Regarding a Class of Nonlocal BVPs for the General Time-Fractional Diffusion Equation

Abstract

A class of initial boundary value problems is here considered for a one-dimensional diffusion equation with a general time-fractional derivative with the Sonin kernel. One of the boundary conditions is in a general non-classical form, which includes no-nlocal cases of integral or multi-point boundary conditions. The problem is studied here by applying spectral projection operators to convert it to a system of relaxation equations in generalized eigenspaces. The uniqueness of the solution is established based on the uniqueness property of the spectral expansion. An algorithm is given for constructing the solution in the form of spectral expansion in terms of the generalized eigenfunctions. Estimates for the time-dependent components in this expansion are established and applied to prove the existence of a solution in the classical sense. The obtained results are applied to a particular case in which the specified boundary conditions lead to two sequences of eigenvalues, one of which consists of triple eigenvalues.

1. Introduction

Diffusion equations with time-fractional derivatives capture well the power-law dependence on the time of the mean squared displacement; see e.g., [1]. During anomalous diffusion in complex systems, transitions between different diffusion regimes in course of time may occur. One way to model such complex behavior is by replacing the relatively simple operators of fractional derivatives by more general operators with different memory kernels, i.e., by employing general fractional derivatives [2,3,4].

In this work, we adopt the definition of general fractional derivative of Caputo type introduced by A. Kochubei in [5]

$${}^C\mathbb{D}_t^{\left(\zeta \right)}f\left(t\right)={\displaystyle \frac{d}{dt}}{\int }_0^t\zeta (t-\tau )f\left(\tau \right)d\tau -\zeta \left(t\right)f\left(0\right),t>0.$$

(1)

Here $\zeta \left(t\right)$ is a non-negative locally integrable Sonin kernel, whose precise properties will be specified later. For an overview on different definitions of general fractional derivatives with Sonin kernels and their basic properties, we refer to [6].

The relevance of the fractional diffusion equation with the general integro-differential operator in time (1) for modeling anomalous diffusion is pointed out in several papers; see e.g., [2,3,7]. It is a generalization of the classical, multi-term, and distributed-order time-fractional diffusion equations, and goes beyond the simple power-law behavior of standard fractional kinetics. Fractional diffusion equations with Sonin kernels can describe complex systems exhibiting varied diffusion regimes through weighted sums of different fractional derivatives or non-power-law memory kernels, allowing for a more comprehensive understanding of anomalous transport in such systems. At the same time, the developed in [5,8,9] general fractional and operational calculi for the operators of the convolution type with Sonin kernels makes the analytical study of the corresponding general fractional diffusion equations feasible.

In [5], the Cauchy problem is studied for the general diffusion equation on an unbounded space domain. Different aspects of differential equations with a general time-fractional derivative with Sonin kernel are discussed in [10,11,12] such as the maximum principles, uniqueness, and existence of a solution to the initial-boundary-value problems with Dirichlet boundary conditions. Optimal estimates for the decay in time of solutions to the general time-fractional diffusion equation on a bounded domain are established in [13] for different memory kernels. Some inverse problems for equations with a general fractional derivative in time are studied in [14,15,16,17]. In the review paper by [18], recent results on direct and inverse problems for general time-fractional diffusion equations are discussed.

Many practical problems involve non-local boundary conditions of various kinds, especially in cases when the data on the boundary cannot be measured directly. The problems with non-local boundary conditions have been studied due to their wide application in the mathematical modeling of different phenomena in physics, engineering, and biology. Starting in 1963 with the work of J.R. Cannon [19], methods of obtaining analytical or numerical solutions for the one-dimensional diffusion equation subject to various non-local boundary conditions are studied by many authors (see, e.g., [20,21,22,23,24,25,26,27,28,29] and the references cited therein). A large number of publications are also devoted to inverse problems for equations with non-local boundary conditions; see, e.g., [16,17,30,31,32,33,34,35].

In the present work, we study the general time-fractional diffusion equation on a bounded interval

$${}^C\mathbb{D}_t^{\left(\zeta \right)}u(x,t)={\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}(x,t),x\in (0,1),t>0,$$

(2)

subject to the initial condition

$$u(x,0)=f\left(x\right)$$

(3)

and the following boundary conditions:

$$u(0,t)=0,{\mathsf{\Phi }}_x\left\{u(x,t)\right\}=0,$$

(4)

where ${\mathsf{\Phi }}_x$ is a continuous linear functional acting in the space of smooth functions $C^1[0,1]$ (the subscript x means that the functional acts with respect to the space variable x). We assume that the functional $\mathsf{\Phi }$ satisfies the property

$$1\in \mathrm{supp}\mathsf{\Phi }.$$

(5)

This assumption is essential in our study, since it ensures the uniqueness property of the corresponding spectral expansion due to a result of N. Bozhinov [36].

Some examples of functionals $\mathsf{\Phi }$ that obey the above requirements are

$$\begin{array}{ccc}\hfill \mathsf{\Phi }\left\{f\right\}& =& {\int }_{x_0}^{1}f\left(x\right)dx,\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill \mathsf{\Phi }\left\{f\right\}& =& a_0{f}^{\prime }\left(x_0\right)+b_{0}f\left(x_0\right)+a_1{f}^{\prime }\left(1\right)+b_{1}f\left(1\right),\hfill \end{array}$$

(7)

where $0\le x_0<1$. The integral boundary condition, corresponding to the functional (6) with $x_0=0$, is referred to as Samarskii–Ionkin condition and appears in early works on heat conduction equation with non-classical boundary conditions; see [20]. The functional (7) corresponds to a multi-point boundary condition. Due to requirement (5), the coefficients in (7) should satisfy $a_1\ne 0$ or/and $b_1\ne 0$. If $a_0=b_0=0$, then (7) defines a boundary condition of Robin-type. If at least one of the coefficients $a_0$, $b_0$ is not zero, then (7) defines a non-local boundary condition.

The function $f\left(x\right)$ in the initial condition (3) is a given sufficiently well-behaved function satisfying the compatibility conditions

$$f\left(0\right)=\mathsf{\Phi }\left\{f\right\}=0.$$

The classical diffusion and wave equations with boundary conditions in the general form (4) are studied in [37,38,39], where Duhamel-type representations of the solutions are derived by employing the convolutional calculus of Dimovski, developed in [37]. The initial-boundary-value problem for the fractional cable equation subject to boundary conditions (4) is studied in [40]. In the works by [41,42], Duhamel-type representations of the solutions are derived for the classical and the time-fractional diffusion equations, respectively, subject to specific non-local multi-point boundary conditions, which are particular cases of (4).

The main aim of the present paper is to propose a constructive approach for solving problem (2)–(4), which is based on generalized eigenfunction expansion. Applying the uniqueness property of this expansion, we prove the uniqueness of the solution. Then, by the use of some properties of the solution to the relaxation equation with a general fractional derivative, we prove estimates for the time-dependent components, which imply that the deduced formal expansion is a solution in the classical sense.

The remainder of the paper is organized as follows. In Section 2, spectral projection operators, corresponding to boundary conditions (4), are defined, and some specific cases of the functional $\mathsf{\Phi }$ are considered. In Section 3, the relaxation equation with the general fractional derivative (1) is discussed, and some properties of its solution are summarized. The spectral projection operators are applied in Section 4 to prove first the uniqueness of the solution to the problem (2)–(4) and then, in addition, an algorithm is given for constructing a formal solution in the form of an expansion in terms of the generalized eigenfunctions. Applying the results of Section 3, it is proven that this is a solution in the classical sense. As an example, in Section 5, the obtained results are applied to a particular problem with a multi-point boundary condition, which leads to two sequences of eigenvalues, consisting of single and triple eigenvalues, respectively. Section 6 contains the concluding remarks.

2. Spectral Projection Operators

In this section, we consider the spectral problem, corresponding to initial-boundary-value problem (2)–(4), and define the corresponding spectral projection operators. For more details, we refer to [37,38,40].

Consider the spatial differential operator $D_x^2=d^2/d{x}^2$ with domain

$$X_{D_x^2}=\{f\in C^2\left([0,1]\right):f\left(0\right)=\mathsf{\Phi }\left\{f\right\}=0\}.$$

The eigenvalues $-{\lambda }_n^2$ of the corresponding spectral problem,

$$y^{″}+{\lambda }^{2}y=0,y\left(0\right)=0,\mathsf{\Phi }\left\{y\right\}=0,$$

(8)

can be obtained from the zeros ${\lambda }_n$ of the entire function

$$E\left(\lambda \right)={\mathsf{\Phi }}_x\left\{{\displaystyle \frac{sin\lambda x}{\lambda }}\right\},$$

referred to as a sine indicatrix of the functional $\mathsf{\Phi }$. Since $E\left(\lambda \right)$ is an even function, for each eigenvalue $-{\lambda }_n^2$, we consider only one of the zeros, ${\lambda }_n$ or $-{\lambda }_n$. Let us take ${\lambda }_n\ge 0$. Assumption (5) implies that the set of zeros of $E\left(\lambda \right)$ is infinite and countable; see [36]. Let ${\kappa }_n$ be the multiplicity of ${\lambda }_n$ as a zero of $E\left(\lambda \right)$, i.e.,

$$E\left({\lambda }_n\right)=E^{\prime }\left({\lambda }_n\right)=\dots =E^{({\kappa }_n-1)}\left({\lambda }_n\right)=0,E^{\left({\kappa }_n\right)}\left({\lambda }_n\right)\ne 0.$$

For any ${\lambda }_n$ there exists a finite sequence of generalized eigenfunctions (also referred to as root functions), the eigenfunction $sin{\lambda }_{n}x$ and ${\kappa }_n-1$ associated eigenfunctions, which span the corresponding ${\kappa }_n$-dimensional eigenspace ${\mathcal{E}}_{{\lambda }_n}$.

Let us note that we do not restrict our considerations to nonzero eigenvalues only. If ${\lambda }_0=0$ is an eigenvalue, then ${\kappa }_0=1$ and the corresponding eigenfunction is x.

The resolvent operator $R_x(-{\lambda }^2)={(D_x^2+{\lambda }^{2}I)}^{-1}$ is well defined for all $\lambda$, such that $E\left(\lambda \right)\ne 0$ and has the explicit representation

$$R_x(-{\lambda }^2)f\left(x\right)={\int }_0^x{\displaystyle \frac{sin\lambda (x-\xi )}{\lambda }}f\left(\xi \right)d\xi -{\mathsf{\Phi }}_{\xi }\left\{{\int }_0^{\xi }{\displaystyle \frac{sin\lambda (\xi -\eta )}{\lambda }}f\left(\eta \right)d\eta \right\}{\displaystyle \frac{sin\lambda x}{\lambda E\left(\lambda \right)}}.$$

(9)

The spectral projection operators $P_{{\lambda }_n}:C^1[0,1]\to {\mathcal{E}}_{{\lambda }_n}$ are defined by

$$P_{{\lambda }_n}:={\displaystyle \frac{1}{\pi i}}{\int }_{{\mathrm{\Gamma }}_{{\lambda }_n}}{R}_x(-{\lambda }^2)\lambda d\lambda ,$$

(10)

where ${\mathrm{\Gamma }}_{{\lambda }_n}$ is a simple contour containing ${\lambda }_n$ and no other zeros of $E\left(\lambda \right)$.

The formal spectral expansion of a function $f\in C\left(\right[0,1\left]\right)$ for eigenvalue problem (8) is said to be the following correspondence:

$$f\left(x\right)\sim \sum _{n=0}^{\infty }{P}_{{\lambda }_n}f.$$

(11)

The formal spectral expansion (11) is said to have the uniqueness property if $P_{{\lambda }_n}f=0$ for $n=0,1,2,\dots$ implies $f=0$ on $[0,1]$. Next, lemma gives a necessary and sufficient condition for the uniqueness of the formal spectral expansion (11), which is established in [36] for a more general case. Here, we use the formulation given in [38], concerning the case considered in the present paper.

Lemma 1. 

A necessary and sufficient condition for the uniqueness property of the formal spectral expansion (11) is the requirement (5), that is, the second end 1 of the interval $[0,1]$ to belong to the support of the functional Φ.

It is worth noting that any continuous linear functional on the space of smooth functions $C^1\left([0,1]\right)$ admits a representation of the form (see, e.g., [37,38]):

$$\mathsf{\Phi }\left\{f\right\}=af\left(0\right)+{\int }_0^1{f}^{\prime }\left(x\right)d\nu \left(x\right),$$

(12)

where $\nu \left(x\right)$ is a function with bounded variation on $[0,1]$ and a is a constant. Indeed, since $f\left(x\right)=f\left(0\right)+{\int }_0^x{f}^{\prime }\left(\xi \right)d\xi$, then

$$\mathsf{\Phi }\left\{f\right\}=\mathsf{\Phi }\left\{1\right\}f\left(0\right)+(\mathsf{\Phi }\circ l_x)\left\{f^{\prime }\right\},$$

where $l_x={\int }_0^x$. Since $\mathsf{\Phi }\circ l_x$ is a continuous linear functional on $C[0,1]$, it can be represented in the form $(\mathsf{\Phi }\circ l_x)\left\{f\right\}={\int }_0^{1}f\left(x\right)d\nu \left(x\right)$, which implies (12). Then property (5) is equivalent to the property that $x=1$ is a growth point of the function $\nu \left(x\right)$ in representation (12).

In general, it is not guaranteed that the series in (11) is convergent. However, if the series in (11) is uniformly convergent on $[0,1]$, then its sum is a continuous function, and the uniqueness property implies that this is exactly the function $f\left(x\right)$, i.e., in this case

$$f\left(x\right)=\sum _{n=0}^{\infty }{P}_{{\lambda }_n}f.$$

Next, we represent the spectral projection operators in a form more convenient for applications. Expression (9), (10), and the Cauchy integral formula yield

$$P_{{\lambda }_n}f\left(x\right)=-2{\mathsf{\Phi }}_{\xi }\{{\int }_0^{\xi }{H}_n(x,\xi -\eta )f\left(\eta \right)d\eta \},$$

(13)

where

$$H_n(x,\xi )={\displaystyle \frac{1}{({\kappa }_n-1)!}}\underset{\lambda \to {\lambda }_n}{lim}{\displaystyle \frac{{\partial }^{{\kappa }_n-1}}{\partial {\lambda }^{{\kappa }_n-1}}}\left[{\displaystyle \frac{sin\lambda \xi sin\lambda x}{\lambda E_n\left(\lambda \right)}}\right],E_n\left(\lambda \right)=E\left(\lambda \right)/{(\lambda -{\lambda }_n)}^{{\kappa }_n}.$$

(14)

Representation (13) implies that $P_{{\lambda }_n}f$ is a linear combination of the functions

$${\phi }_{n,k}\left(x\right)=\left\{\begin{array}{c}{x}^{k}sin{\lambda }_{n}x,k\mathrm{even,},\hfill \\ x^{k}cos{\lambda }_{n}x,k\mathrm{odd},\hfill \end{array}\right.$$

(15)

where $k=0,1,\dots ,{\kappa }_n-1,n\in \mathbb{N}$. Here, and in what follows, $\mathbb{N}$ denotes the set of positive integers. If ${\lambda }_0=0$ is an eigenvalue, then ${\kappa }_0=1$ and the corresponding eigenfunction is ${\phi }_{0,0}\left(x\right)=x$.

The eigenspace ${\mathcal{E}}_{{\lambda }_n}$ is the span of the generalized eigenfunctions, i.e.,

$${\mathcal{E}}_{{\lambda }_n}=\mathrm{span}\{{\phi }_{n,k}\left(x\right),k=0,1,\dots ,{\kappa }_n-1\}$$

and

$$P_{{\lambda }_n}f\left(x\right)=\sum _{k=0}^{{\kappa }_n-1}{f}_{n,k}{\phi }_{n,k}\left(x\right),$$

(16)

where the coefficients $f_{n,k}$ can be found from (13).

For example, in the case of one-dimensional eigenspaces (${\kappa }_n=1$), we have

$$f_{n,0}=-{\displaystyle \frac{2}{{\lambda }_n{E}^{\prime }\left({\lambda }_n\right)}}{\mathsf{\Phi }}_x\left\{{\int }_0^{x}sin{\lambda }_n(x-\xi )f\left(\xi \right)d\xi \right\},n\in \mathbb{N},$$

(17)

and for two-dimensional eigenspaces (${\kappa }_n=2$), Equation (13) yields

$$\begin{array}{cc}\hfill f_{n,0}=& -{\displaystyle \frac{4}{{\lambda }_n{E}^{″}\left({\lambda }_n\right)}}{\mathsf{\Phi }}_x\left\{{\int }_0^x(x-\xi )cos{\lambda }_n(x-\xi )f\left(\xi \right)d\xi \right.\hfill \\ \hfill & -\left.{\displaystyle \frac{3E^{″}\left({\lambda }_n\right)+{\lambda }_n{E}^{‴}\left({\lambda }_n\right)}{3{\lambda }_n{E}^{″}\left({\lambda }_n\right)}}{\int }_0^{x}sin{\lambda }_n(x-\xi )f\left(\xi \right)d\xi \right\},\hfill \\ \hfill f_{n,1}=& -{\displaystyle \frac{4}{{\lambda }_n{E}^{″}\left({\lambda }_n\right)}}{\mathsf{\Phi }}_x\left\{{\int }_0^{x}sin{\lambda }_n(x-\xi )f\left(\xi \right)d\xi \right\}.\hfill \end{array}$$

(18)

Next, some examples of functionals $\mathsf{\Phi }$ are given, leading to one- or two-dimensional eigenspaces. The first two examples are classical.

Example 1. 

Let $\mathsf{\Phi }\left\{y\right\}=y\left(1\right)$. Then the sine indicatrix is $E\left(\lambda \right)=sin\lambda /\lambda$ with single zeros ${\lambda }_n=\pi n,{\kappa }_n=1,n\in \mathbb{N}$. Therefore (17) yields

$$P_{{\lambda }_n}f\left(x\right)=f_{n,0}sin{\lambda }_{n}x,f_{n,0}=2{\int }_0^{1}sin{\lambda }_n\xi f\left(\xi \right)d\xi .$$

(19)

In this case (11) is the well-known sine Fourier expansion.

Example 2. 

Let $\mathsf{\Phi }\left\{y\right\}=y^{\prime }\left(1\right)$. Then $E\left(\lambda \right)=-cos\lambda$ with zeros

$${\lambda }_n=(2n-1)\pi /2,{\kappa }_n=1,n\in \mathbb{N}.$$

(20)

In this case (17) yields Fourier expansion (19) with ${\lambda }_n$ defined in (20).

In the next examples, the functional $\mathsf{\Phi }$ corresponds to a non-local boundary condition. Since multiplication by a constant does not change the homogeneous boundary condition, we consider the form of the functional that satisfies the identity $\mathsf{\Phi }\left\{x\right\}=1$. In the first two examples, the corresponding eigenvalues are double (${\kappa }_n=2$).

Example 3. 

Let $\mathsf{\Phi }\left\{y\right\}=(y^{\prime }\left(0\right)+y^{\prime }\left(1\right))/2$. Then the sine indicatrix is $E\left(\lambda \right)=(1+cos\lambda )/2$. It has double zeros ${\lambda }_n=(2n-1)\pi ,{\kappa }_n=2,n\in \mathbb{N}$. Applying (18) we obtain the corresponding spectral projection operators

$$P_{{\lambda }_n}f\left(x\right)=f_{n,0}sin{\lambda }_{n}x+f_{n,1}xcos{\lambda }_{n}x,n\in \mathbb{N},$$

(21)

where

$$f_{n,0}=4{\int }_0^1(1-\xi )sin{\lambda }_n\xi f\left(\xi \right)d\xi ,f_{n,1}=4{\int }_0^{1}cos{\lambda }_n\xi f\left(\xi \right)d\xi .$$

Example 4. 

Let $\mathsf{\Phi }\left\{y\right\}=2{\int }_0^{1}y\left(x\right)dx$, which corresponds to the Samarskii-Ionkin boundary condition [20]. Then $E\left(\lambda \right)=2(1-cos\lambda )/{\lambda }^2$, which yields ${\lambda }_n=2\pi n,{\kappa }_n=2,n\in \mathbb{N}$. Then (18) implies spectral projection operators of the form (21) with ${\lambda }_n=2\pi n$.

Example 5. 

Let $\mathsf{\Phi }\left\{y\right\}=y\left(1\right)+(y^{\prime }\left(1\right)-y^{\prime }\left(0\right))/\alpha$, $\alpha >0$. A BVP with a boundary condition ${\mathsf{\Phi }}_x\left\{u(x,t)\right\}=0$ is considered in [23,24]. The sine indicatrix is

$$E\left(\lambda \right)={\displaystyle \frac{2}{\alpha }}sin{\displaystyle \frac{\lambda }{2}}\left({\displaystyle \frac{\alpha }{\lambda }}cos{\displaystyle \frac{\lambda }{2}}-sin{\displaystyle \frac{\lambda }{2}}\right).$$

There are two sequences of simple zeros: ${\lambda }_n=2n\pi ,{\kappa }_n=1,n\in \mathbb{N}$, and ${\mu }_l$, satisfying $2l\pi <{\mu }_l<(2l+1)\pi ,{\kappa }_l=1,l=0,1,2,\dots$ The spectral expansion in this case admits the form

$$f\left(x\right)\sim \sum _{n=1}^{\infty }{P}_{{\lambda }_n}f+\sum _{l=0}^{\infty }{P}_{{\mu }_l}f,$$

where the corresponding spectral projection operators $P_{{\lambda }_n}$ and $P_{{\mu }_l}$ are obtained by applying (17).

3. General Fractional Relaxation Equation

In this section, some properties of the solution of the relaxation equation

$${}^C\mathbb{D}_t^{\left(\zeta \right)}v\left(t\right)=-\lambda v\left(t\right)+h\left(t\right),t>0;v\left(0\right)=a\in \mathbb{R},$$

(22)

where $\lambda >0$, are summarized. For more details, we refer to [43], Chapter 5.

In what follows, the following standard notation for the Laplace transform is used:

$$\widehat{u}\left(s\right)={\int }_0^{\infty }{e}^{-st}u\left(t\right)dt.$$

Here, we specify the assumptions on the kernel $\zeta \left(t\right)$ in the definition (1) of the general fractional derivative ${}^C\mathbb{D}_t^{\left(\zeta \right)}$. We suppose that the Laplace transform $\widehat{\zeta }\left(s\right)$ exists for all $s>0$ and obeys the properties

$$\widehat{\zeta }\left(s\right)\in \mathcal{SF}\mathrm{and}\underset{s\to +\infty }{lim}s\widehat{\zeta }\left(s\right)=+\infty ,$$

(23)

where $\mathcal{SF}$ denotes the class of Stieltjes functions. This class consists of all functions defined on ${\mathbb{R}}_{+}=(0,+\infty )$, which admit the representation (see e.g., [5])

$$\phi \left(s\right)={\displaystyle \frac{a}{s}}+b+{\int }_0^{\infty }{e}^{-s\tau }\psi \left(\tau \right)d\tau ,s>0.$$

(24)

where $a,b\ge 0$, the function $\psi$ is completely monotone and the Laplace transform of $\psi$ exists for any $s>0$. Recall that a real-valued infinitely differentiable on ${\mathbb{R}}_{+}$ function $\psi \left(t\right)$ is said to be a completely monotone function ($\psi \in \mathcal{CMF}$) if

$${(-1)}^n{\psi }^{\left(n\right)}\left(t\right)\ge 0,t>0,n\ge 0.$$

For more details on the above classes of functions, we refer to [44].

Consider the kernel $\eta \left(t\right)\in L_{loc}^1\left({\mathbb{R}}_{+}\right)$, which satisfies

$$(\zeta \ast \eta )\left(t\right)\equiv 1,$$

(25)

where ∗ denotes the classical convolution

$$(\zeta \ast \eta )\left(t\right)={\int }_0^t\zeta (t-\tau )\eta \left(\tau \right)d\tau .$$

Pairs of kernels satisfying identity (25) are referred to as Sonin kernels. In a Laplace domain, (25) reads $\widehat{\zeta }\left(s\right)\widehat{\eta }\left(s\right)=1/s$. Assumption (23) guarantees that an associated Sonin kernel $\eta \left(t\right)$ exists and $\eta \left(t\right)\in \mathcal{CMF}$; see e.g., [16].

The unique solution of Equation (22) is given by

$$v\left(t\right)=a{G}_1(t;\lambda )+{\int }_0^{t}G(\tau ;\lambda )h(t-\tau )d\tau ,$$

(26)

where the functions $G_1(t;\lambda )$ and $G(t;\lambda )$ are the fundamental and the impulse-response solutions to Equation (22), corresponding, respectively, to $a=1$, $h\equiv 0$, and $a=0$, $h\left(t\right)=\delta \left(t\right)$—the Dirac delta function. Let us note that the functions $G_1(t;\lambda )$ and $G(t;\lambda )$ are defined via their Laplace transforms with respect to t ($\lambda$ considered as a parameter) as follows:

$${\widehat{G}}_1(s;\lambda )={\displaystyle \frac{g\left(s\right)}{s\left(g\right(s)+\lambda )}},\widehat{G}(s;\lambda )={\displaystyle \frac{1}{g\left(s\right)+\lambda }},g\left(s\right)=s\widehat{\zeta }\left(s\right).$$

(27)

In the next theorem, some properties of the functions $G\left(t\right)$ and $G_1\left(t\right)$ are given. They will be used to find estimates of the terms in the spectral expansion of the solution.

Theorem 1. 

For any $\lambda >0$ the functions $G_1(t;\lambda )$ and $G(t;\lambda )$ are infinitely differentiable and completely monotone in $t>0$ and

$$G_1(0;\lambda )=1;0<G_1(t;\lambda )<1,G(t;\lambda )>0,t>0.$$

(28)

Moreover

$$G_1(t;\lambda )\le {\displaystyle \frac{1}{1+\lambda (1\ast \eta )\left(t\right)}},$$

(29)

where $\eta \left(t\right)$ is the associated Sonin kernel of $\zeta \left(t\right)$, and

$$\lambda {\int }_0^{t}G(\tau ;\lambda )d\tau <1,t>0.$$

(30)

Proof. 

The proofs of all properties, except (29), can be found in [45]. Here, we prove only estimate (29).

The fundamental solution $G_1(t;\lambda )$ is the solution of the equation

$${}^C\mathbb{D}_t^{\left(\zeta \right)}{G}_1(t;\lambda )=-\lambda G_1(t;\lambda ),t>0;G_1(0;\lambda )=1.$$

(31)

Taking into account the definition (1) of the general fractional derivative and the differentiation identity

$${(\zeta \ast v)}^{\prime }\left(t\right)=(\zeta \ast v^{\prime })\left(t\right)+\zeta \left(t\right)v\left(0\right)$$

(32)

for continuously differentiable functions $v\left(t\right)$, it follows that ${}^C\mathbb{D}_t^{\left(\zeta \right)}{G}_1=\zeta \ast G_1^{\prime }$. Let us now apply the convolution operator $\eta \ast$ to both sides of the equation in (31), where $\eta$ is the associated Sonin kernel of $\zeta$. In the left-hand side, we obtain

$$\begin{array}{ccc}\hfill \left(\eta \ast {}^C\mathbb{D}_t^{\left(\zeta \right)}{G}_1\right)\left(t\right)& =& \eta \ast (\zeta \ast G_1^{\prime })=(\eta \ast \zeta )\ast G_1^{\prime }=1\ast G_1^{\prime }\hfill \\ & =& G_1(t;\lambda )-G_1(0;\lambda )=G_1(t;\lambda )-1.\hfill \end{array}$$

Therefore, the function $G_1(t;\lambda )$ satisfies the integral equation

$$G_1(t;\lambda )=1-\lambda {\int }_0^t\eta (t-\tau )G_1(\tau ;\lambda )d\tau ,t>0.$$

(33)

Since $G_1(t;\lambda )$ are positive and decreasing functions, the integral identity (33) yields

$$1=G_1(t;\lambda )+\lambda {\int }_0^t\eta (t-\tau )G_1(\tau ;\lambda )d\tau \ge G_1(t;\lambda )+\lambda G_1(t;\lambda ){\int }_0^t\eta \left(\tau \right)d\tau ,$$

which implies estimate (29). □

For an extended version of the above theorem with detailed proofs, see Theorem 5.8 in [43].

Along with the relaxation functions $G(t;\lambda )$ and $G_1(t;\lambda )$, we define the following sequence of functions:

$$G_{j+1}(t;\lambda )={\int }_0^{t}G(\tau ;\lambda )G_j(t-\tau ;\lambda )d\tau ,j\in \mathbb{N}.$$

(34)

Relations (27) imply the following representation in the Laplace domain

$${\widehat{G}}_j(s;\lambda )={\displaystyle \frac{g\left(s\right)}{s{(g\left(s\right)+\lambda )}^j}},g\left(s\right)=s\widehat{\zeta }\left(s\right),j\in \mathbb{N}.$$

(35)

Based on the properties given in Theorem 1, next, we derive estimates for the functions $G_j(t;\lambda ),j\in \mathbb{N}$.

Theorem 2. 

For any $\lambda >0$ the functions $G_j(t;\lambda )$, $j\in \mathbb{N}$, are positive and continuous in $t\in [0,\infty )$ and satisfy the following estimates:

$$\begin{array}{ccc}\hfill {\lambda }^{j-1}{G}_j(t;\lambda )& \le & 1,t\ge 0,\hfill \end{array}$$

(36)

$$\begin{array}{ccc}\hfill {\lambda }^j{G}_j(t;\lambda )& \le & C_{\epsilon },t\ge \epsilon >0,\hfill \end{array}$$

(37)

where the constant $C_{\epsilon }>0$ does not depend on t or λ.

Proof. 

We prove estimates (36) and (37) by induction. Positivity and continuity of all functions in the sequence follow inductively from the fact that $G(t;\lambda )$ and $G_1(t;\lambda )$ satisfy the same properties.

Let first $t\ge 0$. According to (28), we have $0<G_1(t;\lambda )\le 1$ for $t\ge 0$. Therefore, (36) is proven for $j=1$. This estimate, together with definition (34), yields $j=2$

$$\lambda G_2(t;\lambda )=\lambda {\int }_0^{t}G(\tau ;\lambda )G_1(t-\tau ;\lambda )d\tau \le \lambda {\int }_0^{t}G(\tau ;\lambda )d\tau .$$

To prove estimate (36) for $j=2$, it remains necessary to apply (30).

Now suppose that (36) holds true. This, together with estimate (30) and definition (34), yields

$${\lambda }^j{G}_{j+1}(t;\lambda )=\lambda {\int }_0^{t}G(\tau ;\lambda ){\lambda }^{j-1}{G}_j(t-\tau ;\lambda )d\tau \le \lambda {\int }_0^{t}G(\tau ;\lambda )d\tau \le 1.$$

In this way, we finish the proof of (36).

Let now $t\ge \epsilon$ for some fixed $\epsilon >0$. For the proof of (37), we use estimate (29). We note first that $(1\ast \eta )\left(t\right)$ is strictly positive for all $t>0$. Indeed, since $\eta \in \mathcal{CMF}$, then $(1\ast \eta )\left(t\right)$ is a continuous, non-negative, and non-decreasing function for $t\ge 0$, which is analytic for $t>0$. Therefore, if it vanishes for some $t>0$, it should vanish for all $t>0$, and thus cannot be a Sonin kernel. Let us set

$$C_{\epsilon }={\displaystyle \frac{1}{(1\ast \eta )\left(\epsilon \right)}}.$$

Estimate (29) yields

$$\lambda G_1(t;\lambda )\le {\displaystyle \frac{\lambda }{1+\lambda (1\ast \eta )\left(t\right)}}\le {\displaystyle \frac{\lambda }{1+\lambda (1\ast \eta )\left(\epsilon \right)}}\le {\displaystyle \frac{1}{(1\ast \eta )\left(\epsilon \right)}}=C_{\epsilon }.$$

(38)

Here, we use the fact that the function $f\left(\lambda \right)={\displaystyle \frac{\lambda }{1+c\lambda }}$, where $c>0$ is a constant and is an increasing function of $\lambda$ for all $\lambda >0$. Therefore, $f\left(\lambda \right)\le f(+\infty )=1/c$. In this way, (37) is established for $j=1$. Furthermore, we use (38) and proceed in the same way as in the proof of (36) to deduce by induction (37). □

Let us consider the basic particular case when ${}^C\mathbb{D}_t^{\left(\zeta \right)}$ is the classical Caputo derivative of order $\alpha \in (0,1)$ in which the kernels satisfy

$$\zeta \left(t\right)={\displaystyle \frac{t^{-\alpha }}{\mathrm{\Gamma }(1-\alpha )}},\eta \left(t\right)={\displaystyle \frac{t^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}},g\left(s\right)=s^{\alpha },\alpha \in (0,1),$$

(39)

and the relaxation functions $G_1(t;\lambda )$ and $G(t;\lambda )$ are defined as follows:

$$G_1(t;\lambda )=E_{\alpha }(-\lambda t^{\alpha }),G(t;\lambda )=t^{\alpha -1}{E}_{\alpha ,\alpha }(-\lambda t^{\alpha }),$$

(40)

where

$$E_{\alpha ,\beta }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\mathrm{\Gamma }(\alpha k+\beta )}},E_{\alpha }\left(z\right)=E_{\alpha ,1}\left(z\right),$$

(41)

are Mittag–Leffler functions. In this case, the properties summarized in Theorem 1 reduce to some well-known properties of the Mittag–Leffler functions. Estimate (29), in this case, reads

$$E(-\lambda t^{\alpha })\le {\left(1+\lambda {\displaystyle \frac{t^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}}\right)}^{-1},$$

which is in agreement with the estimate $E_{\alpha }(-x)\le C{(1+x)}^{-1},x\ge 0,$, which is often used in the literature.

In the particular case of Caputo derivative (39), the functions $G_j(t;\lambda ),j\in \mathbb{N},$ are represented in terms of Prabhakar functions as follows:

$$G_j(t;\lambda )=t^{(j-1)\alpha }{E}_{\alpha ,(j-1)\alpha +1}^j(-\lambda t^{\alpha }),j\in \mathbb{N}.$$

(42)

Here, $E_{\alpha ,\beta }^{\delta }\left(z\right)$ denotes the Prabhakar function (see, e.g., [46])

$$E_{\alpha ,\beta }^{\delta }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(\delta \right)}_k}{k!}}{\displaystyle \frac{z^k}{\mathrm{\Gamma }(\alpha k+\beta )}},$$

(43)

where ${\left(\delta \right)}_k$ denotes the Pochhammer symbol

$${\left(\delta \right)}_k=\delta (\delta +1)\dots (\delta +k-1),k\in \mathbb{N},{\left(\delta \right)}_0=1.$$

It is a generalization of the classical Mittag–Leffler functions $E_{\alpha }\left(z\right)$ and $E_{\alpha ,\beta }\left(z\right)$:

$$E_{\alpha }\left(z\right)=E_{\alpha ,1}^1\left(z\right),E_{\alpha ,\beta }\left(z\right)=E_{\alpha ,\beta }^1\left(z\right).$$

Representation (42) can be derived from (35) and the Laplace transform pair [46]

$$\mathcal{L}\left\{t^{\beta -1}{E}_{\alpha ,\beta }^{\delta }(-\lambda t^{\alpha })\right\}\left(s\right)={\displaystyle \frac{s^{\alpha \delta -\beta }}{{(s^{\alpha }+\lambda )}^{\delta }}}.$$

(44)

Taking into account representations (42), the estimates (36) and (37) reduce, in this case, to estimates for particular functions of Prabhakar type, which seem to be new.

4. Uniqueness and Existence of a Classical Solution

Our aim is to construct a solution to problem (2)–(4) in the form of spectral expansion,

$$u(x,t)=\sum _{n=0}^{\infty }{P}_{{\lambda }_n}u,$$

where $P_{{\lambda }_n}$ are the spectral projection operators (10), acting with respect to the spatial variable x. Let us set

$$P_{{\lambda }_n}u=\sum _{k=0}^{{\kappa }_n-1}{A}_{n,k}\left(t\right){\phi }_{n,k}\left(x\right),$$

(45)

where $A_{n,k}\left(t\right)$ are unknown functions, depending only on time, and ${\phi }_{n,k}\left(x\right)$ are the generalized eigenfunctions defined in (15).

We apply the spectral projection operators to Equation (2). Since the function $u(x,t)$ satisfies the boundary conditions (3), it follows (see, e.g., [40]) that

$$P_{{\lambda }_n}\left({\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}\right)={\displaystyle \frac{{\partial }^2}{\partial x^2}}\left(P_{{\lambda }_n}u\right).$$

(46)

Moreover, representations (15) imply the following recurrent relations:

$${\phi }_{n,k}^{″}=-{\lambda }_n^2{\phi }_{n,k}+a_k{\lambda }_n{\phi }_{n,k-1}+b_k{\phi }_{n,k-2},k=0,\dots ,{\kappa }_n-1,$$

(47)

where

$$a_k=2{(-1)}^{k}k,b_k=k(k-1),$$

(48)

and ${\phi }_{n,-1}\equiv {\phi }_{n,-2}\equiv 0.$ Through (46) and (47), we obtain the following system of linear fractional equations for the unknown functions $A_{n,k}\left(t\right)$:

$${}^C\mathbb{D}_t^{\left(\zeta \right)}{A}_{n,k}=-{\lambda }_n^2{A}_{n,k}+a_{k+1}{\lambda }_n{A}_{n,k+1}+b_{k+2}{A}_{n,k+2}$$

(49)

for $k=0,1,\dots ,{\kappa }_n-1$, where we have set $A_{n,{\kappa }_n}\equiv A_{n,{\kappa }_n+1}\equiv 0$. Moreover, the initial condition (3) yields

$$A_{n,k}\left(0\right)=f_{n,k},k=0,1,\dots ,{\kappa }_n-1,$$

(50)

where $f_{n,k}$ are the coefficients in the expansion (16) of the initial function $f\left(x\right)$.

We notice that Equation (49) has the form of a relaxation Equation (22) for the function $A_{n,k}\left(t\right)$, if the functions $A_{n,k+1}\left(t\right)$ and $A_{n,k+2}\left(t\right)$ are already known. In the next theorem, we use the representation of the solution of the relaxation Equation (26) and the uniqueness property of the spectral expansion (Lemma 1) to prove the uniqueness of the solution.

Theorem 3. 

Let $1\in \mathrm{supp}\mathsf{\Phi }$. Then any solution to problem (2)–(4) is unique.

Proof. 

It is sufficient to prove that $f\equiv 0$ implies $u\equiv 0$. From (49), it follows that $A_{n,{\kappa }_n-1}\left(t\right)$ satisfies relaxation Equation (22) with $h\equiv 0$ and $a=0$. This equation has only one solution, and (26) implies that it is trivial, i.e., $A_{n,{\kappa }_n-1}\equiv 0$. Inserting this result in (49), we infer that $A_{n,{\kappa }_n-2}\left(t\right)$ satisfies the same equation. Therefore, $A_{n,{\kappa }_n-2}\equiv 0$. In this way, successively, we obtain $A_{n,k}\equiv 0$ for all $k=0,\dots ,{\kappa }_n-1$. Hence, $P_{{\lambda }_n}u\equiv 0$ for any $n=0,1,2,\dots .$ This, by Lemma 1, implies $u\equiv 0$. □

Next we construct the spectral expansion of the solution and prove that, under appropriate assumptions, it is a solution in the classical sense–that is, $u\in C\left(\right[0,1]\times [0,\infty \left)\right)$ and $u_x,u_{xx},{}^C\mathbb{D}_t^{\left(\zeta \right)}u\in C([0,1]\times (0,\infty ))$.

To find the time-dependent components $A_{n,k}\left(t\right)$ in the representation (45), we solve the system (49) for n fixed and $k={\kappa }_n-1,{\kappa }_n-2,{\kappa }_n-3,\dots ,1,0$ (in descending order).

First, let $k={\kappa }_n-1$. Equation (49) implies that the unknown function $A_{n,{\kappa }_n-1}\left(t\right)$ is the solution of the general relaxation Equation (22) with $h\equiv 0$ and $a=f_{n,{\kappa }_n-1}$. Therefore, (26) yields

$$A_{n,{\kappa }_n-1}\left(t\right)=f_{n,{\kappa }_n-1}{G}_1(t;{\lambda }_n^2),$$

(51)

where $G_1$ is defined in (27). Plugging this result in Equation (49) with $k={\kappa }_n-2$, we deduce that $A_{n,{\kappa }_n-2}\left(t\right)$ satisfies relaxation Equation (22) with $a=f_{n,{\kappa }_n-2}$ and

$$h\left(t\right)=a_{{\kappa }_n-1}{\lambda }_n{A}_{n,{\kappa }_n-1}=a_{{\kappa }_n-1}{f}_{n,{\kappa }_n-1}{\lambda }_n{G}_1(t;{\lambda }_n^2).$$

Therefore, according to (26), the solution is given by

$$A_{n,{\kappa }_n-2}\left(t\right)=f_{n,{\kappa }_n-2}{G}_1(t;{\lambda }_n^2)+a_{{\kappa }_n-1}{f}_{n,{\kappa }_n-1}{\lambda }_n{G}_2(t;{\lambda }_n^2),$$

(52)

where the function $G_2$ is defined in (34).

Inserting the obtained results (51) and (52) for $A_{n,{\kappa }_n-1}\left(t\right)$ and $A_{n,{\kappa }_n-2}\left(t\right)$, respectively, in Equation (49) with $k={\kappa }_n-3$, we deduce that the function $A_{n,{\kappa }_n-3}\left(t\right)$ obeys relaxation Equation (22) with $a=f_{n,{\kappa }_n-3}$ and

$$\begin{array}{ccc}\hfill h\left(t\right)& =& a_{{\kappa }_n-2}{\lambda }_n{A}_{n,{\kappa }_n-2}+b_{{\kappa }_n-1}{A}_{n,{\kappa }_n-1}\hfill \\ & =& \left(a_{{\kappa }_n-2}{f}_{n,{\kappa }_n-2}{\lambda }_n+b_{{\kappa }_n-1}{f}_{n,{\kappa }_n-1}\right)G_1(t;{\lambda }_n^2)+a_{{\kappa }_n-2}{a}_{{\kappa }_n-1}{f}_{n,{\kappa }_n-1}{\lambda }_n^2{G}_2(t;{\lambda }_n^2).\hfill \end{array}$$

According to (26), the solution is

$$\begin{array}{cc}\hfill A_{n,{\kappa }_n-3}\left(t\right)& =f_{n,{\kappa }_n-3}{G}_1(t;{\lambda }_n^2)+\left(a_{{\kappa }_n-2}{f}_{n,{\kappa }_n-2}{\lambda }_n+b_{{\kappa }_n-1}{f}_{n,{\kappa }_n-1}\right)G_2(t;{\lambda }_n^2)\hfill \\ \hfill & +a_{{\kappa }_n-2}{a}_{{\kappa }_n-1}{f}_{n,{\kappa }_n-1}{\lambda }_n^2{G}_3(t;{\lambda }_n^2),\hfill \end{array}$$

(53)

where $G_2$ and $G_3$ are defined in (34). In an analogous way, taking into account that $A_{n,{\kappa }_n-4}\left(t\right)$ obeys Equation (22) with

$$a=f_{n,{\kappa }_n-4},h\left(t\right)=a_{{\kappa }_n-3}{\lambda }_n{A}_{n,{\kappa }_n-3}+b_{{\kappa }_n-2}{A}_{n,{\kappa }_n-2}$$

and, plugging the obtained results (52) and (53) in $h\left(t\right)$, we find the function $A_{n,{\kappa }_n-4}\left(t\right)$. In this way, explicit representations can be derived for all time-dependent coefficients $A_{n,k}\left(t\right)$, $n\ge 0$, $k=0,\dots ,{\kappa }_n-1$, in the spectral expansion of the solution.

Next, we prove that, under some assumptions, the obtained formal expansion is a solution in the classical sense.

Theorem 4. 

Assume the zeros ${\lambda }_n$ of $E\left(\lambda \right)$ are real with multiplicities ${\kappa }_n\le K$ and ${\lambda }_n\to \infty$ as $n\to \infty$. Let $f\in C^2\left([0,1]\right)$, $f\left(0\right)=\mathsf{\Phi }\left\{f\right\}=0$, and $f_{n,k}=O\left(n^{-2}\right)$ as $n\to \infty$ for any $k=0,\dots ,{\kappa }_n-1$, where $f_{n,k}$ are defined by (16). Then, problem (2)–(4) admits a unique classical solution given by the series

$$u(x,t)=\sum _{n=0}^{\infty }\sum _{k=0}^{{\kappa }_n-1}{A}_{n,k}\left(t\right){\phi }_{n,k}\left(x\right).$$

(54)

Here, ${\phi }_{n,k}\left(x\right)$ are the generalized eigenfunctions defined in (15) and the functions $A_{n,k}\left(t\right)$ are defined recursively in descending order k by the relation

$$A_{n,k}\left(t\right)=f_{n,k}{G}_1(t;{\lambda }_n^2)+{\int }_0^{t}G(\tau ;{\lambda }_n^2)\left(a_{k+1}{\lambda }_n{A}_{n,k+1}(t-\tau )+b_{k+2}{A}_{n,k+2}(t-\tau )\right)d\tau ,$$

(55)

where the first two terms $A_{n,{\kappa }_n-1}\left(t\right)$ and $A_{n,{\kappa }_n-2}\left(t\right)$ are given in (51) and (52).

Proof. 

Let us note first that formula (55) gives the solution of Equation (49) with initial condition (50), which is deduced by applying (26). Therefore, the function $u(x,t)$, defined by the formal series expansion (54), satisfies Equation (2). After a direct check, we found that the boundary and initial conditions (3) and (4) are formally satisfied. It remains necessary to prove that this is legitimate, which would hold if the series (54) is uniformly convergent on $[0,1]\times [0,\infty )$ and the series $\sum {\left(P_{{\lambda }_n}u\right)}_x$ and $\sum {\left(P_{{\lambda }_n}u\right)}_{xx}$ are uniformly convergent on $[0,1]\times (0,\infty )$.

Since ${\lambda }_n$ are real, the generalized eigenfunctions ${\phi }_{n,k}\left(x\right)$, defined in (15), are bounded on $[0,1]$.

Let us first estimate the time-dependent components $A_{n,k}\left(t\right)$ for $t\ge 0$. Since the multiplicities of the eigenvalues are bounded (${\kappa }_n\le K$ for all $n=0,1,\dots$), the coefficients $a_k$ and $b_k$, defined in (48), are also bounded. Let us fix n sufficiently large, such that ${\lambda }_n\ge 1$. Estimates (36) for $j=1,2,3$ imply

$$G_1(t;{\lambda }_n^2)\le 1,{\lambda }_n^2{G}_2(t;{\lambda }_n^2)\le 1,{\lambda }_n^4{G}_3(t;{\lambda }_n^2)\le 1,t\ge 0.$$

(56)

Representations (51)–(53) together with estimates (56) yields

$$\begin{array}{ccc}\hfill |A_{n,{\kappa }_n-1}\left(t\right)|& \le & |f_{n,{\kappa }_n-1}|G_1(t;{\lambda }_n^2)\le \left|f_{n,{\kappa }_n-1}\right|,\hfill \\ \hfill |A_{n,{\kappa }_n-2}\left(t\right)|& \le & |f_{n,{\kappa }_n-2}|G_1(t;{\lambda }_n^2)+C\left|f_{n,{\kappa }_n-1}\right|{\lambda }_n{G}_2(t;{\lambda }_n^2)\hfill \end{array}$$

(57)

$$\begin{array}{ccc}& \le & |f_{n,{\kappa }_n-2}|+{\displaystyle \frac{C}{{\lambda }_n}}|f_{n,{\kappa }_n-1}|,\hfill \end{array}$$

(58)

$$\begin{array}{ccc}\hfill |A_{n,{\kappa }_n-3}\left(t\right)|& \le & |f_{n,{\kappa }_n-3}|G_1(t;{\lambda }_n^2)+\left({\displaystyle \frac{|a_{{\kappa }_n-2}|}{{\lambda }_n}}|f_{n,{\kappa }_n-2}|+{\displaystyle \frac{|b_{{\kappa }_n-1}|}{{\lambda }_n^2}}|f_{n,{\kappa }_n-1}|\right){\lambda }_n^2{G}_2(t;{\lambda }_n^2)\hfill \\ & +& {\displaystyle \frac{|a_{{\kappa }_n-2}{a}_{{\kappa }_n-1}|}{{\lambda }_n^2}}\left|f_{n,{\kappa }_n-1}\right|{\lambda }_n^4{G}_3(t;{\lambda }_n^2),\hfill \end{array}$$

$$\begin{array}{ccc}& \le & |f_{n,{\kappa }_n-3}|+{\displaystyle \frac{C}{{\lambda }_n}}|f_{n,{\kappa }_n-2}|+{\displaystyle \frac{C}{{\lambda }_n^2}}\left|f_{n,{\kappa }_n-1}\right|.\hfill \end{array}$$

(59)

Here and below, the letter C stands for different positive constants.

We prove by induction (in descending order of k) that

$$|A_{n,k}\left(t\right)|\le |f_{n,k}|+\sum _{j=k+1}^{{\kappa }_n-1}{\displaystyle \frac{C}{{\lambda }_n^{j-k}}}\left|f_{n,j}\right|,k=0,\dots ,{\kappa }_n-1.$$

(60)

For $k={\kappa }_n-1$, $k={\kappa }_n-2$, and $k={\kappa }_n-3$, this is established in (57)–(59). Furthermore, relation (55) yields

$$|A_{n,k}\left(t\right)|\le |f_{n,k}|G_1(t;{\lambda }_n^2)+{\lambda }_n^2{\int }_0^{t}G(\tau ;{\lambda }_n^2)\left|{\displaystyle \frac{a_{k+1}}{{\lambda }_n}}{A}_{n,k+1}(t-\tau )+{\displaystyle \frac{b_{k+2}}{{\lambda }_n^2}}{A}_{n,k+2}(t-\tau )\right|d\tau ,$$

which, by applying estimate (56) for $G_1(t;{\lambda }_n^2)$ and estimate (60) for $A_{n,k+1}$ and $A_{n,k+2}$, implies

$$|A_{n,k}\left(t\right)|\le |f_{n,k}|+{\lambda }_n^2{\int }_0^{t}G(\tau ;{\lambda }_n^2)d\tau \sum _{j=k+1}^{{\kappa }_n-1}{\displaystyle \frac{C}{{\lambda }_n^{j-k}}}\left|f_{n,j}\right|.$$

In order to obtain (60), it remains to use estimate (30).

Therefore, (60) implies that the series (54) is dominated by the convergent series

$$C\sum _{n=0}^{\infty }\sum _{k=0}^{{\kappa }_n-1}\left|f_{n,k}\right|<\infty .$$

(61)

Using the Weierstrass criterion, the series (54) is absolutely and uniformly convergent in $[0,1]\times [0,\infty )$. This implies that $u\in C\left(\right[0,1]\times [0,\infty \left)\right)$, since any $P_{{\lambda }_n}u\in C([0,1]\times [0,\infty ))$.

For the proof of the uniform convergence for $t>0$ of the series of the derivatives $\sum {\left(P_{{\lambda }_n}u\right)}_x$ and $\sum {\left(P_{{\lambda }_n}u\right)}_{xx}$, we use similar argument. Relation (47) shows that the second space-derivative of $P_{{\lambda }_n}u$ corresponds to a multiplication by (at most) ${\lambda }_n^2$. To compensate for this, we use estimates (37), which hold for $t>0$ and are stronger, compared to (36). The series with first space-derivatives is considered analogously.

This completes the proof of the theorem. □

It is worth noting that, in many particular cases, such as the examples considered in Section 2 and Section 5, the assumption $f\in C^2\left([0,1]\right)$, together with the compatibility conditions $f\left(0\right)=\mathsf{\Phi }\left\{f\right\}=0$, imply the asymptotic behavior of the coefficients $f_{n,k}=O\left(n^{-2}\right)$, $n\to \infty$. This can be proven applying the standard procedure of applying twice integration by parts in the integral representations of the coefficients. However, in the general setting of arbitrary functional $\mathsf{\Phi }$, the problem for sufficient conditions on f that guarantee $u(x,t)$ to be a classical solution is subtle and outside the scope of this paper. Various results on convergence of spectral expansions in root functions can be found in many classical works, such as [47,48]. In the present paper, the asymptotic decay $f_{n,k}=O\left(n^{-2}\right)$, which is sufficient for uniform convergence of the solution series, is assumed in the formulation of Theorem 4.

Let us point out that, in the case of multiple eigenvalues, compared to the better-studied case of single eigenvalues, the following difference is observed. Estimates (60) of the time-dependent components $A_{n,k}\left(t\right)$ in the spectral expansion of the solution show that, under the assumptions of the theorem, the leading term in $A_{n,k}\left(t\right)$ when $n\to \infty$ is $f_{n,k}$, which implies that the spectral expansion of the solution is dominated by the series (61). Therefore, adding more terms due to the multiplicity of eigenvalues does not affect essentially the convergence of the spectral expansion, compared to the case of single eigenvalues.

5. An Example

In the present section, the general results obtained so far are applied to the initial-boundary-value problem for Equation (2) with the following initial condition (3) and boundary conditions:

$$u(0,t)=0,2u(1/2,t)+u(1,t)=0,t\ge 0.$$

(62)

For the initial function $f\left(x\right)$, it is assumed that it satisfies the compatibility conditions

$$f\in C^2\left([0,1]\right),f\left(0\right)=0,2f(1/2)+f\left(1\right)=0.$$

(63)

In this specific case, the spatial differential operator defines two sequences of eigenvalues, one of which consists of triple eigenvalues (${\kappa }_n=3$).

Boundary conditions of the form (62) are considered in [41]. Next, we use some results from [41], concerning the corresponding spectral projection operators.

Let $\mathsf{\Phi }\left\{y\right\}=\left(2y\right(1/2)+y(1\left)\right)/2$. Then the sine indicatrix of the functional $\mathsf{\Phi }$ is

$$E\left(\lambda \right)={\displaystyle \frac{4}{\lambda }}sin{\displaystyle \frac{\lambda }{4}}{cos}^3{\displaystyle \frac{\lambda }{4}}.$$

There are two sequences of eigenvalues. The equation ${\displaystyle {cos}^3\frac{\lambda }{4}=0}$ determines the triple zeros

$${\lambda }_n=2(2n-1)\pi ,{\kappa }_n=3,n\in \mathbb{N},$$

and corresponding triple eigenvalues $-{\lambda }_n^2$. The zeros of ${\displaystyle \frac{4}{\lambda }sin\frac{\lambda }{4}=0}$ are

$${\mu }_l=4l\pi ,{\kappa }_l=1,l\in \mathbb{N},$$

which determine single eigenvalues $-{\mu }_l^2$.

We denote the spectral projection operators, corresponding to the two sequences of eigenvalues by $P_{{\lambda }_n}$ and $Q_{{\mu }_l}$, respectively. Then

$$\begin{array}{ccc}& & P_{{\lambda }_n}f\left(x\right)=f_{n,0}sin{\lambda }_{n}x+f_{n,1}xcos{\lambda }_{n}x+f_{n,2}{x}^{2}sin{\lambda }_{n}x,n\in \mathbb{N},\hfill \\ & & Q_{{\mu }_l}f\left(x\right)=g_{l,0}sin{\mu }_{l}x,l\in \mathbb{N}.\hfill \end{array}$$

The formal spectral expansion (11) in this case admits the form

$$f\left(x\right)=\sum _{n=1}^{\infty }{P}_{{\lambda }_n}f+\sum _{l=1}^{\infty }{Q}_{{\mu }_l}f.$$

(64)

Explicit representations of the coefficients $f_{n,0},f_{n,1},f_{n,2}$, and $g_{n,0}$ are derived in [41]. For completeness, they are given below:

$$\begin{array}{cc}\hfill f_{n,0}& ={\int }_0^1(8{\xi }^2-16\xi +7)f\left(\xi \right)sin{\lambda }_n\xi d\xi -2{\int }_0^{{\displaystyle \frac{1}{2}}}(8{\xi }^2-8\xi +1)f\left(\xi \right)sin{\lambda }_n\xi d\xi ,\hfill \\ \hfill f_{n,1}& =16\left({\int }_0^1(1-\xi )f\left(\xi \right)cos{\lambda }_n\xi d\xi -{\int }_0^{{\displaystyle \frac{1}{2}}}(1-2\xi )f\left(\xi \right)cos{\lambda }_n\xi d\xi \right)\hfill \\ \hfill f_{n,2}& =8\left({\int }_0^{1}f\left(\xi \right)sin{\lambda }_n\xi d\xi -2{\int }_0^{{\displaystyle \frac{1}{2}}}f\left(\xi \right)sin{\lambda }_n\xi d\xi \right)\hfill \\ \hfill g_{l,0}& =2{\int }_0^{{\displaystyle \frac{1}{2}}}f\left(\xi \right)sin{\mu }_l\xi d\xi +{\int }_0^{1}f\left(\xi \right)sin{\mu }_l\xi d\xi .\hfill \end{array}$$

(65)

Taking into account compatibility conditions (63) for the function f, we obtain, after integration by parts, that

$$f_{n,k}=O\left(n^{-2}\right),n\to \infty ,k=0,1,2,g_{l,0}=O\left(l^{-2}\right),l\to \infty .$$

(66)

Therefore, the assumptions of Theorem 4 are satisfied. The unique solution is given by the spectral expansion

$$u(x,t)=\sum _{n=1}^{\infty }{P}_{{\lambda }_n}u+\sum _{l=1}^{\infty }{Q}_{{\mu }_l}u,$$

(67)

where

$$\begin{array}{ccc}\hfill P_{{\lambda }_n}u& =& A_{n,0}\left(t\right)sin{\lambda }_{n}x+A_{n,1}\left(t\right)xcos{\lambda }_{n}x+A_{n,2}\left(t\right)x^{2}sin{\lambda }_{n}x,\hfill \end{array}$$

(68)

$$\begin{array}{ccc}\hfill Q_{{\mu }_l}u& =& B_{l,0}\left(t\right)sin{\mu }_{l}x.\hfill \end{array}$$

(69)

The time-dependent components in (68) and (69) are obtained from (51)–(53) as follows:

$$\begin{array}{ccc}\hfill A_{n,0}\left(t\right)& =& f_{n,0}{G}_1(t;{\lambda }_n^2)+\left(a_1{f}_{n,1}{\lambda }_n+b_2{f}_{n,2}\right)G_2(t;{\lambda }_n^2)+a_1{a}_2{f}_{n,2}{\lambda }_n^2{G}_3(t;{\lambda }_n^2),\hfill \\ \hfill A_{n,1}\left(t\right)& =& f_{n,1}{G}_1(t;{\lambda }_n^2)+a_2{f}_{n,2}{\lambda }_n{G}_2(t;{\lambda }_n^2),\hfill \\ \hfill A_{n,2}\left(t\right)& =& f_{n,2}{G}_1(t;{\lambda }_n^2),\hfill \\ \hfill B_{l,0}\left(t\right)& =& g_{l,0}{G}_1(t;{\mu }_l^2).\hfill \end{array}$$

Let us note that the estimates established in Theorem 2 show that the leading terms as $n\to \infty$ in all four equations above are the terms containing the function $G_1$, i.e., the rest of the terms in $A_{n,0}\left(t\right)$ and $A_{n,1}\left(t\right)$ can be neglected for $n\to \infty$. Therefore, the convergence of the two series in (67), corresponding, respectively, to triple and single eigenvalues, is essentially the same. This example further illustrates the fact that adding more terms due to the multiplicity of eigenvalues does not essentially affect the numerical stability of the spectral expansion of the solution.

As discussed at the end of Section 3, in the particular case of the Caputo derivative (39) for the functions $G_1$, $G_2$, and $G_3$, we have the following representations:

$$G_1(t;\lambda )=E_{\alpha }(-\lambda t^{\alpha }),G_2(t;\lambda )=t^{\alpha }{E}_{\alpha ,\alpha +1}^2(-\lambda t^{\alpha }),G_3(t;\lambda )=t^{2\alpha }{E}_{\alpha ,2\alpha +1}^3(-\lambda t^{\alpha }).$$

Let us note that the Prabhakar functions in $G_2$ and $G_3$ can be represented in terms of Mittag–Leffler functions by applying a reduction formula (e.g., formula (4.4) in [46]).

6. Concluding Remarks

In this paper, we study the general fractional diffusion equation on a bounded space domain, subject to a Dirichlet boundary condition and a boundary condition of a general non-classical form, represented by a continuous linear functional. For the solution to the problem, spectral projection operators are applied. An algorithm for constructing a solution in the form of an expansion in terms of the generalized eigenfunctions is given. The uniqueness property of the corresponding spectral expansion is essentially used to prove the uniqueness of the solution. Estimates for the time-dependent components in the formal spectral expansion are derived and used to prove that, under some assumptions, this expansion provides a solution in the classical sense.

The technique used here can be applied to other kinds of time-fractional differential equations with one classical and one non-classical boundary condition. Applications to some inverse problems for such equations are also feasible; see, e.g., [16,30,32,33].

A limitation of the presented spectral method is that it is confined to linear partial differential equations. Extending the analysis to nonlinear non-local problems would require approximation of the nonlinear problem by appropriate linear ones or completely different approaches, such as those based on fixed-point theorems, as demonstrated in recent work by [49].