\right)$ and $u_0=u_0\left(t\right)$ and also a more complex traveling-wave solution $u_0=\theta (\alpha x+\beta t)$, where $\alpha$ and $\beta$ are arbitrary constants, and the function $\theta \left(y\right)$ satisfies the delay ODE:

$$a{\alpha }^2{\theta }^{″}\left(y\right)-\beta {\theta }^{\prime }\left(y\right)+b\theta \left(y\right)+f\left(\theta \left(y\right)-k\theta (y-\sigma )\right)=0,y=\alpha x+\beta t,\sigma =\beta \tau .$$

Alternatively, the exact solution (43) can be used.

Equation 6. Consider now the equation

$$u_t=a{u}_{xx}+bu+f(u+kw),k>0,$$

(47)

which differs from Equation (42) by the sign of the parameter k in the kinetic term. The following theorem is true.

Theorem 3. 

Let $u_0(x,t)$ be some solution of the nonlinear delay Equation (47), and the function $v=V_2(x,t;b)$ be any τ-antiperiodic solution of the linear diffusion equation with a source (26). Then, the sum

$$u=u_0(x,t)+e^{ct}{V}_2(x,t;b-c),c=\frac{1}{\tau }lnk,$$

(48)

is also a solution of Equation (47). The general form of the function $V_2(x,t;b)$ is defined by Formulas (27) and (28).

Theorem 3 is proved by direct verification and allows one to obtain a wide class of exact solutions to the nonlinear delay Equation (47). As a particular solution $u_0(x,t)$ in (48), one can use, for example, a spatially homogeneous solution $u_0\left(t\right)$, a stationary solution $u_0\left(x\right)$, or a traveling-wave solution $u_0=u_0(\alpha x+\beta t)$.

Equation 7. The more complex nonlinear delay equation

$$u_t=a{u}_{xx}+uf(u-w)+wg(u-w)+h(u-w),w=u(x,t-\tau ),$$

where $f\left(z\right)$, $g\left(z\right)$, $h\left(z\right)$ are arbitrary functions (in this case, one of the two functions f or g can be set equal to zero without loss of generality) admits an exact solution of the form:

$$u=\sum _{n=1}^N[{\phi }_n\left(x\right)cos\left({\beta }_{n}t\right)+{\psi }_n\left(x\right)sin\left({\beta }_{n}t\right)]+t\theta \left(x\right)+\xi \left(x\right),{\beta }_n=\frac{2\pi n}{\tau },$$

where N is any positive integer; the functions ${\phi }_n\left(x\right)$, ${\psi }_n\left(x\right)$, $\theta \left(x\right)$, $\xi \left(x\right)$ are determined from the ODEs:

$$\begin{array}{cc}\hfill a{\phi }_n^{″}+{\phi }_n[f\left(\tau \theta \right)+g\left(\tau \theta \right)]-{\beta }_n{\psi }_n& =0,\hfill \\ \hfill a{\psi }_n^{″}+{\psi }_n[f\left(\tau \theta \right)+g\left(\tau \theta \right)]+{\beta }_n{\phi }_n& =0,\hfill \\ \hfill a{\theta }^{″}+\theta [f\left(\tau \theta \right)+g\left(\tau \theta \right)]& =0,\hfill \\ \hfill a{\xi }^{″}+\xi f\left(\tau \theta \right)+(\xi -\tau \theta )g\left(\tau \theta \right)+h\left(\tau \theta \right)-\theta & =0.\hfill \end{array}$$

Note that the third nonlinear equation admits the trivial solution $\theta =0$; in this case, the remaining equations become linear ODEs with constant coefficients.

Equation 8. The nonlinear delay equation

$$u_t=a{u}_{xx}+uf(u-kw)+wg(u-kw)+h(u-kw),k>0,$$

(49)

where $f\left(z\right)$, $g\left(z\right)$, $h\left(z\right)$ are arbitrary functions, has an exact solution of the form:

$$\begin{array}{cc}\hfill u& =e^{ct}\left\{\theta \left(x\right)+\sum _{n=1}^N[{\phi }_n\left(x\right)cos\left({\beta }_{n}t\right)+{\psi }_n\left(x\right)sin\left({\beta }_{n}t\right)]\right\}+\xi \left(x\right),\hfill \\ \hfill c& =\frac{1}{\tau }lnk,{\beta }_n=\frac{2\pi n}{\tau },\hfill \end{array}$$

where N is any positive integer; the functions $\theta \left(x\right)$, ${\phi }_n\left(x\right)$, ${\psi }_n\left(x\right)$, $\xi \left(x\right)$ are determined from the ODEs:

$$\begin{array}{cc}\hfill & a{\theta }^{″}+\theta \left[f\left(\eta \right)+\frac{1}{k}g\left(\eta \right)-c\right]=0,\eta =(1-k)\xi ,\hfill \\ \hfill & a{\phi }_n^{″}+{\phi }_n\left[f\left(\eta \right)+\frac{1}{k}g\left(\eta \right)-c\right]-{\beta }_n{\psi }_n=0,\hfill \\ \hfill & a{\psi }_n^{″}+{\psi }_n\left[f\left(\eta \right)+\frac{1}{k}g\left(\eta \right)-c\right]+{\beta }_n{\phi }_n=0,\hfill \\ \hfill & a{\xi }^{″}+\xi [f\left(\eta \right)+g\left(\eta \right)]+h\left(\eta \right)=0.\hfill \end{array}$$

Equation 9. The nonlinear delay equation

$$u_t=a{u}_{xx}+uf(u+kw)+wg(u+kw)+h(u+kw),k>0,$$

where $f\left(z\right)$, $g\left(z\right)$, $h\left(z\right)$ are arbitrary functions, admits an exact solution of the form:

$$\begin{array}{cc}\hfill u& =e^{ct}\sum _{n=1}^N[{\phi }_n\left(x\right)cos\left({\beta }_{n}t\right)+{\psi }_n\left(x\right)sin\left({\beta }_{n}t\right)]+\xi \left(x\right),\hfill \\ \hfill c& =\frac{1}{\tau }lnk,{\beta }_n=\frac{\pi (2n-1)}{\tau },\hfill \end{array}$$

where N is any positive integer; the functions ${\phi }_n\left(x\right)$, ${\psi }_n\left(x\right)$, $\xi \left(x\right)$ are determined from the ODEs:

$$\begin{array}{cc}\hfill & a{\phi }_n^{″}+{\phi }_n\left[f\left(\eta \right)-\frac{1}{k}g\left(\eta \right)-c\right]-{\beta }_n{\psi }_n=0,\hfill \\ \hfill & a{\psi }_n^{″}+{\psi }_n\left[f\left(\eta \right)-\frac{1}{k}g\left(\eta \right)-c\right]+{\beta }_n{\phi }_n=0,\hfill \\ \hfill & a{\xi }^{″}+\xi [f\left(\eta \right)+g\left(\eta \right)]+h\left(\eta \right)=0,\eta =(1+k)\xi .\hfill \end{array}$$

Note that the last equation is isolated (i.e., does not depend on the functions of other equations).

Equation 10. The nonlinear delay equation

$$u_t=a{u}_{xx}+uf(u^2+w^2)+wg(u^2+w^2),w=u(x,t-\tau ),$$

where $f\left(z\right)$ and $g\left(z\right)$ are arbitrary functions, has an exact solution of the form:

$$\begin{array}{cc}\hfill & u={\phi }_n\left(x\right)cos\left({\lambda }_{n}t\right)+{\psi }_n\left(x\right)sin\left({\lambda }_{n}t\right),\hfill \\ \hfill & {\lambda }_n=\frac{\pi (2n+1)}{2\tau },n=0,\pm 1,\pm 2,\dots ,\hfill \end{array}$$

where the functions ${\phi }_n={\phi }_n\left(x\right)$ and ${\psi }_n={\psi }_n\left(x\right)$ are determined from the nonlinear system of ODEs:

$$\begin{array}{cc}\hfill a{\phi }_n^{″}+{\phi }_{n}f({\phi }_n^2+{\psi }_n^2)+{(-1)}^{n+1}{\psi }_{n}g({\phi }_n^2+{\psi }_n^2)-{\lambda }_n{\psi }_n& =0,\hfill \\ \hfill a{\psi }_n^{″}+{\psi }_{n}f({\phi }_n^2+{\psi }_n^2)+{(-1)}^n{\phi }_{n}g({\phi }_n^2+{\psi }_n^2)+{\lambda }_n{\phi }_n& =0.\hfill \end{array}$$

4. Methods for Numerical Integration of Delay RDEs

4.1. Preliminary Remarks. Time-Domain Decomposition Method

Problem for the delay RDE. Consider the initial-boundary value problem for the quasilinear RDE with a constant delay:

$$u_t=a{u}_{xx}+f(u,w),0<x<L,t>0,$$

(50)

where $u=u(x,t)$, $w=u(x,t-\tau )$, $a>0$, with the initial conditions:

$$u(x,t)=g(x,t),0<x<L,-\tau \le t\le 0,$$

(51)

and the homogeneous boundary conditions of the first kind:

$$u(0,t)=u(L,t)=0,t>0.$$

(52)

Remark 10. 

The problem that is described by the delay RDE (50) with the initial data (51) and the inhomogeneous boundary conditions of the first kind

$$u(0,t)=h_1\left(t\right),u(L,t)=h_2\left(t\right),t>0,$$

where $h_1\left(t\right)$ and $h_2\left(t\right)$ are predefined functions, by substituting $u=U+h_1\left(t\right)+\frac{x}{L}[h_2\left(t\right)-h_1\left(t\right)]$ is reduced to an initial-boundary value problem with delay with homogeneous boundary conditions for the function $U=U(x,t)$. In this case, the transformed equation will explicitly depend on the variables x and t, which is an insignificant complication of the problem.

Time-domain decomposition method. In the case of a constant delay, one can decompose the considered time interval $0\le t\le T$ by dividing $[0,T]$ into several segments of equal length: $[0,\tau ],[\tau ,2\tau ],\dots$ (see, for example, [160]). Then, problem (50)–(52) on the segment $0<t\le \tau$ takes the form:

$$\begin{array}{cc}\hfill & u_t=a{u}_{xx}+f(u,g),0<x<L,0<t\le \tau ,\hfill \\ \hfill & u(x,0)=g(x,0),0<x<L,\hfill \\ \hfill & u(0,t)=u(L,t)=0,0\le t\le \tau .\hfill \end{array}$$

(53)

Here, we take into account that $w(x,t)\equiv g(x,t)$ for $0<t\le \tau$. Let the function $u_1(x,t)$ be obtained as a solution to problem (53), which is a problem without delay. Then, we can proceed to solving the subproblem on the next segment $\tau <t\le 2\tau$, which is written as follows:

$$\begin{array}{cc}\hfill & u_t=a{u}_{xx}+f(u,u_1),0<x<L,\tau <t\le 2\tau ,\hfill \\ \hfill & u(x,\tau )=u_1(x,\tau ),0<x<L,\hfill \\ \hfill & u(0,t)=u(L,t)=0,\tau <t\le 2\tau .\hfill \end{array}$$

(54)

Here, $w(x,t)\equiv u_1(x,t)$ for $\tau <t\le 2\tau$. Arguing further in a similar way, we can eventually construct a solution to the original problem (50)–(52). Each subproblem is a problem without delay, which can be solved by any known analytical or numerical method for PDEs without delay (for example, finite-difference or finite-element methods).

Remark 11. 

The time domain decomposition method is a natural generalization of the method of steps, which is used to solve the Cauchy problems for delay ODEs (see [2]). If $f(u,w)=f_1\left(w\right)u+f_0\left(w\right)$ in Equation (50), then the subproblems on all the intervals $[0,\tau ],[\tau ,2\tau ],\dots$ will be linear.

4.2. Method of Lines. Numerical Integration Using Mathematica

Preliminary remarks. Formulation of the problem. At the moment, the theory for solving delay ODEs is quite well developed in comparison with the theory of solving delay RDEs. This applies to both analytical (see, for example, [1,2,19,161,162,163,164,165,166]) and numerical methods (see, for example, [167,168,169]). In addition, widespread software such as Maple, Mathematica, and MATLAB allow solving first-order delay ODEs [135,136,137]. Therefore, it is useful to reduce the delay PDE to a system of delay ODEs and then solve it, not the original equation. This approach is often implemented using the method of lines [170]. A large number of programs that analyze models for delay PDEs using the method of lines are contained in the book [35].

Let us introduce the spatial grid: $x_n=nh$, $n=0,1,\dots ,N$, where $h=L/N$ is the stepsize, N is the number of spatial intervals, and reduce problem (50)–(52) to the system of ODEs by approximating the spatial derivative with the finite difference and writing the equation at the point $x_n$:

$$\begin{array}{cc}\hfill & {\left(u_n\right)}_t^{\prime }=a{\delta }_{xx}{u}_n+f(u_n,w_n),n=1,\dots ,N-1,0<t\le T;\hfill \\ \hfill & u_0\left(t\right)=u_N\left(t\right)=0,0\le t\le T;\hfill \\ \hfill & u_n\left(t\right)=g_n\left(t\right),n=1,\dots ,N-1,-\tau \le t\le 0,\hfill \end{array}$$

(55)

where ${\delta }_{xx}{u}_n=h^{-2}(u_{n+1}-2u_n+u_{n-1})$. System (55) contains $N-1$ unknown functions $u_n\left(t\right)$ and the same number of equations, as well as two known functions $u_0\left(t\right)$ and $u_N\left(t\right)$.

Remark 12. 

Higher accuracy can be achieved if we discretize using Chebyshev–Gauss–Lobatto points [171,172,173,174].

The main problem with this approach is that the resulting system of delay ODEs often turns out to be stiff. A system is called stiff if it describes processes occurring on very different time scales [175,176]. When solving such problems numerically, restrictions on the stepsize are imposed, not to improve accuracy but to ensure the stability of the algorithm [175], and usually only excessively small steps are suitable. Solving such a system requires the development and application of special methods with increased stability [167,177]. Usually, these are algorithms from the class of implicit Runge–Kutta [168,178,179] and BDF methods [175,176,180]. The schemes of Rosenbrock type [181,182,183,184] are also suitable.

In Mathematica, stiff ODE systems with a constant delay (including those with several delays) are solved numerically using the command NDSolve [135,185,186]. Without additional options, NDSolve uses an adaptive procedure, in which, during the calculation, methods are changed, and values of parameters are selected automatically. Using the option Method of the command NDSolve, one can manually specify one of the built-in methods for solving stiff ODE systems, namely the implicit Runge–Kutta [187,188] method or the implicit multi-step method based on the backward differentiation formula (BDF method) [189]. To describe these methods, we will use the Cauchy problem in vector form:

$$\begin{array}{cc}\hfill & {\mathit{u}}_t^{\prime }=\mathit{f}(t,\mathit{u},\mathit{u}(t-\tau )),0<t\le T;\hfill \\ \hfill & \mathit{u}\left(t\right)=\mathit{g}\left(t\right),-\tau \le t\le 0,\hfill \end{array}$$

(56)

where $\mathit{u}={(u_0,\dots ,u_N)}^T$, $\mathit{g}={(g_0,\dots ,g_N)}^T$, $\mathit{f}={(f_0,\dots ,f_N)}^T$ are column vectors. Problem (55) is a special case of problem (56).

Implicit Runge–Kutta method. Implicit Runge–Kutta schemes for stiff ODE systems (56) are defined by the formulas [168,187,188]:

$$\begin{array}{cc}\hfill & {\mathit{u}}_{k+1}={\mathit{u}}_k+h_{k+1}\sum _{m=1}^M{c}_m{\mathit{r}}_{k+1}^{\left(m\right)},k=0,\dots ,K-1,\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill & {\mathit{r}}_{k+1}^{\left(m\right)}=\mathit{f}\left(t_k+{\alpha }_m{h}_{k+1},{\mathit{u}}_k+h_{k+1}\sum _{j=1}^{M_{*}}{\beta }_{mj}{\mathit{r}}_{k+1}^{\left(j\right)},\mathit{u}(t_k+{\alpha }_m{h}_{k+1}-\tau )\right),\hfill \end{array}$$

(58)

where ${\mathit{r}}_k^{\left(m\right)}={(r_{1,k}^{\left(m\right)},r_{2,k}^{\left(m\right)},\dots ,r_{N,k}^{\left(m\right)})}^T$ is a column vector of auxiliary functions $r_{n,k}^{\left(m\right)}$, each of which corresponds to the unknown function $u_n$, time layer $t_k$, and stage m, $n=1,\dots ,N$, $k=1,\dots ,K$, $m=1,\dots ,M$; $h_{k+1}=t_{k+1}-t_k$ is the stepsize, $c_m$ are the weights of the quadrature formula ($0\le c_m\le 1$), ${\alpha }_m$ are the coefficients defining the nodes of the quadrature formula, ${\beta }_{mj}$ are the weights of intermediate quadrature formulas. If the value $t_k+{\alpha }_m{h}_{k+1}-\tau$ lies outside the grid points, the delay function values $\mathit{u}(t_k+{\alpha }_m{h}_{k+1}-\tau )$ are calculated using interpolation on the interval $[t_q,t_{q+1}]$, where $q>0$ is such an integer that $t_q\le t_k+{\alpha }_m{h}_{k+1}-\tau \le t_{q+1}$. If $t_k+{\alpha }_m{h}_{k+1}-\tau =t_q$, the delay function values $\mathit{u}(t_k+{\alpha }_m{h}_{k+1}-\tau )$ coincide with the values ${\mathit{u}}_q$, which were calculated earlier on the layer $t_q$. Different Runge–Kutta methods are generated by different quadrature formulas, which are defined by sets of coefficients ${\beta }_{mj}$, $c_m$, and ${\alpha }_m$.

The method based on Formulas (57) and (58) is an implicit M-stage method. For $M_{*}=m-1$, we have an explicit method for ODE systems. In the case of $M_{*}=m$, the values ${\mathit{r}}_k^{\left(m\right)}$ are determined successively from separate nonlinear equations. In the case of $M_{*}=M$, the values ${\mathit{r}}_k^{\left(m\right)}$ must be calculated simultaneously for all stages from the system of $N\times M$ equations, which, by default, is solved in Mathematica by Newton’s method (see the description of this method in [176]).

To successfully solve stiff problems, it is important to choose the appropriate coefficients of quadrature formulas. In Mathematica, by default, coefficient values are determined automatically. However, one can choose the type of these coefficients manually using the setting Coefficients of the option Method of the command NDSolve [188]. For example, these may be the coefficients based on the Lobatto quadrature formulas [190,191]. The first and last nodes of the Lobatto quadrature formula coincide with the beginning and end of the integration interval; therefore, ${\alpha }_1=0$, ${\alpha }_M=1$. The remaining coefficients ${\alpha }_m$ are the zeros of the derivatives of the Legendre polynomials:

$$\frac{d^{M-2}}{d{\alpha }_m^{M-2}}({\alpha }_m^{M-1}{({\alpha }_m-1)}^{M-1})=0.$$

(59)

As a result, quadrature formulas of order $2M-2$ are obtained. The weights $c_1,\dots ,c_M$ and the coefficients ${\beta }_{mj}$ of the Lobatto quadrature formulas are determined from the conditions:

$$\begin{array}{cc}\hfill \sum _{m=1}^M{c}_m{\alpha }_m^{\gamma -1}& =\frac{1}{\gamma },\gamma =1,\dots ,2M-2;\hfill \\ \hfill \sum _{j=1}^M{\beta }_{mj}{\alpha }_j^{\gamma -1}& =\frac{{\alpha }_m^{\gamma }}{\gamma },m=1,\dots ,M,\gamma =1,\dots ,M-1;\hfill \\ \hfill {\beta }_{m1}& =c_1,m=1,\dots ,M.\hfill \end{array}$$

(60)

In Mathematica, the stepsize $h_k$ of schemes (57) and (58) is determined automatically based on the estimate of the local error of solution [187]. For this, the solutions obtained by the original method with the order of approximation p and weights $c_m$ are compared with the solutions of the auxiliary method with order $\widehat{p}$ and weights ${\widehat{c}}_m$ (by default $\widehat{p}=p-1$). In this case, the coefficients ${\alpha }_m$ and ${\beta }_{mj}$ of both methods coincide, and hence the values ${\mathit{s}}_k^{\left(m\right)}$ also coincide, which eliminates the need to solve the nonlinear system a second time (58).

Remark 13. 

In problems with solutions reaching high-order absolute values (see, for example, test problem 2 below), the commandNDSolvewith the Runge–Kutta method chosen can take minutes and tens of minutes to calculate a solution. The method running time can be significantly reduced to several seconds by increasing the allowable absolute and relative errors using the options AccuracyGoal → q and PrecisionGoal → p. Given the values of q and p, the program will try to ensure that the error of the numerical solution does not exceed the value${10}^{-q}+{10}^{-p}\left|x\right|$.

BDF method. The BDF method is built into Mathematica as part of the IDA package from SUNDIALS (SUite of Nonlinear and DIfferential/ALgebraic equation Solvers), developed at the Center for Applied Scientific Computing of Lawrence Livermore National Laboratory [189]. The program code of the IDA methods (see user manual [180]) is based on DASPK [192,193] that are Fortran codes for solving large-scale differential–algebraic systems.

The M-step BDF method for system (56) is based on the formula [175,176,180]:

$${\alpha }_0{\mathit{u}}_k-h_k\mathit{f}(t_k,{\mathit{u}}_k,\mathit{u}(t_k-\tau ))=-\sum _{m=1}^M{\alpha }_m{\mathit{u}}_{k-m}.$$

(61)

If the value $t_k-\tau$ lies outside the grid, the values $\mathit{u}(t_k-\tau )$ are calculated by interpolation on the interval $[t_q,t_{q+1}]$, where $q>0$ is an integer such that $t_q\le t_k-\tau \le t_{q+1}$. If $t_k-\tau =t_q$, the values $\mathit{u}(t_k-\tau )$ coincide with the values ${\mathit{u}}_q$, which were calculated earlier on the layer $t_q$. The system of nonlinear algebraic Equations (61) can be solved by one or another iterative method, for example, Newton’s method.

In order for the BDF method to have the p-th order of approximation, one should set ([176] p. 255):

$${\alpha }_0=-\sum _{m=1}^M{\alpha }_m,\sum _{m=1}^{M}m{\alpha }_m=-1,\sum _{m=1}^M{m}^j{\alpha }_m=0,j=2,3,\dots ,p.$$

(62)

The highest achievable order of approximation of the M-step BDF method is M.

Assuming $M=1$ in (61) and (62), we obtain the formula for the first-order implicit Euler method:

$${\mathit{u}}_k={\mathit{u}}_{k-1}+h_k\mathit{f}(t_k,{\mathit{u}}_k,\mathit{u}(t_k-\tau )).$$

For $M=2$, $M=3$, and $M=4$, we have the relations ([176] p. 256):

$$\begin{array}{cc}\hfill & 3{\mathit{u}}_k-4{\mathit{u}}_{k-1}+{\mathit{u}}_{k-2}=2h_k\mathit{f}(t_k,{\mathit{u}}_k,\mathit{u}(t_k-\tau )),\hfill \\ \hfill & 11{\mathit{u}}_k-18{\mathit{u}}_{k-1}+9{\mathit{u}}_{k-2}-2{\mathit{u}}_{k-3}=6h_k\mathit{f}(t_k,{\mathit{u}}_k,\mathit{u}(t_k-\tau )),\hfill \\ \hfill & 25{\mathit{u}}_k-48{\mathit{u}}_{k-1}+36{\mathit{u}}_{k-2}-16{\mathit{u}}_{k-3}+3{\mathit{u}}_{k-4}=12h_k\mathit{f}(t_k,{\mathit{u}}_k,\mathit{u}(t_k-\tau )),\hfill \end{array}$$

determining BDF methods of the second, third, and fourth order of approximation, respectively.

In Mathematica, the BDF method on every time layer computes an estimate ${\mathit{E}}_k$ of the local truncation error, the stepsize $h_k$ and order M are automatically chosen so that $\parallel {\mathit{E}}_k/bm{\omega }_k\parallel <1$, where the n-th component ${\omega }_{n,k}$ of the vector ${\mathsf{\omega }}_k$ is given by

$${\omega }_{n,k}=\frac{1}{{10}^{-p}\left|u_{n,k}\right|+{10}^{-q}}.$$

The values of the constants p and q are found using the options PrecisionGoal $\to p$ and AccuracyGoal $\to q$ of the command NDSolve. By default, the norm $\parallel ·\parallel$ is automatically chosen by the command NDSolve depending on the method of integration (but can be set manually). For the BDF method, this is the norm of the form ${\parallel x\parallel }_2=\sqrt{\sum |x_i{|}^2}$ [194].

The steps $h_k$ are chosen automatically by the command NDSolve. The maximum number of steps the algorithm uses to construct a solution is estimated by the value of the initial step [195]. This can lead to problems if, for example, the solution grows exponentially without limit (see, for example, test problem 2). One can remove the restriction on the maximum number of steps using the option MaxSteps $\to \infty$ inside the command NDSolve.

Algorithm of the numerical integration of the delay reaction–diffusion problems by the method of lines using Mathematica. The flowchart of the numerical integration of the initial-boundary value problem (55) using Mathematica is shown in Figure 2. The algorithm can be described as the following sequence:

$1^{\circ }$.

Formulate the problem consisting of the equation, initial, and boundary conditions.

$2^{\circ }$.

Choose the number of spatial steps N.

$3^{\circ }$.

Apply the method of lines and obtain the system of $N-1$ delay ODEs and $N-1$ initial conditions (and two algebraic relations on the edge of the region).

$4^{\circ }$.

Choose the time interval $0<t\le T$ of integration.

$5^{\circ }$.

Solve the system from $3^{\circ }$ using methods of the command NDSolve.

$6^{\circ }$.

In case of errors during calculation, try to reduce the time interval from $4^{\circ }$ and obtain the solution on a shorter time interval.

$7^{\circ }$.

As a result, the values of the unknown function on all time layers, the values of absolute and relative errors (if the exact solution is known), plots and animations of the numerical solution (together with the exact solution) are obtained.

Figure 2. The flowchart of the numerical integration of delay reaction–diffusion problems by the method of lines.

Figure 2. The flowchart of the numerical integration of delay reaction–diffusion problems by the method of lines.

Remark 14. 

In addition to a uniform grid in x, one can also use non-uniform grids. For non-uniform grids with the variable step $h_n=x_n-x_{n-1}$, the second derivative $u_{xx}$ is approximated as:

$$u_{xx}\approx \frac{2}{h_n+h_{n+1}}\left(\frac{u_{n+1}-u_n}{h_{n+1}}-\frac{u_n-u_{n-1}}{h_n}\right),$$

where $\sum h_N=L$ (for $0\le x\le L$).

Remark 15. 

The method of lines can also be used for the numerical integration of the more complex delay hyperbolic problems of a fairly general form [120]:

$$\begin{array}{cc}\hfill & \epsilon u_{tt}+\sigma u_t={\left[p(x,u)u_x\right]}_x+q(x,u,w)u_x+f(x,u,w),t>0,0\le x\le L;\hfill \\ \hfill & u(x,t)={\phi }_0(x,t),u_t(x,t)={\phi }_1(x,t),-\tau \le t\le 0;\hfill \\ \hfill & u(0,t)={\psi }_0\left(t\right),u(1,t)={\psi }_1\left(t\right),t>0,\hfill \end{array}$$

(63)

where $w=u(x,t-\tau )$; functions p, q, and f can also depend on t explicitly. This equation includes, as special cases, delay reaction–diffusion equations ($\epsilon =0$, $\sigma =1$), delay equations of Klein–Gordon type ($\epsilon =1$, $\sigma =0$), nonlinear delay telegraph equations ($\epsilon =1$, $\sigma \ne 0$).

In the cases of 2D and 3D delay RDEs, the terms ${\left[p(x,u)u_x\right]}_x$ and $q(x,u,w)u_x$ are replaced, respectively, by $\nabla ·\left[p\right(\mathbf{x},u)\nabla u]$ and $\mathbf{q}(\mathbf{x},u,w)·\nabla u$. In particular, in the two-dimensional case on uniform grids, the Laplace operator ($\mathsf{\Delta }u=u_{xx}+u_{yy}$) is approximated as follows:

$$\mathsf{\Delta }u\approx \frac{1}{h_1^2}(u_{n+1,m}-2u_{n,m}+u_{n-1,m})+\frac{1}{h_2^2}(u_{n,m+1}-2u_{n,m}+u_{n,m-1}),$$

where $u_{n,m}=u(x_n,y_m,t)$, $x_n=n{h}_1$, $y_m=m{h}_2$ ($n=0,1,\dots ,N$; $m=0,1,\dots ,M$), and $h_1=L_1/N$, $h_2=L_2/M$ are stepsizes over spatial coordinates x and y.

4.3. Finite-Difference Methods

Explicit finite-difference scheme. Optimization of data storage in RAM. Consider the uniform spatial grid $x_n=nh(n=0,1,2,\dots ,N)$, where $h=L/N$ is the spatial stepsize. Let s be the time stepsize such that $Ms=\tau$, where M is an integer. We approximate the time derivative by a finite difference of the form:

$$u_t\approx s^{-1}(u_{n,k+1}-u_{n,k}),$$

and the spatial derivative, by the second order finite difference:

$$u_{xx}\approx {\delta }_{xx}{u}_{n,k}=h^{-2}(u_{n+1,k}-2u_{n,k}+u_{n-1,k}).$$

Thus, problem (50)–(52) has the following finite-difference scheme form:

$$\begin{array}{cc}\hfill & u_{n,k+1}=u_{n,k}+as{\delta }_{xx}{u}_{n,k}+sf(u_{n,k},u_{n,k-M}),\hfill \\ \hfill & n=1,2,\dots ,N-1,k=0,\dots ,K-1;\hfill \\ \hfill & u_{0,k}=u_{N,k}=0,k=0,1,\dots ,K;\hfill \\ \hfill & u_{n,k}=g_{n,k},n=0,1,\dots ,N,k=-M,\dots ,0.\hfill \end{array}$$

(64)

As seen from (64), to calculate the values on layer $k+1$, one needs the data not only from the previous layer k but also from layer $k-M$. Therefore, it is necessary to keep in RAM the values from all the time layers within the delay range, as these values are used to obtain the solution. The process is complicated by the need to control the condition

$$s<\frac{h^2}{2\parallel u(x,t)\parallel }$$

so that the explicit scheme (64) is stable.

When solving problems with delay, it is required to store a sufficiently large amount of data with the possibility of constant access to it. Most often, the amount of data exceeds the amount of fast cache memory of the CPU, which means that it is necessary to interact with the RAM. This means that the speed of calculations depends on the speed of interaction with the RAM. Note that low read/write speeds of external drives make them even more useless for our purposes.

In [196], the algorithm that can significantly reduce the need for RAM is discussed. It is proposed to store data, not from all time layers but only from some so called base layers, and to restore intermediate values using interpolation. During the calculation, the number of base time layers within the delay range can be changed to maintain a balance between the accuracy of the algorithm and the amount of data stored and depends on the smoothness of the function. The type of interpolation is selected based on the parameters and properties of a particular model.

The smoothness of the function is determined by the formula [196]:

$$\Gamma (t+s)=\underset{1\le n\le N}{max}\left|\frac{u(x_n,t+s)-u(x_n,t)}{u(x_n,t)}\right|.$$

Let us introduce a parameter p such that each p-th layer within the delay range is a base layer. To minimize interpolation errors, but maintain a sufficiently high computational speed, it is recommended to choose the values of p from a range from 1 to 20. If the values of the function u change too quickly, then to maintain the required accuracy of the algorithm, all layers should be saved, i.e., $p=1$. If the function changes slowly ($\Gamma <0.01$), one should take $p=20$. In this case, the proposed algorithm demonstrates the best efficiency. In [196], the empirical rule for calculating the optimal number of base layers for reaction–diffusion problems is proposed:

$$p=\tau s^{-1}{[1+exp(2-50\Gamma \left(t\right))]}^{-1}.$$

The total number of stored layers q in the delay range depends on the parameter p and the time stepsize s. In the simplest case, when p and s are fixed, one can use the formula $q=\tau p^{-1}{s}^{-1}$.

Let ${\widehat{t}}_j$ be the time of the j-th base layer ($j=1,2,\dots ,q$), ${\widehat{u}}_{n,j}$ be the value of the function u on the j-th base layer. To restore the values of the intermediate layers from the q base layers, i.e., to find the values of $u(x_n,t)$ for $t_{k-M}\le t\le t_{k-1}$, one can use the Newton interpolation polynomial [196]:

$$\begin{array}{c}\phantom{ww}u(x_n,t)=R\left({\widehat{u}}_{n,1}\right)+(t-{\widehat{t}}_1)R({\widehat{u}}_{n,1},{\widehat{u}}_{n,2})+\dots \hfill \\ \hfill +(t-{\widehat{t}}_1)(t-{\widehat{t}}_2)\dots (t-{\widehat{t}}_{q-1})R({\widehat{u}}_{n,1},\dots {\widehat{u}}_{n,q}),\phantom{ww}\end{array}$$

where the divided differences $R(\dots )$ are determined by the formulas:

$$\begin{array}{cc}\hfill & R\left({\widehat{u}}_{n,1}\right)={\widehat{u}}_{n,1},\hfill \\ \hfill & R({\widehat{u}}_{n,1},{\widehat{u}}_{n,2})=\frac{{\widehat{u}}_{n,2}-{\widehat{u}}_{n,1}}{{\widehat{t}}_2-{\widehat{t}}_1},\hfill \\ \hfill & R({\widehat{u}}_{n,1},\dots {\widehat{u}}_{n,q})=\frac{R({\widehat{u}}_{n,2},\dots {\widehat{u}}_{n,q})-R({\widehat{u}}_{n,1},\dots {\widehat{u}}_{n,q-1})}{t_q-t_1}.\hfill \end{array}$$

It is important that the described algorithm does not depend on the numerical scheme, but only establishes the rules of data storage. Therefore, it can be easily generalized for use with other difference schemes.

Implicit finite-difference scheme. Using the time decomposition, problem (50)–(52) is solved step by step on the intervals $[0,\tau ]$, $[\tau ,2\tau ]$, ⋯ The uniform grid is constructed according to the rules: $x_n=nh$ ($n=0,1,2,\dots ,N$), $t_k=ks$ ($k=0,1,2,\dots ,K$), where $h=L/N$ is the spatial stepsize, $s=\tau /M$ is the time stepsize, M is an integer. The time derivative is approximated by the finite difference:

$$u_t\approx s^{-1}(u_{n,k+1}-u_{n,k}),$$

and the spatial derivative by the second-order finite difference:

$$u_{xx}\approx {\delta }_{xx}{u}_{n,k+1}=h^{-2}(u_{n+1,k+1}-2u_{n,k+1}+u_{n-1,k+1}).$$

Given that $w(x,t)=g(x,t)$ for $0<t\le \tau$, the implicit finite-difference scheme for problem (50)–(52) on this interval has the form:

$$\begin{array}{cc}\hfill & (1-as{\delta }_{xx})u_{n,k+1}=u_{n,k}+sf(u_{n,k+1},g_{n,k+1-M}),\hfill \\ \hfill & n=1,2,\dots ,N-1,k=0,\dots ,K-1;\hfill \\ \hfill & u_{0,k+1}=u_{N,k+1}=0,k=0,1,\dots ,K;\hfill \\ \hfill & u_{n,0}=g_{n,0},n=0,1,\dots ,N.\hfill \end{array}$$

(65)

On the next interval $\tau <t\le 2\tau$, the values of $w(x,t)$ will also be known and equal to the corresponding values of $u(x,t)$ calculated on the previous interval $0<t\le \tau$. Performing the calculations on successive intervals, we obtain a solution for the en–time interval. In [160], the conditions for the uniqueness of the solution of scheme (65) were established. In [197], the convergence of this scheme with order $h^2+s$ was proved and its stability was studied.

Remark 16. 

Scheme (65) is a nonlinear system of difference equations, which can be solved by iterative methods, for example, the Picard–Schwarz method [160] or the method of upper and lower solutions [198,199,200,201].

Finite-difference scheme with weights. Consider problem (50)–(52) in the region $Q=\{0\le x\le L,0\le t\le T\}$. Construct a uniform grid according to the rules: $x_n=nh$ ($n=0,1,2,\dots ,N$), $t_k=ks$ ($k=0,1,2,\dots ,K$), where $h=L/N$ is the spatial stepsize, and $s=\tau /M$ is the time stepsize, M is an integer.

Denoting $f_{n,k}=f(u_{n,k},w_{n,k}\left(t\right))$, we write the scheme with weights (see, for example, [202]):

$$\begin{array}{cc}\hfill & (1-\sigma as{\delta }_{xx})u_{n,k+1}=[1+(1-\sigma )as{\delta }_{xx}]u_{n,k}+s{f}_{n,k},\hfill \\ \hfill & n=1,2,\dots ,N-1,k=0,1,\dots ,K-1;\hfill \\ \hfill & u_{0,k}=u_{N,k}=0,k=0,1,\dots ,K;\hfill \\ \hfill & u_{n,k}=g_{n,k},n=0,1,\dots ,N,k=-M,\dots ,-1,0,\hfill \end{array}$$

(66)

where $0\le \sigma \le 1$. The weight value $\sigma =0$ corresponds to the explicit scheme. For $0<\sigma \le 1$, we have a tridiagonal system of linear algebraic equations, which is solved by the tridiagonal matrix algorithm. For $\sigma \ge 1/2$, the scheme is unconditionally stable (more on stability, including for $\sigma <1/2$, see [202]).

Remark 17. 

A related scheme with weights for a hyperbolic delay equation is considered, for example, in [203], and for a parabolic equation with a delay in the diffusion term in [204].

Higher-order finite-difference schemes. For sufficiently smooth solutions of Equation (50), one can use the following stable multi-step finite-difference scheme of order $h^4+s^2$ [205]:

$$\begin{array}{cc}\hfill & (\mathcal{A}-\frac{1}{2}as{\delta }_{xx})u_{n,k+1}=(\mathcal{A}+\frac{1}{2}as{\delta }_{xx})u_{n,k}\hfill \\ \hfill & +s\mathcal{A}f(\frac{3}{2}{u}_{n,k}-\frac{1}{2}{u}_{n,k-1},\frac{1}{2}{u}_{n,k+1-M}+\frac{1}{2}{u}_{n,k-M}),\hfill \\ \hfill & n=1,2,\dots ,N-1,k=0,1,\dots ,N-1,\hfill \end{array}$$

where the difference operator $\mathcal{A}$ is defined as

$$\mathcal{A}{u}_{n,k}=\frac{1}{12}(u_{n-1,k}+10u_{n,k}+u_{n+1,k}).$$

Remark 18. 

For related, more complex delay RDEs, similar multi-step higher-order finite-difference schemes are given in [206,207].

Two special finite-difference schemes for a linear problem. In [208], two special finite-difference schemes for problem (50)–(52) with the linear kinetic function $f(u,w)=bw$ are considered. The proposed schemes have the same stability region, as the original problem. This is provided by the choice of spatial and time stepsizes, as well as a special method of approximation.

Both schemes use the same grid. The spatial stepsize is calculated by the formula $\tilde{h}=2sin\left(\frac{1}{2}h\right)$, where h is the standard stepsize of a rectangular grid; time stepsize is $s=\tau /(M-\epsilon )$, where M is an integer, $0\le \epsilon <1$. Note that the intervals $[0,\tau ]$, $[\tau ,2\tau ]$, … contain, in the general case, a non-integer number of time steps in contrast to the schemes considered above.

To build the first scheme, the trapezoid rule is applied, when the spatial derivative and the delay term are calculated as average values from two adjacent time layers:

$$\begin{array}{cc}\hfill u_{xx}& \approx \frac{1}{2}{\tilde{\delta }}_{xx}(u_{n,k+1}+u_{n,k}),bw\approx \frac{b}{2}(w_{n,k+1}+w_{n,k}),\hfill \end{array}$$

where ${\tilde{\delta }}_{xx}{u}_{n,k}={\tilde{h}}^{-2}(u_{n+1,k}-2u_{n,k}+u_{n-1,k})$. The grid function $w_{n,k}$ approximates the function $w(x,t)$ in the point $(x_n,t_k)$ using the linear interpolation:

$$w_{n,k}=\epsilon u_{n,k-M+1}+(1-\epsilon )u_{n,k-M}.$$

(67)

Then, Equation (50) is approximated as follows:

$$(2-as{\tilde{\delta }}_{xx})u_{n,k+1}=(2+as{\tilde{\delta }}_{xx})u_{n,k}+bs[\epsilon u_{n,k-M+2}+u_{n,k-M+1}+(1-\epsilon )u_{n,k-M}].$$

The second finite-difference scheme is based on the use of the second-order backward differentiation formula for approximating the time derivative and has the form:

$$\frac{1}{2s}(3u_{n,k+2}-4u_{n,k+1}+u_{n,k})=a{\tilde{\delta }}_{xx}{u}_{n,k+2}+b{w}_{n,k+2}.$$

Using linear interpolation (67) for $w_{n,k+2}$ and rearranging the terms, we have

$$(3-2as{\tilde{\delta }}_{xx})u_{n,k+2}=4u_{n,k+1}-u_{n,k}+2bs[\epsilon u_{n,k-M+3}+(1-\epsilon )u_{n,k-M+2}].$$

Here, it is necessary to have the values $u_{n,1}$ ($n=1,\dots ,N-1$), which can be obtained using the first difference scheme.

5. Construction, Choice, and Use of Test Problems for Delay RDEs

5.1. Basic Principles for Choosing Test Problems

For nonlinear delay RDEs, when choosing test problems designed to check the adequacy and evaluate the accuracy of the corresponding numerical and approximate analytical methods, it is useful to be guided by the following principles.

$1^{\circ }$.

The most reliable test problems are obtained by using exact solutions of delay RDEs.

$2^{\circ }$.

It is preferable to choose simple test problems, the solutions of which are expressed in terms of elementary functions.

$3^{\circ }$.

It is preferable to choose test problems containing free parameters (that can be arbitrarily varied over a wide range) or arbitrary functions.

$4^{\circ }$.

One can choose test problems from a wider class of similar equations (there is no need to use exact solutions of the equation under consideration, which are not always possible to obtain).

$5^{\circ }$.

To check the adequacy and evaluate the accuracy of numerical methods, it is better to use several different test problems.

$6^{\circ }$.

Testing of numerical methods should begin with simple test problems that have monotonic solutions with small gradients of the unknowns.

$7^{\circ }$.

Numerical methods should be tested on test problems with large gradients of the unknowns in the initial or boundary conditions (for example, for rapidly oscillating initial conditions).

$8^{\circ }$.

It is useful to test numerical methods on rapidly growing solutions at sufficiently large times.

$9^{\circ }$.

If possible, the accuracy of numerical methods should be checked on test problems near the critical values of the parameters and independent variables that determine the singular points of the equation, unstable solutions, or solutions with large gradients.

It is important to emphasize that well-chosen test problems make it possible to compare and improve ‘workable’ numerical methods and filter out unsuitable ones.

Remark 19. 

For nonlinear RDEs with delay, one cannot limit oneself to test problems obtained using exact solutions of the simpler nonlinear RDEs without delay.

5.2. Construction of Test Problems

Examples of exact solutions of delay RDEs that can be used to formulate test problems. To construct test problems, one should use the known exact solutions of nonlinear delay RDEs. A large number of such solutions can be found in ([120], Sections 3 and 4).

Consider the nonlinear delay RDE

$$\begin{array}{c}\hfill u_t=a{u}_{xx}+bu[1-s(u-kw)],w=u(x,t-\tau ),\end{array}$$

(68)

which depends on five parameters $a>0$, b, k, s, $\tau >0$ and is a special case of Equation (49) for $g\left(z\right)=h\left(z\right)\equiv 0$, $f\left(z\right)=b(1-sz)$. Let us choose its parameters so that it has the stationary solutions $u_0=0$ and $u_0=1$. The trivial stationary solution $u_0=0$ already exists for any values of the parameters. For the second stationary solution to exist, we substitute $u=1$ into (68) and, after elementary transformations, we have $s=1/(1-k)$. As a result, we arrive at the equation

$$u_t=a{u}_{xx}+bu\left(1-\frac{u-kw}{1-k}\right),w=u(x,t-\tau ),$$

(69)

which can be written in the alternative form:

$$u_t=a{u}_{xx}+bu[1-({\sigma }_{1}u+{\sigma }_{2}w)],{\sigma }_1+{\sigma }_2=1,$$

where ${\sigma }_1=1/(1-k)$.

For $k\to 0$, Equation (69) becomes Fisher–KPP equation, and for $k\to \pm \infty$, it becomes the diffusion logistic equation with delay. Below are two groups of the simplest exact solutions to Equation (69).

(i)

Solutions for $k>0$ $(k\ne 1)$:

$$\begin{array}{cccccc}\hfill u& =e^{ct}[Acos\left(\gamma x\right)+Bsin\left(\gamma x\right)],\hfill & & \gamma =\sqrt{(b-c)/a}\hfill & & \mathrm{for}b>c;\hfill \\ \hfill u& =e^{ct}(A{e}^{-\gamma x}+B{e}^{\gamma x}),\hfill & & \gamma =\sqrt{(c-b)/a}\hfill & & \mathrm{for}b<c;\hfill \\ \hfill u& =e^{ct}(Ax+B),\hfill & & \hfill & & \mathrm{for}b=c;\hfill \\ \hfill u& =1+e^{ct}[Acos\left(\gamma x\right)+Bsin\left(\gamma x\right)],\hfill & & \gamma =\sqrt{-c/a}\hfill & & \mathrm{for}c<0;\hfill \\ \hfill u& =1+e^{ct}(A{e}^{-\gamma x}+B{e}^{\gamma x}),\hfill & & \gamma =\sqrt{c/a}\hfill & & \mathrm{for}c>0,\hfill \end{array}$$

(70)

where ${\displaystyle c=(lnk)/\tau }$; A, B are arbitrary constants.

(ii)

Solutions for $k<0$:

$$\begin{array}{cc}\hfill & u=A_n{e}^{ct\mp {\lambda }_{n}x}cos({\beta }_{n}t\mp {\gamma }_{n}x+C_n),{\beta }_n=\frac{\pi (2n-1)}{\tau },\hfill \\ \hfill & {\gamma }_n=\frac{{\beta }_n}{2a{\lambda }_n},{\lambda }_n={\left(\frac{\sqrt{{(b-c)}^2+{\beta }_n^2}-b+c}{2a}\right)}^{1/2};\hfill \\ \hfill & u=1+A_n{e}^{ct\mp {\lambda }_{n}x}cos({\beta }_{n}t\mp {\gamma }_{n}x+C_n),{\beta }_n=\frac{\pi (2n-1)}{\tau },\hfill \\ \hfill & {\gamma }_n=\frac{{\beta }_n}{2a{\lambda }_n},{\lambda }_n={\left(\frac{\sqrt{c^2+{\beta }_n^2}+c}{2a}\right)}^{1/2},\hfill \end{array}$$

(71)

where ${\displaystyle c=(ln|k\left|\right)/\tau }$; $A_n$, $C_n$ are arbitrary constants; $n=1,2,\dots$

To test numerical methods of nonlinear delay RDEs, one can choose the above exact solutions of Equation (69). Let us note the qualitative features of some solutions that are useful for comparison with the results of numerical calculations.

The first and fourth solutions from group (i) are periodic in spatial variable x. These solutions are useful as test solutions for initial-boundary value problems with boundary conditions of the first and second kind on the interval $0\le x\le m\pi /\gamma$ ($m=1,2,\dots$). By appropriately choosing the values of the free constants A and B, one can make the unknown function on the boundary equal to zero or one (for boundary conditions of the first kind) or obtain the zero derivative with respect to x on the boundary (for boundary conditions of the second kind). In the case of problems with mixed boundary conditions, it is convenient to consider these solutions on the intervals $0\le x\le \frac{1}{2}m\pi /\gamma$ ($m=1,2,\dots$). The initial conditions at $-\tau \le t\le 0$ (or $0\le t\le \tau$) are determined from the solutions used as test problems. It is useful to compare the numerical and exact solutions of test problems for k close to unity (when the solutions change little in time) and for sufficiently large k (when the solutions change quickly).

Solutions from group (ii) for $k=-1$ are periodic in time and for $\tau \to 0$ rapidly oscillate in time and space. Such solutions are useful for estimating the accuracy of numerical methods for problems with large gradients.

The test problems are formulated as follows: the chosen equation and its exact solution are supplemented with initial conditions at $-\tau \le t\le 0$ and boundary conditions at $x=0$ and $x=L$, which are obtained from the used exact solution. Some test problems formulated in this way are given below.

Test problems for delay RDEs. Using the known exact solutions, we formulate several model test problems for delay RDEs, which can be used to check the adequacy and evaluate the accuracy of numerical methods. All test problems contain free parameters.

Test problem 1. Let us assume the following in the first formula of (70):

$$A=1,B=2,b=(lnk)/\tau +a{\pi }^2/4,c=(lnk)/\tau ,k>0,$$

(72)

and obtain the exact solution of Equation (69):

$$\begin{array}{c}\hfill u=U_1(x,t)\equiv e^{ct}[cos(\pi x/2)+2sin(\pi x/2)],c=(lnk)/\tau .\end{array}$$

(73)

Substituting $-\tau \le t\le 0$ into it, and then $x=0$ and $x=1$, we find the initial conditions:

$$\begin{array}{c}\hfill u(x,t)=e^{ct}[cos(\pi x/2)+2sin(\pi x/2)],-\tau \le t\le 0,\end{array}$$

(74)

and the boundary conditions:

$$u(0,t)=e^{ct},t>0;u(1,t)=2e^{ct},t>0.$$

(75)

As a result, we have a test problem, which is described by Equation (69) with the parameter b from (72), the initial conditions (74), and the boundary conditions (75). The exact solution of this test problem is given by Formula (73), where $0\le x\le 1$, $t>0$.

Remark 20. 

Other test problems can be obtained in a similar way, using exact solutions (70) and (71) of the RDE (69).

Test problem 2. It can be shown by direct verification that Equation (68) for

$$b=-4a,k=e^{5a\tau },s=\frac{3}{2(1-k)}$$

(76)

admits the exact solution

$$u=U_2(x,t)\equiv {cosh}^{-2}\left(x\right)+e^{ct}{cosh}^3\left(x\right),c=(lnk)/\tau .$$

(77)

Substituting $-\tau \le t\le 0$ into (77), and then $x=0$ and $x=1$, we obtain the initial conditions:

$$u(x,t)={cosh}^{-2}\left(x\right)+e^{ct}{cosh}^3\left(x\right),-\tau \le t\le 0,$$

(78)

and boundary conditions:

$$u(0,t)=1+e^{ct},t>0;u(1,t)={cosh}^{-2}\left(1\right)+e^{ct}{cosh}^3\left(1\right),t>0.$$

(79)

Thus, we have a test problem, which is described by Equation (68) with the parameters (76), the initial conditions (78), and boundary conditions(79). The exact solution of this test problem is given by Formula (77), where $0\le x\le 1$, $t>0$.

Now, consider the nonlinear five-parameter delay RDE:

$$\begin{array}{cc}\hfill & u_t=a{u}_{xx}+bu-s{(u-kw)}^2,w=u(x,t-\tau ),\hfill \end{array}$$

(80)

which is a special case of Equation (42) for $f\left(z\right)=-s{z}^2$ and becomes Fisher–KPP equations in the degenerate cases $k=0$ or $\tau =0$.

Test problem 3. Let us assume that

$$k>0,k\ne 1,b=(lnk)/\tau -a,s=b/{(1-k)}^2.$$

(81)

Then, the test problem, described by Equations (80) and (81), initial conditions

$$u(x,t)=U_3(x,t)\equiv 1+\frac{e^{ct+1}}{e^2-1}(e^x-e^{-x}),c=\frac{lnk}{\tau },-\tau \le t\le 0,$$

(82)

and boundary conditions

$$u(0,t)=1,t>0;u(1,t)=1+e^{ct},t>0,$$

(83)

has the exact solution $u=U_3(x,t)$ in the region $0\le x\le 1$, $t>0$.

The solution $u=U_3(x,t)$ was obtained using Formula (43), where $\psi \equiv 1$, and $\phi$ is the corresponding solution of the linear ODE (44).

Test problem 4. Let us assume that

$$k>0,k\ne 1,b=4a{\pi }^2+(lnk)/\tau -1/\left(a{\tau }^2\right),s=b/{(1-k)}^2.$$

(84)

Then, the test problem, described by Equations (80) and (84), initial conditions

$$\begin{array}{c}u(x,t)=U_4(x,t)\equiv 1+e^{ct-\lambda x}cos(\beta t-2\pi x),-\tau \le t\le 0,\\ c=(lnk)/\tau ,\lambda =1/\left(a\tau \right),\beta =4\pi /\tau ,\end{array}$$

(85)

and boundary conditions

$$u(0,t)=1+e^{ct}cos\left(\beta t\right),t>0;u(1,t)=1+e^{ct-\lambda }cos\left(\beta t\right),t>0,$$

(86)

has the exact solution $u=U_4(x,t)$ in the region $0\le x\le 1$, $t>0$.

The solution $u=U_4(x,t)$ was obtained using Formula (46), where $u_0(x,t)\equiv 1$ and $V_1(x,t;b-c)$ is determined by Formulas (22) and (23) with $A_2=1$ and all the other constants $A_n$, $B_n$, $C_n$, $D_n$ equal to zero.

5.3. Comparison of Numerical and Exact Solutions of Nonlinear Delay RDEs

Preliminary remarks. Numerical solutions of all test problems were obtained in Mathematica by the method of lines in combination with the second-order implicit Runge–Kutta method (with the coefficients of the Lobatto type) and the BDF method. The calculations were carried out on the interval $0\le t\le T=50\tau$ for three delays $\tau =0.05$, $\tau =0.1$, and $\tau =0.5$ (sometimes additional values $\tau =1$ and $\tau =5$ were taken). Some test problems could not be solved on such a large interval. The integration procedure was interrupted with an error with an indication of the interruption time. Nevertheless, in most cases, an adequate numerical solution of the problem was obtained after shortening the time interval in a suitable way.

The absolute and relative errors of the numerical solution $u_{n,k}=u_h(x_n,t_k)$ of a test problem for a delay RDE are, respectively, determined by the formulas:

$${\sigma }_{\mathrm{a}}=\underset{\begin{array}{c}n,k\end{array}}{max}|u_{\mathrm{e}}-u_{n,k}|,{\sigma }_{\mathrm{r}}=\underset{\begin{array}{c}n,k\end{array}}{max}|(u_{\mathrm{e}}-u_{n,k})/u_{\mathrm{e}}|,$$

where $u_{\mathrm{e}}=u_{\mathrm{e}}(x_n,t_k)$ is the value of the exact solution of the test problem on the time layer $t_k$ at the point $x_n$.

Comparison of exact and numerical solutions of test problems. In the previous section, four test problems for nonlinear delay RDEs (delay equations of the Fisher–KPP type) were formulated. In this section, we discuss the results of the numerical integration of the test problems and compare the obtained numerical solutions with the exact solutions of these problems. The numbering and wording of the test problems considered below are the same as the numbering and wording of the test problems given in the previous section.

Test problem 1. The exact solution $u=U_1(x,t)$ of test problem 1 for $a=1$, $k=0.5$, and $s=0.2$ is monotonically decaying in time. The relative error of the numerical solution becomes noticeable only at sufficiently large times, when the solution is practically equal to zero. For the numerical methods used, with an increase in the delay $\tau$ (from 0.05 to 5), the time interval on which the methods operate with a small relative error increases. At $N=100$, the numerical solution obtained by the second-order implicit Runge–Kutta method begins to differ from the exact solution when the solution reaches absolute values of the order of ${10}^{-5}$ for $\tau =0.05$ and ${10}^{-20}$ for $\tau =5$; the numerical solution by the BDF method begins to differ, reaching absolute values of the order of ${10}^{-6}$ for $\tau =0.05$ and ${10}^{-10}$ for $\tau =5$.

The numerical solutions of the ODE system with $N=100$ of test problem 1, which were obtained by the second-order implicit Runge–Kutta method, and the corresponding exact solutions are shown in Figure 3 for $a=1$, $k=0.5$, and $s=0.2$ and the delays $\tau =0.05$ and $\tau =0.5$ (using a logarithmic scale along the u-axis). The results obtained by the BDF method look similar and are therefore omitted.

The absolute errors of the numerical solutions obtained by the second-order implicit Runge–Kutta and BDF methods for five different delay times are presented in

Table 1. The absolute errors of the numerical solutions of test problem 1 for $a=1$, $k=0.5$, $s=0.2$ and five different delays $\tau$ on the interval $0\le t\le T=50\tau$.

Method	N	$\mathit{\tau }=0.05$	$\mathit{\tau }=0.1$	$\mathit{\tau }=0.5$	$\mathit{\tau }=1$	$\mathit{\tau }=5$
Second-order Runge–Kutta	10	$2.8\times {10}^{-4}$	$4.6\times {10}^{-4}$	$9.9\times {10}^{-4}$	$1.2\times {10}^{-3}$	$1.4\times {10}^{-3}$
	50	$1.3\times {10}^{-5}$	$2.0\times {10}^{-5}$	$4.1\times {10}^{-5}$	$4.8\times {10}^{-5}$	$5.6\times {10}^{-5}$
	100	$4.5\times {10}^{-6}$	$6.1\times {10}^{-6}$	$1.2\times {10}^{-5}$	$1.3\times {10}^{-5}$	$1.4\times {10}^{-5}$
BDF	10	$2.8\times {10}^{-4}$	$4.6\times {10}^{-4}$	$9.9\times {10}^{-4}$	$1.2\times {10}^{-3}$	$1.4\times {10}^{-3}$
	50	$1.3\times {10}^{-5}$	$1.9\times {10}^{-5}$	$4.0\times {10}^{-5}$	$4.7\times {10}^{-5}$	$5.6\times {10}^{-5}$
	100	$2.8\times {10}^{-6}$	$4.7\times {10}^{-6}$	$9.9\times {10}^{-6}$	$1.2\times {10}^{-5}$	$1.4\times {10}^{-5}$

.

Test problem 2. The exact solution $u=U_2(x,t)$ of test problem 2 for $a=1$ grows exponentially in time. The program execution is interrupted with an error when the solution reaches high order values. For the second-order Runge–Kutta method, these are values of the order of ${10}^8$. For $\tau =0.05$, the numerical solution can be obtained on the en–interval up to $T=50\tau$; for $\tau =0.1$, up to $T=35\tau$; and for $\tau =0.5$, up to $T=7.3\tau$. The BDF method works better: until the solution reaches values of the order of ${10}^{13}$. For $\tau =0.05$ and $\tau =0.1$, the numerical solution can be obtained on the en–interval up to $T=50\tau$ and for $\tau =0.5$, up to $T=11.5\tau$. We also note that the BDF method allows one to obtain a solution in just a few seconds, while the second-order implicit Runge–Kutta method takes minutes and tens of minutes (see Remark 13 about a possible reduction in the calculation time).

The numerical solutions of the ODE system with $N=100$ of test problem 2, which were obtained by the BDF method, and the corresponding exact solutions are shown in Figure 4 for $a=1$ and the delays $\tau =0.05$ and $\tau =0.5$ (using a logarithmic scale along the u-axis). The results obtained by the Runge–Kutta method look similar and are therefore omitted.

The relative errors of the numerical solutions obtained by the second-order implicit Runge–Kutta and BDF methods are presented in

Table 2. The relative errors of the numerical solutions of test problem 2 for $a=1$ and three different delays $\tau$.

Method	N	$\mathit{\tau }=0.05$	$\mathit{\tau }=0.1$	$\mathit{\tau }=0.5$
		$T=50\tau$	$T=35\tau$	$T=7.3\tau$
Second-order Runge–Kutta	10	$1.8\times {10}^{-3}$	$1.9\times {10}^{-3}$	$2.5\times {10}^{-3}$
	50	$7.5\times {10}^{-5}$	$7.8\times {10}^{-5}$	$1.0\times {10}^{-4}$
	100	$2.0\times {10}^{-5}$	$2.0\times {10}^{-5}$	$2.6\times {10}^{-5}$
BDF	10	$1.8\times {10}^{-3}$	$1.9\times {10}^{-3}$	$2.5\times {10}^{-3}$
	50	$7.3\times {10}^{-5}$	$7.6\times {10}^{-5}$	$1.0\times {10}^{-4}$
	100	$1.8\times {10}^{-5}$	$1.9\times {10}^{-5}$	$2.5\times {10}^{-5}$

.

Test problem 3. The numerical solutions of the ODE system with $N=100$ of test problem 3, which were obtained by the second-order implicit Runge–Kutta method, and the corresponding exact solutions are shown in Figure 5 for $a=1$, $k=0.5$ and the moderate delay $\tau =0.5$. The results obtained by the BDF method look similar and are therefore omitted. Both methods demonstrate a good approximation of the exact solution over the en–interval $0\le t\le T=50\tau$.

Table 3. The relative errors of the numerical solutions of test problem 3 for $a=1$, $k=0.5$, and three different delays $\tau$ on the interval $0\le t\le T=50\tau$.

Method	N	$\mathit{\tau }=0.5$	$\mathit{\tau }=1$	$\mathit{\tau }=5$
Second-order Runge–Kutta	10	$7.1\times {10}^{-5}$	$5.4\times {10}^{-5}$	$4.8\times {10}^{-5}$
	50	$2.0\times {10}^{-6}$	$1.9\times {10}^{-6}$	$1.9\times {10}^{-6}$
	100	$2.5\times {10}^{-7}$	$3.7\times {10}^{-7}$	$5.1\times {10}^{-7}$
BDF	10	$7.2\times {10}^{-5}$	$5.5\times {10}^{-5}$	$4.8\times {10}^{-5}$
	50	$2.8\times {10}^{-6}$	$2.2\times {10}^{-6}$	$1.9\times {10}^{-6}$
	100	$6.3\times {10}^{-7}$	$5.1\times {10}^{-7}$	$5.1\times {10}^{-7}$

presents the relative errors of the numerical solutions of test problem 3 obtained by the BDF and the second-order implicit Runge–Kutta methods at moderate and large delays on the interval $0\le t\le T=50\tau$.

For $a=1$, $k=0.5$, and sufficiently small delay times $\tau =0.05$ and $\tau =0.1$, the numerical solution of test problem 3 begins to deviate strongly from the exact solution in the region where the stationary state is reached. After that, the program execution is interrupted with an error. This happens due to the fact that the parameter b, which is included in the equation of the test problem and defined in (81), grows indefinitely as $O\left({\tau }^{-1}\right)$ as $\tau \to$0. As a result, the stationary solution $u=1$, to which the solution of the problem under consideration tends as $t\to \infty$, becomes unstable for small $\tau$ (in the linear approximation, the proof of this fact is given below).

To illustrate the described situation, consider the dependences on time of the numerical and exact solutions for a fixed $x=0.5$ (see Figure 6), for $a=1$, $k=0.5$, and the small delays $\tau =0.05$ and $\tau =0.1$. The choice of the midpoint $x=0.5$ is due to the fact that it was at this point that the maximum deviation of the numerical solution from the exact one was observed. For $\tau =0.05$, both the numerical methods used adequately work only on a very short initial segment (slightly before reaching the asymptote); then, the second-order implicit Runge–Kutta method gives a downward nonmonotone oscillating curve that has nothing to do with the exact solution, and the BDF method leads to a curve that strongly deviates from the exact solution, rising sharply upwards. For $\tau =0.1$, both the Runge–Kutta method and the BDF method provide a fairly accurate approximation of the exact solution over a fairly significant time interval (reaching the asymptote), and the BDF method has a slightly larger time range of applicability. The time dependence of the error of the numerical solution obtained by the BDF method is monotonic, in contrast to the nonmonotonic dependence for the Runge–Kutta method. In both cases, the errors increase sharply sometime after the steady state is established. Let us show that the values $\tau =0.05$ and $\tau =0.1$ lie inside the region of instability of the stationary solution of the considered delay equation.

Equation (80) of test problem 3 has the form

$$\begin{array}{c}\hfill u_t=a{u}_{xx}+bu-\frac{b}{{(1-k)}^2}{(u-kw)}^2,b=\frac{lnk}{\tau }-a,\end{array}$$

(87)

and admits the stationary solution $u_0=1$ (the solution of the test problem under consideration asymptotically tends to this solution as $t\to \infty$). To analyze the linear stability/instability of the solution $u_0=1$, consider perturbed solutions of the form [134]:

$$\begin{array}{c}\hfill u=1+\delta e^{-\lambda t}sin\left(\pi nx\right),n=1,2,\dots ,\end{array}$$

(88)

where $\delta$ is a small parameter, $\lambda$ is the spectral parameter to be determined. On the boundaries $x=0$ and $x=1$ of the considered region, the perturbed solution (88) is equal to 1 for any t. Substituting (88) into Equation (87). Discarding terms of order ${\delta }^2$ and higher, after reduction by $sin\left(\pi nx\right)$, we obtain the characteristic equation for the $\lambda$:

$$\begin{array}{c}\hfill \lambda -a{\left(\pi n\right)}^2-\frac{b(1+k)}{1-k}+\frac{2bk}{1-k}{e}^{\lambda \tau }=0,b=\frac{lnk}{\tau }-a.\end{array}$$

(89)

For $a=n=1$, $k=0.5$, and $\tau =0.05$, the characteristic Equation (89) has the negative root $\lambda \approx -27.0213$. It follows that the second term in Formula (88) grows exponentially as $t\to \infty$ and the considered stationary solution of the delay RDE (80) is unstable in the linear approximation for $\tau =0.05$. As the delay increases to ${\tau }_{*}\approx 0.09153$ (the values of other parameters do not change), there are also one or two real negative roots of the transcendental Equation (89), while for $\tau >{\tau }_{*}$, this equation has no real negative roots.

For $a=n=1$, $k=0.5$, $\tau =0.1$, the characteristic Equation (89) has the complex root with the negative real part $\mathrm{Re}\lambda =-4.38498$. Therefore, the considered stationary solution of the delay RDE (80) is also unstable in the linear approximation for $\tau =0.1$.

Remark 21. 

The value $\tau =0.5$ lies outside the region of instability of the stationary solution of test problem 3. In this case, the characteristic Equation (89) has the root with $Re\lambda =0.0895394$, and, as noted earlier, numerical methods work well.

Test problem 4. The numerical solutions of the ODE system with $N=100$ of test problem 4, which were obtained by the second-order implicit Runge–Kutta method, and the corresponding exact solutions are shown in Figure 7 for $a=1$, $k=0.5$, and the delay $\tau =0.5$. The results obtained by the BDF method look similar and are therefore omitted.

The relative errors of the numerical solutions of test problem 4 at moderate and large delays on the interval $0\le t\le T=50\tau$ are identical to two significant digits for the BDF and the second-order implicit Runge–Kutta methods. Values of the errors are presented in

Table 4. The relative errors of the numerical solutions of test problem 4 for $a=1$, $k=0.5$ and three different delays $\tau$ on the interval $0\le t\le T=50\tau$.

N	$\mathit{\tau }=0.5$	$\mathit{\tau }=1$	$\mathit{\tau }=5$
10	$7.3\times {10}^{-3}$	$1.2\times {10}^{-2}$	$7.4\times {10}^{-2}$
50	$2.9\times {10}^{-4}$	$5.1\times {10}^{-4}$	$2.9\times {10}^{-3}$
100	$7.3\times {10}^{-5}$	$1.3\times {10}^{-4}$	$7.3\times {10}^{-4}$

.

Note that for delays of the order of unity (for example, $\tau =0.5$), the oscillations of the solution are not high-frequency and do not lead to problems in the numerical integration of the considered test problem, in contrast to the case of small delays, which is discussed below.

For small delays $\tau =0.05$ and $\tau =0.1$ and for $a=1$ and $k=0.5$, the numerical solution of the ODE system of test problem 4 with $N=100$ could not be obtained either by the Runge–Kutta or the BDF method. This is primarily due to the fact that the parameter b, which is included in the equation of the test problem and defined in (84), grows indefinitely as $O\left({\tau }^{-2}\right)$ as $\tau \to 0$ (in this case, the coefficient b grows much faster than in test problem 3). An additional complicating factor for small delays is the rapid oscillations of the solution in a small neighborhood of the left boundary $x=0$, whereas in the rest of the considered region $u\approx 1$. Obtaining adequate numerical results in such cases is possible only at small spatial stepsizes in the region of the boundary layer type, where the solution changes rapidly. Using a variable spatial stepsize, that is, a different number of ODEs for regions with and without fast oscillations, in the method of lines is difficult, because it is difficult to determine in advance in which region high-frequency oscillations occur. The use of a small stepsize in the en–area of calculations is associated with an excessive increase in the running time of the method.

Remark 22. 

The combination of the method of lines and the BDF method built into Mathematica is used in Section 2.1 to solve the initial-boundary value problem for the diffusion system of the Lotka–Volterra type.

6. Brief Conclusions

The phenomena associated with delay occur in biology, biomedicine, ecology, physics, chemistry, chemical technology, control theory, etc. Usually, models describing such phenomena are based on delay ODEs. However, as shown in the paper, in many cases, delay PDEs, namely delay RDEs, are more useful. Based on the analysis of the extensive literature data, it has been established that for the majority of interesting cases considered, the diffusion term can be given a clear and well-defined physical meaning. It is important that such equations have a number of physical features associated with the possible non-smoothness and instability of their solutions.

The paper has discussed the concept of the ‘exact solution’ of nonlinear delay RDEs. Nonlinear delay RDEs containing one or more arbitrary functions and methods allowing one to construct exact solutions for such equations have been considered. Exact solutions represented in elementary functions and containing free parameters have been obtained. Such exact solutions have been used to formulate test problems for checking the accuracy of the numerical methods.

It has been shown that the use of the method of lines for integrating delay RDEs, i.e., approximating the spatial derivatives by the corresponding finite differences, leads to the replacement of the original delay RDE with an approximate system of delay ODEs, which makes it possible to use the standard numerical methods of the modern software, such as Mathematica. The Mathematica command ‘NDSolve’ allows one to solve systems of delay ODEs by different methods, such as the implicit Runge–Kutta and BDF methods. However, with this approach, questions related to the accuracy and stability of the numerical integration of the original equation remain unexplored.

To interpret the results of the numerical integration, determine their adequacy to the considered processes, and evaluate the limits of applicability depending on the values of the parameters of the problem, the paper has proposed an approach based on a comparison of exact and numerical solutions of specially formulated test problems. Test initial-boundary value problems are formulated on the basis of nonlinear delay RDEs of a general form in which the arbitrary functions are specified. The initial and boundary conditions are chosen so that the known exact solutions of these equations satisfy these conditions (the uniqueness of the solutions is not investigated). On the basis of a linear analysis of the stability of the simplest stationary solutions of the formulated problem, the parametric regions of its stability/instability are determined. In these regions, numerical solutions of the corresponding initial-boundary value problems can be obtained and compared with the exact solution. The absolute or relative calculation error and the values of the parameters for which the numerical method can be applied can also be established. To sum up, this approach allows one to test the adequacy and estimate the accuracy of the numerical methods by the comparison of the exact and numerical solutions of test problems.