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Article

Comparative Study of Some Numerical Methods for the Burgers–Huxley Equation

by
Appanah Rao Appadu
1,*,
Bilge İnan
2 and
Yusuf Olatunji Tijani
1
1
Department of Mathematics and Applied Mathematics, Nelson Mandela University, Port-Elizabeth 6031, South Africa
2
Department of Mathematics, Faculty of Muallim Rıfat Education, Kilis 7 Aralık University, 79000 Kilis, Turkey
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(11), 1333; https://doi.org/10.3390/sym11111333
Submission received: 19 September 2019 / Revised: 17 October 2019 / Accepted: 19 October 2019 / Published: 24 October 2019

Abstract

:
In this paper, we construct four numerical methods to solve the Burgers–Huxley equation with specified initial and boundary conditions. The four methods are two novel versions of nonstandard finite difference schemes (NSFD1 and NSFD2), explicit exponential finite difference method (EEFDM) and fully implicit exponential finite difference method (FIEFDM). These two classes of numerical methods are popular in the mathematical biology community and it is the first time that such a comparison is made between nonstandard and exponential finite difference schemes. Moreover, the use of both nonstandard and exponential finite difference schemes are very new for the Burgers–Huxley equations. We considered eleven different combination for the parameters controlling diffusion, advection and reaction, which give rise to four different regimes. We obtained stability region or condition for positivity. The performances of the four methods are analysed by computing absolute errors, relative errors, L 1 and L errors and CPU time.

Graphical Abstract

1. Introduction

Numerical and mathematical analysis are of significant importance for the solution and understanding of problems in science and engineering. Such problems are usually expressed using differential equations. Many numerical and mathematical methods use geometrical and analytical properties of mathematical and/or numerical problems.
Non-linearities exist in almost every branch of science and engineering ranging from biological systems, organic and inorganic chemistry, population dynamics, astrophysics, robotics, biomolecular engineering and zoology [1]. Non-linear partial differential equations (NLPDE) are the resulting equations for many physical phenomena, which in turn has motivated many researchers to study these alluring non-linear problems. Since mathematicians regard existence as one of the prime attributes of studies [2], the existence, uniqueness, symmetry and integrability have been studied for many evolution parabolic equations (see [3,4,5]). Unfortunately, exact solution rarely exists for many of these non-linear problems. However, there are some non-linear partial differential equations that become integrable after some symbolic transformation. In this case, the analytical solution becomes obtainable. In this regard, many powerful techniques have been introduced, amongst them are Lie group method [6], Jacobi elliptic functions method [7], function expansion method [8,9], homogeneous balance method [10], solitary wave ansatz method [11], and Hirota bilinear method [12], to mention a few. The Burgers–Huxley (BH) equation, which can be seen as an archetypal equation for describing the interaction among reaction mechanism, convection effects and diffusion transport, has been widely studied by researchers. Many techniques for obtaining solution to the Burgers–Huxley (BH) equation have been provided, one of them being from the work of Ismail et al. [13]. They provided a solution to the generalised Burgers–Huxley equation using the Adomian decomposition method (ADM). Batiha et al. [14], Biazar and Mohammadi [15], Yaghouti and Zabili [16], Molabahramia and Khani [17], Mittal and Jiwari [18], Javidi [19] and Tomasiello [20] obtained solutions to the Burgers–Huxley equation using variational iteration method (VIM), differential transform method (DTM), Laplace decomposition method, homotopy analysis method (HAM), differential quadrature method, spectral collocation and IDQ methods, respectively. Ray and Gupta [21,22] constructed a novel Haar wavelet collocation method and used it for the numerical solutions of the Burgers, Boussinesq–Burgers, Huxley and Burgers–Huxley equations. Discontinuity is the major drawback of the Haar wavelet method. The approximate analytical methods are plagued with many downsides, which include slow convergence at large t, high computer memory usage and difficulty in finding a closed form formula for the obtained series expression. It has been duly noted that the use of standard time integration techniques such as forward or backward Euler and Runge–Kutta methods to solve such differential models often lead to numerical instabilities and chaotic solution due to selection of discretisation parameters [23]. Kyrychko et al. [24] presented a traveling wave analysis for the extended-Burgers–Huxley equation. The existence of traveling wave solution was also validated for singularly perturbed Burgers–Huxley equation.
Among various techniques for solving partial differential equations especially in mathematical biology, the NSFD methods have been proved to be one of the most efficient approaches in recent years [25,26] due to positive definiteness under some conditions, boundedness of solution, monotonicity of the solutions and properties such as special solutions with predetermined stability. Nonstandard (NSFD) methods have been used since 1994 and, in the early stages, the linear advection and advection-diffusion equations were discretised using this type of method. Mickens and Gumel [27] presented a detailed explanation on the nonstandard technique for the Burgers-Fisher equation and the paper was published in 2002. Zhang et al. [28] constructed a scheme based on exact solution and the nonstandard finite difference schemes for the Burgers and Burgers–Fisher equation; their paper was published in 2014. Recently, Agbavon and Appadu [29] analysed four nonstandard schemes for the FitzHugh–Nagumo equation. Interested readers can check the following references for works where nonstandard finite difference scheme are employed (Appadu et al. [30], Appadu [31], Agbavon et al. [32] Chapwanya et al. [33], Mickens [34], Jordan [35], and Aderogba and Chapwanya [36]. The construction of NSFD methods for the Burgers–Huxley equation is very recent. Zibaei et al. [37] presented the exact and nonstandard schemes for the Burgers–Huxley equation and their paper was published in 2016. It is noteworthy to mention that the application of NSFD schemes are not limited to areas of Mathematical Biology; see the works of Oluwaseye and Talitha [38], Dai [39] and Diaz et al. [40].
Another class of schemes known as exponential finite difference methods have been used to solve the Burgers type equations. Some of the good points of these methods to mathematical biologists are the distinctive features of computational efficiency, lesser computational time, ability to preserve physical properties of the differential equation [41,42]. The exponential scheme was originally developed by Bhattacharry [43] to solve the heat equation and the paper was published in 1985. İnan and Bahadir [44] employed implicit and fully implicit exponential finite difference method to solve Burgers equation and their paper was published in 2013. Bahadir [45] obtained a numerical solution for the small time Korteweg–de Vries (KdV) equation using the exponential scheme. Macías-Díaz constructed a modified exponential method that preserves structural properties of the solutions of the Burgers–Huxley equation [42]. More generally, recently in 2017, Burgers–Huxley equations were solved using an explicit exponential finite difference method constructed by İnan [46]. In addition, Burgers equation was solved with exponential method modified with Padé approximation by İnan and Macías-Díaz [47] and by İnan [48]. The quite good accuracy of the exponential finite difference schemes and the nature of some of these alluring non-linear problems has made these methods quite popular for the Burgers-type equations.
The paper is organised as follows. In Section 2, we have a short overview of the Burgers–Huxley equation. Section 3 describes the numerical experiment. In Section 4 and Section 5, we construct two versions of non-standard finite difference schemes and two versions of exponential finite difference schemes and study some of their properties. The results are presented in Section 6. Section 7, which is the concluding part, highlights the salient features of this paper and possible future extension of the present study. MATLAB R2015a and Fortran computing platforms were used for simulations. A very short version of this paper has been accepted for publication as [49].

2. The Burgers–Huxley Equation

In recent time, partial differential equations containing nonlinear diffusion such as Equation (1) played an important role in non-linear physics, physiology and nerve propagation. The generalised Burgers–Huxley equation describes a wide class of physical non-linear phenomena in biology. It is well-known that the generalised Burgers–Huxley equation has a traveling wave solution with properties like boundedness, monotonicity and positivity (see [24,50,51]).
The generalised Burgers–Huxley equation is given by
u t = 2 u x 2 α u δ u x + β u ( 1 u δ ) ( u δ γ ) .
In population dynamics, u ( x , t ) represent the population density, γ is the species carrying capacity, α stands for the speed of advection and β is a parameter which describe nonlinear source.
When certain condition is imposed on the parameter, the generalised Burgers–Huxley equation is reduced to many parabolic evolution equations of physical insight. When δ = 1 and β = 0 , Equation (1) is reduced to the Burgers equation, which can be used to study sound waves in viscous medium [52]. The FitzHugh–Nagumo equation is obtained when δ = 1 and α = 0 ; this equation has enormous applications in neurophysiology, logistic population growth and auto catalytic reaction [29]. When δ = 1 , α = 0 and γ = 1 , we have the Newell–Whitehead–Segel equation [53]. For α = 0 and β = 0 , Equation (1) is reduced to the well-known heat conduction equation. When α = 0 , the Huxley equation, which describes nerve pulse propagation, liquid crystal wall motion and nerve fiber, is obtained. We note that, for δ = 1 , Equation (1) gives the Burgers–Huxley equation.

3. Numerical Experiment

We solve the Burgers–Huxley equation
u t + α u u x 2 u x 2 = β u 1 u u γ , = β ( 1 + γ ) u 2 β γ u β u 3 ,
subject to the following initial and boundary conditions [54]:
u ( x , 0 ) = γ 2 + γ 2 tanh ( A 1 x ) , u ( 0 , t ) = γ 2 + γ 2 tanh ( A 1 A 2 ) t , u ( 1 , t ) = γ 2 + γ 2 tanh ( A 1 ( 1 A 2 t ) ) ,
where α > 0 , β > 0 , 0 < γ < 1 , x [ 0 , 1 ] and t [ 0 , 10 ] . The exact solution is given by [54]
u ( x , t ) = γ 2 + γ 2 tanh ( A 1 ( x A 2 t ) ) ,
where
A 1 = α + α 2 + 8 β 8 γ and A 2 = α γ 2 ( 2 γ ) ( α + α 2 + 8 β ) 4 .
We note that the initial condition u ( x , 0 ) is non-negative i.e., u ( x , 0 ) 0 . In this work, we consider 11 different cases:
(1)
α = 0.5 , β = 0.5 , γ = 0.001 .
(2)
α = 0.5 , ( β > α ) β = 2.0 , γ = 0.001 .
(3)
α = 0.5 , ( β > > α ) β = 10.0 , γ = 0.001 . (singularly perturbed case)
(4)
α > β ( α = 2.0 ) , β = 0.5 , γ = 0.001 .
(5)
α = 0.5 , β = 0.5 , γ = 0.5 .
(6)
α = 0.5 , β > α ( β = 2.0 ) , γ = 0.5 .
(7)
α = 0.5 , ( β > > α ) β = 10.0 , γ = 0.5 . (singularly perturbed case)
(8)
α > β ( α = 2.0 ) , β = 0.5 , γ = 0.5 .
(9)
α = β = 1.0 , γ = 0.001 ([13,14]).
(10)
α = 0.5 , β = 0.5 , γ = 0.001 and k = 0.1 (FIEFDM)
(11)
α = 0.5 , β = 10.0 , γ = 0.5 and k = 0.1 (FIEFDM)
We chose the first eight cases to consider four different regimes, i.e., when α = β , β > α , β > > α (singularly perturbed) and α > β and with two different values of γ , namely γ = 0.001 and γ = 0.5 . We also considered Case 9 so that we can compare our results with other methods in [13,14]. The case β > > α is a very challenging one. In [32], the partial differential equation
u t 2 u x 2 = β u 1 u
was solved using β = 10 4 and the initial condition consisted of exponential function. This is quite a challenging numerical experiment. Cases 10 and 11 were chosen to check the effectiveness of the fully implicit exponential finite difference method at other values of k.
We obtained numerical solutions for the numerical experiment using non-standard finite difference, and explicit and fully implicit exponential finite difference methods. The solution domains are discretised into cells as ( x j , t n ) where x j = j h , ( j = 1 , 2 , , N ) and t n = n k , ( n = 1 , 2 , ) , h = Δ x = 1 0 N 1 is the spatial mesh size and we choose h = 0.1 for all computations in this paper. The temporal step size is k = Δ t and obtained by applying positivity condition or from stability analysis. U j n represents numerical solution while u ( x , t ) denotes the exact solution at point x j , t n .

4. Nonstandard Finite Difference Scheme

The pioneering work on non-standard finite difference scheme can be traced back to the works of Ronald Mickens [26,55,56,57]. The NSFD scheme is designed to resolve the issue of numerical instabilities and/or chaotic behaviour problems, which mostly plague many numerical methods. The concepts generally work on the principle of dynamical consistency which vary from one system to another. There are some major rules to be followed in the construction of such methods. They are listed below.
  • Non-local representation of linear and non-linear terms on the computational grid; E.g., u n 2 U n U n + 1 , u n 2 U n + 1 + 2 U n + U n 1 4 U n , u n 3 2 U n 3 U n 2 U n + 1 etc.
  • Use of numerator and denominator functions ψ ( h ) and ϕ ( k ) , respectively with the property
    lim h 0 ψ ( h ) = h , and lim k 0 ϕ ( k ) = k ,
    where ψ ( h ) = h + O ( h 2 ) and ϕ ( k ) = k + O ( k ) 2 .
  • The difference equation should have the same order as the original differential equation. In general, when the order of the difference equation is larger than the order of the differential equation, spurious solutions will appear [58].
  • The discrete approximation should preserve some important properties of the corresponding differential equation. Properties such as boundedness and positivity should be preserved.
The scheme constructed by Zibaei et al. [37] for Equation (2) is given by
U j n + 1 U j n Φ = U j + 1 n 2 U j n + U j 1 n Ψ α U j n + 1 U j n U j 1 n Γ + β ( 1 + γ ) ( U j 1 n ) 2 β γ U j n + 1 + β 1 2 ( U j 1 n ) 3 3 2 ( U j 1 n ) 2 U j n + 1 ,
where
Φ = 1 e 2 A 1 A 2 k 2 A 1 A 2 , Γ = e 2 A 1 h 1 2 A 1 , Ψ = Γ 2 .
A 1 is described in Equation (5) and A 2 = α + 4 2 γ γ A 1 .
We propose two versions of NSFD scheme as NSFD1 and NSFD2.

4.1. NSFD1 Scheme

Following the rules above, we construct NSFD1 scheme for Equation (2) as
U j n + 1 U j n ϕ ( k ) = U j + 1 n 2 U j n + U j 1 n [ ψ ( h ) ] 2 α U j n + 1 U j n U j 1 n ψ ( h ) + β ( 1 + γ ) 2 ( U j n ) 2 U j n U j n + 1 β γ U j n + 1 β U j n + 1 ( U j n ) 2 .
The denominator functions are defined as
ϕ ( k ) = e β k 1 β , ψ ( h ) = e β h 1 β .
We have approximated the linear and non-linear terms as follows:
u 3 ( x j , t n ) U j n + 1 ( U j n ) 2 , u 2 ( x j , t n ) 2 ( U j n ) 2 U j n U j n + 1 , u ( x j , t n ) U j n + 1 .
Equation (8) above is an explicit scheme. By making U j n + 1 the subject and noting that R = ϕ ( k ) [ ψ ( h ) ] 2 and r = ϕ ( k ) ψ ( h ) , we have
U j n + 1 = ( 1 2 R ) U j n + R ( U j + 1 n + U j 1 n ) + 2 ϕ ( k ) β ( 1 + γ ) ( U j n ) 2 1 + α r ( U j n U j 1 n ) + ϕ ( k ) β γ + Φ ( k ) β ( 1 + γ ) U j n + ϕ ( k ) β ( U j n ) 2 .
We then proceed to check positivity and boundedness of the scheme given by Equation (10).
Theorem 1
(Dynamical Consistency). If 1 2 R 0 , the numerical solution from NSFD1 satisfies
0 U j n γ , 0 U j n + 1 γ ,
for all considered values of n and j.
Proof. 
For positivity, we require 1 2 R 0 . Substituting the appropriate form for R, we obtain
e β k 1 β β e β h 1 2 1 2 ,
which gives
k 1 β ln 1 + ( e β h 1 ) 2 2 β .
On substituting h = 0.1 and evaluating for some different values of β , we obtain
(a)
k 5.251 × 10 3 for β = 0.5 .
(b)
k 5.515 × 10 3 for β = 1.0 .
(c)
k 6.090 × 10 3 for β = 2.0 .
(d)
k 1.377 × 10 2 for β = 10.0 .
For boundedness of NSFD1, we assume 0 U j n γ for all considered values of n and j. Therefore,
( U j n + 1 γ ) 1 + α r ( U j n U j 1 n ) + ϕ ( k ) β γ + ϕ ( k ) β ( 1 + γ ) U j n + ϕ ( k ) β ( U j n ) 2 = ( 1 2 R ) U j n + R ( U j + 1 n + U j 1 n ) + 2 ϕ ( k ) β ( 1 + γ ) ( U j n ) 2 γ α r γ ( U j n U j 1 n ) ϕ ( k ) β γ 2 ϕ ( k ) β γ ( 1 + γ ) U j n ϕ ( k ) β γ ( U j n ) 2 2 β ϕ ( k ) β ( U j n ) 2 + 2 ϕ ( k ) β γ ( U j n ) 2 α r ( U j n U j 1 n ) ϕ ( k ) β γ 2 ϕ ( k ) β γ ( U j n ) ϕ ( k ) β γ 2 U j n ϕ ( k ) β γ ( U j n ) 2 = ϕ ( k ) β γ ( U j n ) 2 + ϕ ( k ) β 2 ( U j n ) 2 γ 2 γ U j n α r γ ( U j n U j 1 n ) ϕ ( k ) β γ 2 U j n β ϕ ( k ) γ U j n ( U j n γ ) α r γ ( U j n U j 1 n ) 0 .
Using Equation (13), we have U j n + 1 γ . Thus, the NSFD1 scheme is bounded, provided the values from the initial conditions are non-negative and bounded. □

4.2. NSFD2 Scheme

Using other non-local representations for the linear and non-linear terms as
u 3 ( x j , t n ) 2 U j n + 1 ( U j n ) 2 ( U j n ) 3 , u ( x j , t n ) U j n + 1 ,
we propose the following scheme to discretise Equation (2):
U j n + 1 U j n ϕ ( k ) = U j + 1 n 2 U j n + U j 1 n [ ψ ( h ) ] 2 α U j n + 1 U j n U j 1 n ψ ( h ) + β ( 1 + γ ) ( U j n ) 2 β γ U j n + 1 2 β U j n + 1 ( U j n ) 2 + β ( U j n ) 3 .
By making U j n + 1 the subject of the equation, we obtain
U j n + 1 = ( 1 2 R ) U j n + R ( U j + 1 n + U j 1 n ) + β ϕ ( k ) [ ( 1 + γ ) ( U j n ) 2 + ( U j n ) 3 ] 1 + α r ( U j n U j 1 n ) + ϕ ( k ) β γ + 2 ϕ ( k ) β ( U j n ) 2 ,
We then proceed to check positivity and boundedness of NSFD2 scheme.
Theorem 2
(Dynamical Consistency). If 1 2 R 0 , the numerical solution from NSFD2 satisfies
0 U j n γ , 0 U j n + 1 γ ,
for all relevant values of n and j.
Proof. 
For positivity, we require 1 2 R 0 . We have the same condition as for NSFD1 scheme. For boundedness, we note that 0 U j n γ , for all values of n and j. We have
( U j n + 1 γ ) 1 + α r ( U j n U j 1 n ) + ϕ ( k ) β γ + 2 ϕ ( k ) β ( U j n ) 2 = ( 1 2 R ) U j n + R ( U j + 1 n + U j 1 n ) + β ϕ ( k ) ( 1 + γ ) ( U j n ) 2 + ( U j n ) 3 γ γ r α ( U j n U j 1 n ) γ 2 ϕ ( k ) β 2 ϕ ( k ) β γ ( U j n ) 2 = β ϕ ( k ) ( U j n ) 2 + β ϕ ( k ) γ ( U j n ) 2 + β ϕ ( k ) ( U j n ) 3 α r γ ( U j n U j 1 n ) β γ 2 ϕ ( k ) 2 ϕ ( k ) β γ ( U j n ) 2 = β ϕ ( k ) ( U j n ) 2 + β ϕ ( k ) ( U j n ) 3 α r γ ( U j n U j 1 n ) β γ 2 ϕ ( k ) ϕ ( k ) β γ ( U j n ) 2 = β ϕ ( k ) ( U j n ) 2 U j n γ α r γ ( U j n U j 1 n ) + β ϕ ( k ) ( U j n ) 2 γ 2 = β ϕ ( k ) ( U j n ) 2 + β ϕ ( k ) ( U j n + γ ) ( U j n γ ) α r γ ( U j n U j 1 n ) 0 .
Hence, U j n + 1 γ . Hence, we conclude that NSFD2 scheme is bounded. The positivity conditions guarantee non-negativity of the solution. □

5. Exponential Finite Difference Methods

In the construction of exponential finite difference schemes, we use the following difference operators:
δ t U j n = U j n + 1 U j n k ,
δ x 1 U j n = U j + 1 n U j 1 n 2 h ,
δ x 2 U j n = U j + 1 n 2 U j n + U j 1 n h 2 ,
δ x 1 U j n + 1 = U j + 1 n + 1 U j 1 n + 1 2 h ,
δ x 2 U j n + 1 = U j + 1 n + 1 2 U j n + 1 + U j 1 n + 1 h 2 ,
for each j 1 , 2 , , N and n 1 , 2 , . In addition, we introduce the new discrete operator
Λ U j n = ln U j n + 1 ln U j n k = 1 k ln U j n + 1 U j n .

5.1. Explicit Exponential Finite Difference Method (EEFDM)

An explicit exponential finite difference method was proposed for the generalised Burgers–Huxley equation and numerical solutions for δ = 1 were presented by İnan in [46]. When Equation (2) is rearranged, the following equation is obtained
u t = β u 1 u u γ α u u x + 2 u x 2 .
Dividing by u, we obtain
ln u t = 1 u β u 1 u u γ α u u x + 2 u x 2 ,
and using finite difference approximations for derivatives, we obtain
Λ U j n = 1 U j n β U j n 1 U j n U j n γ α U j n δ x 1 U j n + δ x 2 U j n .
Finally, using Equations (18), (19) and (22), a single expression for EEFDM scheme is
U j n + 1 = U j n exp k U j n β U j n 1 U j n U j n γ α U j n U j + 1 n U j 1 n 2 h + U j + 1 n 2 U j n + U j 1 n h 2 .
To obtain the stability of EEFDM, we consider the corresponding standard finite difference scheme given by
U j n + 1 U j n k = U j + 1 n 2 U j n + U j 1 n h 2 α U j n U j + 1 n U j 1 n 2 h + β ( 1 + γ ) ( U j n ) 2 β γ U j n β ( U j n ) 3 .
which can be rewritten in the form
U j n + 1 = U j n + k U j + 1 n 2 U j n + U j 1 n h 2 k α U j n U j + 1 n U j 1 n 2 h + k β ( 1 + γ ) ( U j n ) 2 k β γ U j n k β ( U j n ) 3 .
We follow the idea of Taha and Ablowitz [59] by using the freezing coefficients method and Von Neumann stability analysis. We obtain the amplification factor as:
ξ = 1 I k α h U m a x sin w + k h 2 ( 2 cos w 2 ) + k β U m a x ( 1 + γ ) k β γ k β U m a x 2 .
Since 0 U γ , it follows that U m a x = γ . On simplification, we obtain
ξ = 1 4 k h 2 sin 2 w 2 2 + k α γ h sin w 2 .
Stability is guaranteed when 0 < ξ 1 for w = [ π , π ] . Table 1 shows as follow:
We fix h = 0.1 and we obtain the 3D plots of | ξ | vs. k vs. ω [ π , π ] . As shown in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9.

5.2. Fully Implicit Exponential Finite Difference Method (FIEFDM)

We rearrange Equation (2) to obtain
u t = 2 u x 2 α u u x + β u 1 u u γ .
Dividing by u gives
ln u t = 1 u 2 u x 2 α u u x + β u 1 u u γ ,
and using finite difference approximations for derivatives, we obtain following equation.
Λ U j n = 1 U j n β U j n + 1 1 U j n + 1 U j n + 1 γ α U j n + 1 δ x 1 U j n + 1 + δ x 2 U j n + 1 .
Finally, using Equations (20)–(33) gives
U j n + 1 = U j n exp ( k U j n [ U j + 1 n + 1 2 U j n + 1 + U j 1 n + 1 h 2 α U j n + 1 U j + 1 n + 1 U j 1 n + 1 2 h + β U j n + 1 ( 1 U j n + 1 ) ( U j n + 1 γ ) ] ) ,
which is valid for values of j lying in the interval 1 j N 1 . Equation (34) is a system of nonlinear difference equations. Let us consider these nonlinear systems of equations in the form
F ( V ) = 0 ,
where F = f 1 , f 2 , , f N 1 T and V = U 1 n + 1 , U 2 n + 1 , , U N 1 n + 1 T .
Newton’s method applied to Equation (35) results in the following iteration:
  • Set V ( 0 ) , an initial guess.
  • For m = 0 , 1 , 2 , until convergence do:
Solve J ( V ( m ) ) Ω ( m ) = F ( V ( m ) ) ;
Set V ( m + 1 ) = V ( m ) + Ω ( m ) where J ( V ( m ) ) is the Jacobian matrix which is evaluated analytically. The solution at the previous time-step is taken as the initial estimate. The Newton’s iteration at each time-step is stopped when
F ( V ( m ) ) 10 5 .
We choose 10 5 as tolerance. It is not easy to choose a tolerance less than 10 5 due to excessive computational time.

6. Numerical Results

We performed the numerical experiment described in Section 3 to check the effectiveness of the two NSFD schemes and two exponential schemes. The accuracy of the proposed methods was measured using the absolute error, relative error, L 1 and L errors.
Absolute Error = u x j , t n U x j , t n ,
Relative Error = u x j , t n U x j , t n u ( x j , t n ) ,
L 1 = h j = 1 N u x j , t n U x j , t n ,
and
L = max u x j , t n U x j , t n .
All simulations were performed with h = 0.1 ; the value of k = 0.005 was carefully chosen to preserve the positivity and boundedness condition and/or the stability condition for the two NSFD schemes and two exponential finite difference methods. Comparison of the numerical solutions, exact solution, absolute error, relative error, L 1 and L errors, CPU time for various values of α , β and γ are made in Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15, Table 16, Table 17, Table 18, Table 19, Table 20, Table 21, Table 22, Table 23, Table 24, Table 25, Table 26, Table 27, Table 28, Table 29, Table 30, Table 31, Table 32, Table 33, Table 34, Table 35, Table 36, Table 37 and Table 38 and in Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27, Figure 28 and Figure 29. We also present results using implicit exponential finite difference scheme using α = 0.5 , β = 0.5 , γ = 0.001 and α = 0.5 , β = 10.0 , γ = 0.5 using temporal step size k = 0.1 and spatial step size h = 0.1 .
  • In Case 1, parameters are chosen as α = β = 0.5 and γ = 0.001 . Table 2, Table 3, Table 4 and Table 5 display the results for Case 1 at t = 1.0 and t = 10.0 .
  • In Case 2, parameters are chosen as α = 0.5 , β = 2.0 ( β > α ) and γ = 0.001 . Table 6, Table 7, Table 8 and Table 9 display the results for Case 2 at t = 1.0 and t = 10.0 .
  • In Case 3, parameters are chosen as α = 0.5 ( β > > α ) , β = 10.0 and γ = 0.001 . Table 10, Table 11, Table 12 and Table 13 display the results for Case 3 at t = 1.0 and t = 10.0 .
  • In Case 4, parameters are chosen as α = 2.0 ( α > β ) , β = 0.5 and γ = 0.001 . Table 14, Table 15, Table 16 and Table 17 display the results for Case 4 at t = 1.0 and t = 10.0 .
  • In Case 5, parameters are chosen as α = β = 0.5 and γ = 0.5 . Table 18, Table 19, Table 20 and Table 21 display the results for Case 5 at t = 1.0 and t = 10.0 .
  • In Case 6, parameters are chosen as α = 0.5 , β = 2.0 ( β > α ) and γ = 0.5 . Table 22, Table 23, Table 24 and Table 25 display the results for Case 6 at t = 1.0 and t = 10.0 .
  • In Case 7, parameters are chosen as α = 0.5 ( β > > α ) , β = 10.0 and γ = 0.5 . Table 26, Table 27, Table 28 and Table 29 display the results for Case 7 at t = 1.0 and t = 10.0 .
  • In Case 8, parameters are chosen as α = 2.0 ( α > β ) , β = 0.5 and γ = 0.5 . Table 30, Table 31, Table 32 and Table 33 display the results for Case 8 at t = 1.0 and t = 10.0 .
  • In Case 9, we compare the absolute errors obtained by our four methods with other methods in [13,14]. Parameters are chosen as α = β = 1.0 and γ = 0.001 . Table 34 display the results for Case 9 at t = 0.05 , t = 0.1 and t = 1.0 .
  • In Cases 10 and 11, we compare the accuracy of the fully implicit exponential scheme by chosen parameters as α = 0.5 , β = 0.5 ( β = α ) , γ = 0.001 and k = 0.1 and α = 0.5 , β = 10.0 ( β > > α ) , γ = 0.5 and k = 0.1 , respectively. Table 35, Table 36, Table 37 and Table 38 display the results for Cases 10 and 11 at t = 1.0 and t = 10.0 . We note that, for Cases 1–9, h = 0.1 and k = 0.005 .
Case 1: α = β = 0.5 and γ = 0.001 .
Case 2: α = 0.5 , β = 2.0 ( β > α ) and γ = 0.001
Case 3: α = 0.5 , β = 10.0 ( β > > α ) , and γ = 0.001 . (singularly perturbed)
Case 4: α = 2.0 ( α > β ) , β = 0.5 and γ = 0.001 .
Case 5: α = β = 0.5 and γ = 0.5
Case 6: α = 0.5 , β = 2.0 ( β > α ) and γ = 0.5 .
Case 7: α = 0.5 , β = 10.0 ( β > > α ) , and γ = 0.5 . (singularly perturbed)
Case 8: α = 2.0 ( α > β ) , β = 0.5 and γ = 0.5
Case 9: α = 1.0 ( α = β ) , β = 1.0 and γ = 0.001
In cases 10 and 11, we test the effectiveness of the fully implicit exponential finite difference scheme (FIEFDM) by taking a less refined time step k = 0.1 for computations.
Case 10: α = β = 0.5 ( β = α ) , γ = 0.001 and k = 0.1
Case 11: α = 0.5 , β = 10.0 ( β > > α ) , γ = 0.5 and k = 0.1
We next present plots of the initial, exact and numerical solutions for the eleven cases in Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27, Figure 28 and Figure 29.

7. Conclusions

In this paper, we have obtained numerical solutions to the Burgers–Huxley equation with specified initial and boundary conditions using two novel non-standard finite difference and two exponential finite difference schemes. These types of schemes are very recent for such partial differential equations. The positivity condition of the two NSFD schemes are dependent on k , β and h. We considered eleven test cases, which made use of two values of γ = 0.001 and 0.5. The last two test cases are for FIEFDM only, where we used a larger time step size.
For γ = 0.001 , we observe that, for the three different regimes ( α = β , α < β , α > β , ) , all the four schemes are very efficient. The relative error is of order 10 5 . The CPU time for the FIEFDM is slightly larger than the other three explicit schemes. For γ = 0.5 , we observe that, for all the three different regimes, the four schemes perform quite well and the relative error is of order 10 2 , 10 3 and 10 4 . We also compared our four methods with adomian decomposition and variational iteration method for the case α = β = 1.0 , γ = 0.001 and our methods slightly performed better.
In reference to Figure 14 and Figure 15, we can observe that EEFDM amd FIEFDM are much better than NSFD1 and NSFD2 for the case α = 0.5 , β = 10.0 , γ = 0.001 at time t = 1.0 .
If we refer to Figure 22 and Figure 23, we can observe that EEFDM and FIEFDM are much better than NSFD1, NSFD2 for the case α = 0.5 , β = 10.0 , γ = 0.5 at time t = 1.0 .
There is not much difference in performance of FIEFDM at k = 0.1 as compared to k = 0.005 , hence a larger time step size can be used.
We would like to extend this study to solve generalised Burgers–Huxley equation using a more challenging numerical experiment, possibly one which consists of a shock-like profile.

Author Contributions

These authors contributed equally to this work.

Funding

A. R. Appadu is grateful to Nelson Mandela University for seed funding which enabled him to present his work at ICNAAM 2019. Yusuf O. Tijani is grateful to Nelson Mandela University for providing some funding to cover first year registration fees at the university.

Acknowledgments

The authors would like to thank the three anonymous reviewers for the insightful comments and suggestions, which have enabled the authors to considerably improve the paper.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:
NSFDNonstandard Finite Difference
EEFDMExplicit Exponential Finite Difference Method
FIEFDMFully Implicit Exponential Finite Difference Method
MATLABMatrix Laboratory

References

  1. Albowitz, M.J.; Clarkson, P.A. Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform; University Press: Cambridge, UK, 1990. [Google Scholar]
  2. Lawrence, C.E. Partial Differential Equation; Graduate Studies in Mathematics; American Mathematical Society: Berkeley, CA, USA, 1997; pp. 1–662. [Google Scholar]
  3. Estevez, P.G. Non-classical symmetries and the singular manifold method: The Burgers and the Burgers–Huxley equations. J. Phys. A Math. Gen. 1994, 27, 2113–2127. [Google Scholar] [CrossRef]
  4. Estevez, P.G.; Gordoa, P.R. Nonclassical symmetries and the singular manifold method: Theory and six examples. Stud. Appl. Math. 1995, 95, 73–113. [Google Scholar] [CrossRef]
  5. Chou, M.H.; Lin, Y.T. Exotic dynamic behavior of the forced FitzHugh-Nagumo equations. Comput. Math. Appl. 1996, 32, 109–124. [Google Scholar] [CrossRef] [Green Version]
  6. Olver, P. Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics; Springer: New York, NY, USA, 1994; Volume 107. [Google Scholar]
  7. Zhao, X.; Tang, D.; Wang, L. New soliton-like solutions for KdV equation with variable coefficient. Phys. Lett. A 2005, 346, 228–291. [Google Scholar] [CrossRef]
  8. Yomba, E. The extended Fan’s sub-equation method and its applications to KdV-MkdV, BKK and variant Boussinesq equations. Phys. Lett. A 2005, 336, 463–476. [Google Scholar] [CrossRef]
  9. Wazwaz, A.M. Solutions and periodic solutions for the fifth-order KdV equation. Appl. Math. Lett. 2006, 19, 1162–1167. [Google Scholar] [CrossRef]
  10. Wang, M. Exact solutions for a compound KdV-Burgers equation. Phys. Lett. A 1996, 213, 79–287. [Google Scholar] [CrossRef]
  11. Biswas, A. Anjan S1-soliton solution of the K (m, n) equation with generalized evolution. Phys. Lett. A 2008, 25, 4601–4602. [Google Scholar] [CrossRef]
  12. Yu, X.; Gao, Y.T.; Sun, Z.Y.; Liu, Y. N-soliton solutions, Backlund transformation and Lax pair for a generalized variable-coefficient fifth-order Korteweg–de Vries equation. Phys. Scr. 2010, 81, 045402. [Google Scholar] [CrossRef]
  13. Ismail, H.N.A.; Raslan, K.; Abd Rabboh, A.A. Adomian decomposition method for burgers—Huxley and burgers—Fisher equations. Appl. Math. Comput. 2004, 1, 291–301. [Google Scholar]
  14. Batiha, M.S.M.; Noorani, B.; Hashim, I. Application of variational iteration method to the generalized burgers–huxley equation. Chaos Solitons Fractals 2008, 36, 660–663. [Google Scholar] [CrossRef]
  15. Biazar, J.; Mohammadi, F. Application of differential transform method to the generalized burgers–huxley equation. Appl. Appl. Math. 2010, 5, 1726–1750. [Google Scholar]
  16. Yaghouti, M.R.; Zabihi, A. Application of Laplace decomposition method for burgers–huxley and burgers–fisher equations. J. Math. Model. 2013, 1, 41–67. [Google Scholar]
  17. Molabahramia, A.; Khan, F. The homotopy analysis method to solve the burgers–huxley equation. Nonlinear Anal. Real World Appl. 2009, 10, 589–600. [Google Scholar] [CrossRef]
  18. Mittal, R.C.; Jiwari, R. A numerical study of burger–huxley equation by differential quadrature method. J. Appl. Math. Mech. 2009, 5, 1–9. [Google Scholar]
  19. Javidi, M. A numerical solution of the generalized burgers–huxley equation by spectral collocation method. Appl. Math. Comput. 2006, 178, 338–344. [Google Scholar]
  20. Tomasiello, S. Numerical solutions of the burgers–huxley equation by the IDQ method. Int. J. Comput. Math. 2010, 87, 129–140. [Google Scholar] [CrossRef]
  21. Ray, S.S.; Gupta, A. Comparative analysis of variational iteration method and Haar wavelet method for the numerical solutions of Burgers–Huxley and Huxley equations. J. Math. Chem. 2014, 52, 1060–1080. [Google Scholar]
  22. Ray, S.S.; Gupta, A. Application of novel schemes based on Haar wavelet collocation method for numerical simulations of Burger and Boussinesq–Burger equations. Appl. Math. Inf. Sci. 2016, 10, 1513–1524. [Google Scholar] [CrossRef]
  23. Chen, Z.; Gumel, A.B.; Micken, R.E. Nonstandard discretizations of the generalized nagumo reaction-diffusion equation. Numer. Methods Partial Differ. Equ. 2003, 19, 363–379. [Google Scholar] [CrossRef]
  24. Kyrychko, Y.N.; Bartuccelli, M.; Blysuss, K.B. Persistence of travelling wave solutions of a fourth order diffusion system. J. Comput. Appl. Math. 2005, 176, 433–443. [Google Scholar] [CrossRef]
  25. Anguelov, R.; Lubuma, J.M.S. Contributions to the Mathematics of the Nonstandard Finite Difference Method and Applications. Numer. Methods Partial Differ. Equ. 2001, 17, 518–543. [Google Scholar] [CrossRef]
  26. Mickens, R.E. Application of Nonstandard Finite Difference Scheme; World Scientific: Singapore, 2000; pp. 1–261. [Google Scholar]
  27. Mickens, R.E.; Gumel, A.B. Construction and analysis of a non-standard finite difference scheme for the Burgers-Fisher equation. J. Sound Vib. 2002, 4, 791–797. [Google Scholar] [CrossRef]
  28. Zhang, L.; Wang, L.; Xiaohu, D. Exact finite difference scheme and nonstandard finite difference scheme for burgers and burgers-fisher equations. J. Appl. Math. 2014, 597926. [Google Scholar] [CrossRef]
  29. Agbavon, K.M.; Appadu, A.R. Construction and analysis of some nonstandard finite difference methods for the Fitzhugh-Nagumo equation. Numer. Methods Partial Differ. Equ. In Review.
  30. Appadu, A.R.; Lubuma, J.M.S.; Mphephu, N. Computational study of three numerical methods for some linear and nonlinear advection-diffusion-reaction problems. Prog. Comput. Fluid Dyn. 2017, 17, 114–129. [Google Scholar] [CrossRef]
  31. Appadu, A.R. Numerical solution of the 1D advection-diffusion equation using standard and nonstandard finite difference schemes. J. Appl. Math. 2013, 2013, 734374. [Google Scholar] [CrossRef]
  32. Agabvon, K.M.; Appadu, A.R.; Khumalo, M. On the numerical solution of fishers equation with coefficient of diffusion term much smaller than coefficient of reaction term. Adv. Differ. Equ. 2019, 146, 33. [Google Scholar] [CrossRef]
  33. Chapwanya, M.; Appadu, A.R.; Jejeniwa, M.O.; Lubuma, J.M.S. An explicit nonstandard finite difference scheme for the Fitzhugh-Nagumo equations. Int. J. Comput. Math. 2019, 96, 1993–2009. [Google Scholar] [CrossRef]
  34. Mickens, R.E. Nonstandard finite difference schemes for reaction-diffusion equations. Numer. Methods Partial Differ. Equ. 1999, 15, 201–215. [Google Scholar] [CrossRef]
  35. Jordan, P.M. A nonstandard finite difference scheme for nonlinear heat transfer in a thin finite rod. J. Differ. Equ. Appl. 2003, 9, 1015–1021. [Google Scholar] [CrossRef]
  36. Aderogba, A.A.; Chapwanya, M. Positive and bounded nonstandard finite difference scheme for the hodgkin–huxley equations. Jpn. J. Ind. Appl. Math. 2018, 35, 773–785. [Google Scholar] [CrossRef]
  37. Zibaei, M.; Zeinadini, S.; Namjoo, M. Numerical solutions of Burgers–Huxley equation by exact finite difference and NSFD schemes. J. Differ. Equ. Appl. 2016, 22, 1098–1113. [Google Scholar] [CrossRef]
  38. Oluwaseye, A.; Talitha, W. Nonstandard finite difference scheme for a Tacoma Narrows Bridge model. Appl. Math. Model. 2018, 62, 223–236. [Google Scholar]
  39. Dai, W. Nonstandard finite difference schemes for solving nonlinear micro heat transport equations in double-layered metal thin films exposed to ultrashort pulsed lasers. In Advances in the Applications of Nonstandard Finite Difference Schemes; Mickens, R.E., Ed.; World Scientific: Hackensack, NJ, USA, 2005; pp. 191–248. [Google Scholar]
  40. Díaz-Rodríguez, M.; González-Parra, G.; Arenas, A.J. Nonstandard numerical schemes for modeling a 2-DOF serial robot with rotational spring-damper-actuators. Int. J. Numer. Methods Biomed. Eng. 2011, 27, 1211–1224. [Google Scholar] [CrossRef]
  41. Inan, B.; Bahadir, A.R. A numerical solution of the Burgers equation using a Crank–Nicolson exponential finite difference method. J. Math. Comput. Sci. 2014, 4, 849–860. [Google Scholar]
  42. Macías-Díaz, J.E. A modified exponential method that preserves structural properties of the solutions of the Burgers-Huxley equation. Int. J. Comput. Math. 2017. [Google Scholar] [CrossRef]
  43. Bhattacharya, M.C. An explicit conditionally stable finite difference equation for heat conduction problems. Int. J. Numer. Method Eng. 1985, 21, 239–265. [Google Scholar] [CrossRef]
  44. Inan, B.; Bahadir, A.R. Numerical solution of the one-dimensional burgers equation: Implicit and fully implicit exponential finite difference methods. Pramana J. Phys. 2018, 81, 547–556. [Google Scholar] [CrossRef]
  45. Bahadir, A.R. Exponential finite-difference method applied to korteweg-de vries equation for small times. Appl. Math. Comput. 2005, 160, 675–682. [Google Scholar]
  46. İnan, B. Finite difference methods for the generalized huxley and burgers-huxley equations. Kuwait J. Sci. 2017, 44, 20–27. [Google Scholar]
  47. İnan, B. An exponential finite difference method based on Padé approximation. Celal Bayar Univ. Sci. 2017, 13, 71–80. [Google Scholar] [CrossRef]
  48. Macías-Díaz, J.E.; İnan, B. Numerical efficiency of some exponential methods for an advection-diffusion equation. Int. J. Comput. Math. 2019, 96, 1005–1029. [Google Scholar] [CrossRef]
  49. Appadu, A.R.; İnan, B.; Tijani, Y.O. Comparison of some numerical methods for the Burgers-Huxley equation. To appear in AIP Conference Proceedings.
  50. Griffiths, W.G.; Schiesser, E.W. Traveling Wave Analysis of Partial Differential Equation: Numerical and Analytical Methods with MATLAB and Maple; Academic Press: Cambridge, MA, USA, 2011; p. 461. ISBN 9780123846525. [Google Scholar]
  51. Deng, X. Travelling wave solutions for the generalized Burgers–Huxley equation. Appl. Math. Comput. 2008, 204, 733–737. [Google Scholar] [CrossRef]
  52. Burgers, J.M. A mathematical model illustrating the theory of turbulence. In Advances in Applied Mechanics; Academic Press: New York, NY, USA, 1948; Volume 1, pp. 171–199. [Google Scholar]
  53. Nourazar, S.S.; Soori, M.; Nazari-Golshan, A. On the Exact Solution of Newell-Whitehead-Segel Equation Using the Homotopy Perturbation Method. Aust. J. Basic Appl. Sci. 2011, 5, 1400–1411. [Google Scholar]
  54. Wang, Z.S.; Zhuo, X.Y.; Lu, Y.K. Solitary wave solutions of the generalised Burgers-Huxley equation. J. Phys. A Math. Gen. 1990, 23, 271–274. [Google Scholar] [CrossRef]
  55. Mickens, R.E. Pitfalls in the numerical integration of differential equations. In Analytical Techniques for Material Characterization; Collins, W.E., Chowdari, B.V.R., Radhakrishna, S., Eds.; World Scientific: Singapore, 1987; Volume 17, pp. 123–143. [Google Scholar]
  56. Mickens, R.E.; Smith, A. Finite-Difference Model of Ordinary Differential Equations: Influence of Denominator Function. J. Franklin Inst. 1990, 327, 143–149. [Google Scholar] [CrossRef]
  57. Mickens, R.E. Lie methods in mathematical modeling: Difference equation models of differential equations. Math. Comput. Model. 1988, 11, 528–530. [Google Scholar] [CrossRef]
  58. Hilderband, F.B. Finite-Difference Equations and Simulations; Prentice-Hall: Englewood Cliffs, NJ, USA, 1968; p. 338. [Google Scholar]
  59. Taha, T.; Ablowitz, M. Analytical and numerical aspects of certain nonlinear evolution equations III. Numerical Korteweg-de Vries equation. J. Comput. Phys. 1984, 2, 231–235. [Google Scholar] [CrossRef]
Figure 1. Plot of | ξ | vs. k vs. ω for α = 0.5 , β = 0.5 , γ = 0.001 .
Figure 1. Plot of | ξ | vs. k vs. ω for α = 0.5 , β = 0.5 , γ = 0.001 .
Symmetry 11 01333 g001
Figure 2. Plot of | ξ | vs. k vs. ω for α = 0.5 , β = 2.0 , γ = 0.001 .
Figure 2. Plot of | ξ | vs. k vs. ω for α = 0.5 , β = 2.0 , γ = 0.001 .
Symmetry 11 01333 g002
Figure 3. Plot of | ξ | vs. k vs. ω for α = 0.5 , β = 10.0 , γ = 0.001 .
Figure 3. Plot of | ξ | vs. k vs. ω for α = 0.5 , β = 10.0 , γ = 0.001 .
Symmetry 11 01333 g003
Figure 4. Plot of | ξ | vs. k vs. ω for α = 2.0 , β = 0.5 , γ = 0.001 .
Figure 4. Plot of | ξ | vs. k vs. ω for α = 2.0 , β = 0.5 , γ = 0.001 .
Symmetry 11 01333 g004
Figure 5. Plot of | ξ | vs. k vs. ω for α = 0.5 , β = 0.5 , γ = 0.5 .
Figure 5. Plot of | ξ | vs. k vs. ω for α = 0.5 , β = 0.5 , γ = 0.5 .
Symmetry 11 01333 g005
Figure 6. Plot of | ξ | vs. k vs. ω for α = 0.5 , β = 2.0 , γ = 0.5 .
Figure 6. Plot of | ξ | vs. k vs. ω for α = 0.5 , β = 2.0 , γ = 0.5 .
Symmetry 11 01333 g006
Figure 7. Plot of | ξ | vs. k vs. ω for α = 0.5 , β = 10.0 , γ = 0.5 .
Figure 7. Plot of | ξ | vs. k vs. ω for α = 0.5 , β = 10.0 , γ = 0.5 .
Symmetry 11 01333 g007
Figure 8. Plot of | ξ | vs. k vs. ω for α = 2.0 , β = 0.5 , γ = 0.5 .
Figure 8. Plot of | ξ | vs. k vs. ω for α = 2.0 , β = 0.5 , γ = 0.5 .
Symmetry 11 01333 g008
Figure 9. Plot of | ξ | vs. k vs. ω for α = 1.0 , β = 1.0 , γ = 0.001 .
Figure 9. Plot of | ξ | vs. k vs. ω for α = 1.0 , β = 1.0 , γ = 0.001 .
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Figure 10. Comparison between Initial, Exact, NSFD1 and NSFD2 profiles for Case 1 at t = 1.0 .
Figure 10. Comparison between Initial, Exact, NSFD1 and NSFD2 profiles for Case 1 at t = 1.0 .
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Figure 11. Comparison between Initial, Exact, EEFDM and FIEFDM profiles for Case 1 at t = 1.0 .
Figure 11. Comparison between Initial, Exact, EEFDM and FIEFDM profiles for Case 1 at t = 1.0 .
Symmetry 11 01333 g011
Figure 12. Comparison between Initial, Exact, NSFD1 and NSFD2 profiles for Case 2 at t = 1.0 .
Figure 12. Comparison between Initial, Exact, NSFD1 and NSFD2 profiles for Case 2 at t = 1.0 .
Symmetry 11 01333 g012
Figure 13. Comparison between Initial, Exact, EEFDM and FIEFDM profiles for Case 2 at t = 1.0 .
Figure 13. Comparison between Initial, Exact, EEFDM and FIEFDM profiles for Case 2 at t = 1.0 .
Symmetry 11 01333 g013
Figure 14. Comparison between Initial, Exact, NSFD1 and NSFD2 profiles for Case 3 at t = 1.0 .
Figure 14. Comparison between Initial, Exact, NSFD1 and NSFD2 profiles for Case 3 at t = 1.0 .
Symmetry 11 01333 g014
Figure 15. Comparison between Initial, Exact, EEFDM and FIEFDM profiles for Case 3 at t = 1.0 .
Figure 15. Comparison between Initial, Exact, EEFDM and FIEFDM profiles for Case 3 at t = 1.0 .
Symmetry 11 01333 g015
Figure 16. Comparison between Initial, Exact, NSFD1 and NSFD2 profiles for Case 4 at t = 1.0 .
Figure 16. Comparison between Initial, Exact, NSFD1 and NSFD2 profiles for Case 4 at t = 1.0 .
Symmetry 11 01333 g016
Figure 17. Comparison between Initial, Exact, EEFDM and FIEFDM profiles for Case 4 at t = 1.0 .
Figure 17. Comparison between Initial, Exact, EEFDM and FIEFDM profiles for Case 4 at t = 1.0 .
Symmetry 11 01333 g017
Figure 18. Comparison between Initial, Exact, NSFD1 and NSFD2 profiles for Case 5 at t = 1.0 .
Figure 18. Comparison between Initial, Exact, NSFD1 and NSFD2 profiles for Case 5 at t = 1.0 .
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Figure 19. Comparison between Initial, Exact, EEFDM and FIEFDM profiles for Case 5 at t = 1.0 .
Figure 19. Comparison between Initial, Exact, EEFDM and FIEFDM profiles for Case 5 at t = 1.0 .
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Figure 20. Comparison between Initial, Exact, NSFD1 and NSFD2 profiles for Case 6 at t = 1.0 .
Figure 20. Comparison between Initial, Exact, NSFD1 and NSFD2 profiles for Case 6 at t = 1.0 .
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Figure 21. Comparison between Initial, Exact, EEFDM and FIEFDM profiles for Case 6 at t = 1.0 .
Figure 21. Comparison between Initial, Exact, EEFDM and FIEFDM profiles for Case 6 at t = 1.0 .
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Figure 22. Comparison between Initial, Exact, NSFD1 and NSFD2 profiles for Case 7 at t = 1.0 .
Figure 22. Comparison between Initial, Exact, NSFD1 and NSFD2 profiles for Case 7 at t = 1.0 .
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Figure 23. Comparison between Initial, Exact, EEFDM and FIEFDM profiles for Case 7 at t = 1.0 .
Figure 23. Comparison between Initial, Exact, EEFDM and FIEFDM profiles for Case 7 at t = 1.0 .
Symmetry 11 01333 g023
Figure 24. Comparison between Initial, Exact, NSFD1 and NSFD2 profiles for Case 8 at t = 1.0 .
Figure 24. Comparison between Initial, Exact, NSFD1 and NSFD2 profiles for Case 8 at t = 1.0 .
Symmetry 11 01333 g024
Figure 25. Comparison between Initial, Exact, EEFDM and FIEFDM profiles for Case 8 at t = 1.0 .
Figure 25. Comparison between Initial, Exact, EEFDM and FIEFDM profiles for Case 8 at t = 1.0 .
Symmetry 11 01333 g025
Figure 26. Comparison between Initial, Exact, NSFD1 and NSFD2 profiles for Case 9 at t = 1.0 .
Figure 26. Comparison between Initial, Exact, NSFD1 and NSFD2 profiles for Case 9 at t = 1.0 .
Symmetry 11 01333 g026
Figure 27. Comparison between Initial, Exact, EEFDM and FIEFDM profiles for Case 9 at t = 1.0 .
Figure 27. Comparison between Initial, Exact, EEFDM and FIEFDM profiles for Case 9 at t = 1.0 .
Symmetry 11 01333 g027
Figure 28. Comparison between Initial, Exact and FIEFDM profiles for Case 10 at t = 1.0 .
Figure 28. Comparison between Initial, Exact and FIEFDM profiles for Case 10 at t = 1.0 .
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Figure 29. Comparison between Initial, Exact and FIEFDM profiles for Case 11 at t = 1.0 .
Figure 29. Comparison between Initial, Exact and FIEFDM profiles for Case 11 at t = 1.0 .
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Table 1. Range of values of k for stability of EEFDM with h = 0.1 .
Table 1. Range of values of k for stability of EEFDM with h = 0.1 .
CasesParameter ValuesCondition for Stability
1 α = 0.5 , β = 0.5 , γ = 0.001 k 0.005
2 α = 0.5 , β = 2.0 , γ = 0.001 k 0.005
3 α = 0.5 , β = 10.0 , γ = 0.001 k 0.005
4 α = 2.0 , β = 0.5 , γ = 0.001 k 0.005
5 α = 0.5 , β = 0.5 , γ = 0.5 k 0.005
6 α = 0.5 , β = 2.0 , γ = 0.5 k 0.005
7 α = 0.5 , β = 10.0 , γ = 0.5 k 0.005
8 α = 2.0 , β = 0.5 , γ = 0.5 k 0.005
9 α = 1.0 , β = 1.0 , γ = 0.001 k 0.005
Table 2. A comparison between the exact and the numerical solutions at some values of x.
Table 2. A comparison between the exact and the numerical solutions at some values of x.
txExactNSFD1NSFD2EEFDMFIEFDM
1 0.1 5.000858 × 10 4 5.000764 × 10 4 5.000764 × 10 4 5.000768 × 10 4 5.000769 × 10 4
0.5 5.001249 × 10 4 5.000985 × 10 4 5.000985 × 10 4 5.000998 × 10 4 5.000998 × 10 4
0.9 5.001640 × 10 4 5.001545 × 10 4 5.001545 × 10 4 5.001549 × 10 4 5.001549 × 10 4
10 0.1 5.007711 × 10 4 5.007616 × 10 4 5.007616 × 10 4 5.007621 × 10 4 5.007621 × 10 4
0.5 5.008102 × 10 4 5.007838 × 10 4 5.007838 × 10 4 5.007851 × 10 4 5.007851 × 10 4
0.9 5.008492 × 10 4 5.008397 × 10 4 5.008397 × 10 4 5.008402 × 10 4 5.008402 × 10 4
Table 3. The absolute errors at some values of x for each of the numerical schemes.
Table 3. The absolute errors at some values of x for each of the numerical schemes.
txNSFD1NSFD2EEFDMFIEFDM
1 0.1 9.509935 × 10 9 9.509930 × 10 9 9.048885 × 10 9 9.048222 × 10 9
0.5 2.641711 × 10 8 2.641710 × 10 8 2.513647 × 10 8 2.513446 × 10 8
0.9 9.510585 × 10 9 9.510580 × 10 9 9.049488 × 10 9 9.048757 × 10 9
10 0.1 9.510514 × 10 9 9.510509 × 10 9 9.049214 × 10 9 9.048760 × 10 9
0.5 2.641900 × 10 8 2.641898 × 10 8 2.513754 × 10 8 2.513624 × 10 8
0.9 9.511164 × 10 9 9.511159 × 10 9 9.049817 × 10 9 9.049302 × 10 9
Table 4. The relative errors at some values of x for each of the numerical schemes.
Table 4. The relative errors at some values of x for each of the numerical schemes.
txNSFD1NSFD2EEFDMFIEFDM
1 0.1 1.901660 × 10 5 1.901659 × 10 5 1.809466 × 10 5 1.809333 × 10 5
0.5 5.282102 × 10 5 5.282100 × 10 5 5.026038 × 10 5 5.025637 × 10 5
0.9 1.901493 × 10 5 1.901492 × 10 5 1.809304 × 10 5 1.809158 × 10 5
10 0.1 1.899174 × 10 5 1.899173 × 10 5 1.807056 × 10 5 1.806967 × 10 5
0.5 5.275251 × 10 5 5.275249 × 10 5 5.019375 × 10 5 5.019115 × 10 5
0.9 1.899007 × 10 5 1.899006 × 10 5 1.806894 × 10 5 1.806792 × 10 5
Table 5. L 1 and L error norms with CPU time taken for the four numerical methods.
Table 5. L 1 and L error norms with CPU time taken for the four numerical methods.
tSchemes L 1 Error L Error CPU Time
1NSFD1 1.743535 × 10 8 2.641711 × 10 8 0.0642
NSFD2 1.743534 × 10 8 2.641710 × 10 8 0.0649
EEFDM 1.659010 × 10 8 2.513647 × 10 8 0.0660
FIEFDM 1.658878 × 10 8 2.513446 × 10 8 0.0683
10NSFD1 1.743654 × 10 8 2.641900 × 10 8 0.2261
NSFD2 1.743653 × 10 8 2.641898 × 10 8 0.2247
EEFDM 1.659078 × 10 8 2.513754 × 10 8 0.2210
FIEFDM 1.658990 × 10 8 2.513624 × 10 8 0.2341
Table 6. A comparison between the exact and the numerical solutions at some values of x.
Table 6. A comparison between the exact and the numerical solutions at some values of x.
txExactNSFD1NSFD2EEFDMFIEFDM
1 0.1 5.004115 × 10 4 5.003626 × 10 4 5.003626 × 10 4 5.003715 × 10 4 5.003715 × 10 4
0.5 5.004997 × 10 4 5.003639 × 10 4 5.003639 × 10 4 5.003886 × 10 4 5.003886 × 10 4
0.9 5.005880 × 10 4 5.005391 × 10 4 5.005391 × 10 4 5.005480 × 10 4 5.005480 × 10 4
10 0.1 5.039160 × 10 4 5.038671 × 10 4 5.038671 × 10 4 5.038760 × 10 4 5.038760 × 10 4
0.5 5.040428 × 10 4 5.038684 × 10 4 5.038684 × 10 4 5.038931 × 10 4 5.038931 × 10 4
0.9 5.040926 × 10 4 5.040436 × 10 4 5.040436 × 10 4 5.040525 × 10 4 5.040525 × 10 4
Table 7. The absolute errors at some values of x for each of the numerical schemes.
Table 7. The absolute errors at some values of x for each of the numerical schemes.
txNSFD1NSFD2EEFDMFIEFDM
1 0.1 4.891509 × 10 8 4.891498 × 10 8 4.001139 × 10 8 4.000776 × 10 8
0.5 1.358748 × 10 7 1.358745 × 10 7 1.111457 × 10 7 1.111355 × 10 7
0.9 4.891869 × 10 8 4.891858 × 10 8 4.001405 × 10 8 4.000958 × 10 8
10 0.1 4.892413 × 10 8 4.892402 × 10 8 4.001053 × 10 8 4.000788 × 10 8
0.5 1.359053 × 10 7 1.359050 × 10 7 1.111440 × 10 7 1.111369 × 10 7
0.9 4.892767 × 10 8 4.892756 × 10 8 4.001314 × 10 8 4.000966 × 10 8
Table 8. The relative errors at some values of x for each of the numerical schemes.
Table 8. The relative errors at some values of x for each of the numerical schemes.
txNSFD1NSFD2EEFDMFIEFDM
1 0.1 9.774973 × 10 5 9.774951 × 10 5 7.995698 × 10 5 7.994972 × 10 5
0.5 2.714782 × 10 4 2.714776 × 10 4 2.220695 × 10 4 2.220490 × 10 4
0.9 9.772245 × 10 5 9.772223 × 10 5 7.993409 × 10 5 7.992517 × 10 5
10 0.1 9.708786 × 10 5 9.708764 × 10 5 7.939920 × 10 5 7.939394 × 10 5
0.5 2.696512 × 10 4 2.696506 × 10 4 2.205220 × 10 4 2.205079 × 10 4
0.9 9.706089 × 10 5 9.706068 × 10 5 7.937658 × 10 5 7.936967 × 10 5
Table 9. L 1 and L error norms with CPU time taken for the four numerical methods.
Table 9. L 1 and L error norms with CPU time taken for the four numerical methods.
tSchemes L 1 Error L Error CPU Time (Sec)
1NSFD1 8.967845 × 10 8 1.358748 × 10 7 0.0641
NSFD2 8.967825 × 10 8 1.358745 × 10 7 0.0643
EEFDM 7.335633 × 10 8 1.111457 × 10 7 0.0645
FIEFDM 7.334929 × 10 8 1.111355 × 10 7 0.0654
10NSFD1 8.969751 × 10 8 1.359053 × 10 7 0.2322
NSFD2 8.969731 × 10 8 1.359053 × 10 7 0.2273
EEFDM 7.335505 × 10 8 1.111440 × 10 7 0.2237
FIEFDM 7.335000 × 10 8 1.111369 × 10 7 0.2321
Table 10. A comparison between the exact and the numerical solutions at some values of x.
Table 10. A comparison between the exact and the numerical solutions at some values of x.
txExactNSFD1NSFD2EEFDMFIEFDM
1 0.1 5.022873 × 10 4 5.016837 × 10 4 5.016837 × 10 4 5.020743 × 10 4 5.020743 × 10 4
0.5 5.024987 × 10 4 5.008301 × 10 4 5.008301 × 10 4 5.019071 × 10 4 5.019071 × 10 4
0.9 5.027102 × 10 4 5.021065 × 10 4 5.021065 × 10 4 5.024972 × 10 4 5.024972 × 10 4
10 0.1 5.223822 × 10 4 5.217618 × 10 4 5.217619 × 10 4 5.221696 × 10 4 5.221696 × 10 4
0.5 5.225932 × 10 4 5.208699 × 10 4 5.208699 × 10 4 5.220027 × 10 4 5.220026 × 10 4
0.9 5.228042 × 10 4 5.221838 × 10 4 5.221838 × 10 4 5.225917 × 10 4 5.225916 × 10 4
Table 11. The absolute errors at some values of x for each of the numerical schemes.
Table 11. The absolute errors at some values of x for each of the numerical schemes.
txNSFD1NSFD2EEFDMFIEFDM
1 0.1 6.036257 × 10 7 6.036201 × 10 7 2.129791 × 10 7 2.129933 × 10 7
0.5 1.668652 × 10 6 1.668637 × 10 6 5.916245 × 10 7 5.917030 × 10 7
0.9 6.036927 × 10 7 6.036870 × 10 7 2.129928 × 10 7 2.129952 × 10 7
10 0.1 6.203521 × 10 7 6.203444 × 10 7 2.125747 × 10 7 2.125934 × 10 7
0.5 1.723336 × 10 6 1.723315 × 10 6 5.905050 × 10 7 5.905967 × 10 7
0.9 6.204109 × 10 7 6.204031 × 10 7 2.125841 × 10 7 2.125916 × 10 7
Table 12. The relative errors at some values of x for each of the numerical schemes.
Table 12. The relative errors at some values of x for each of the numerical schemes.
txNSFD1NSFD2EEFDMFIEFDM
1 0.1 1.201754 × 10 3 1.201743 × 10 3 4.240185 × 10 4 4.240467 × 10 4
0.5 3.320709 × 10 3 3.320679 × 10 3 1.177365 × 10 3 1.177521 × 10 3
0.9 1.200876 × 10 3 1.200865 × 10 3 4.236891 × 10 4 4.236939 × 10 4
10 0.1 1.187545 × 10 3 1.187530 × 10 3 4.069333 × 10 4 4.069691 × 10 4
0.5 3.297662 × 10 3 3.297621 × 10 3 1.129952 × 10 3 1.130127 × 10 3
0.9 1.186698 × 10 3 1.186683 × 10 3 4.066228 × 10 4 4.066371 × 10 4
Table 13. L 1 and L error norms with CPU time taken for the four numerical methods.
Table 13. L 1 and L error norms with CPU time taken for the four numerical methods.
tSchemes L 1 Error L Error CPU Time (Sec)
1NSFD1 1.102963 × 10 6 1.668652 × 10 6 0.0475
NSFD2 1.102954 × 10 6 1.668637 × 10 6 0.0748
EEFDM 3.904728 × 10 7 4.320404 × 10 7 0.1122
FIEFDM 3.905126 × 10 7 4.320876 × 10 7 0.1620
10NSFD1 1.137391 × 10 6 1.723336 × 10 6 0.2023
NSFD2 1.137376 × 10 6 1.723315 × 10 6 0.3251
EEFDM 3.897320 × 10 7 4.312213 × 10 7 0.2729
FIEFDM 3.897805 × 10 7 5.905967 × 10 7 0.3018
Table 14. A comparison between the exact and the numerical solutions at some values of x.
Table 14. A comparison between the exact and the numerical solutions at some values of x.
txExactNSFD1NSFD2EEFDMFIEFDM
1 0.1 5.000267 × 10 4 5.000196 × 10 4 5.000196 × 10 4 5.000200 × 10 4 5.000200 × 10 4
0.5 5.000473 × 10 4 5.000280 × 10 4 5.000280 × 10 4 5.000290 × 10 4 5.000290 × 10 4
0.9 5.000680 × 10 4 5.000611 × 10 4 5.000611 × 10 4 5.000614 × 10 4 5.000614 × 10 4
10 0.1 5.002138 × 10 4 5.002121 × 10 4 5.002121 × 10 4 5.002124 × 10 4 5.002124 × 10 4
0.5 5.002397 × 10 4 5.002205 × 10 4 5.002205 × 10 4 5.002214 × 10 4 5.002214 × 10 4
0.9 5.002604 × 10 4 5.002535 × 10 4 5.002535 × 10 4 5.002539 × 10 4 5.002538 × 10 4
Table 15. The absolute errors at some values of x for each of the numerical schemes.
Table 15. The absolute errors at some values of x for each of the numerical schemes.
txNSFD1NSFD2EEFDMFIEFDM
1 0.1 6.922896 × 10 9 6.922895 × 10 9 6.585668 × 10 9 6.585338 × 10 9
0.5 1.923270 × 10 8 1.923269 × 10 8 1.829585 × 10 8 1.829491 × 10 8
0.9 6.924789 × 10 9 6.924788 × 10 9 6.587425 × 10 9 6.587092 × 10 9
10 0.1 6.923333 × 10 9 6.923332 × 10 9 6.585922 × 10 9 6.585749 × 10 9
0.5 1.923411 × 10 8 1.923411 × 10 8 1.829667 × 10 8 1.829624 × 10 8
0.9 6.925227 × 10 9 6.925226 × 10 9 6.587680 × 10 9 6.587504 × 10 9
Table 16. The relative errors at some values of x for each of the numerical schemes.
Table 16. The relative errors at some values of x for each of the numerical schemes.
txNSFD1NSFD2EEFDMFIEFDM
1 0.1 1.384506 × 10 5 1.384505 × 10 5 1.317064 × 10 5 1.316998 × 10 5
0.5 3.846175 × 10 5 3.846175 × 10 5 3.658823 × 10 5 3.658636 × 10 5
0.9 1.384770 × 10 5 1.384769 × 10 5 1.317306 × 10 5 1.317239 × 10 5
10 0.1 1.384060 × 10 5 1.384060 × 10 5 1.316608 × 10 5 1.316573 × 10 5
0.5 3.844979 × 10 5 3.844978 × 10 5 3.657580 × 10 5 3.657494 × 10 5
0.9 1.384324 × 10 5 1.384324 × 10 5 1.316850 × 10 5 1.316815 × 10 5
Table 17. L 1 and L error norms with CPU time taken for the four numerical methods.
Table 17. L 1 and L error norms with CPU time taken for the four numerical methods.
tSchemes L 1 Error L Error CPU Time (Sec)
1NSFD1 1.269362 × 10 8 1.923270 × 10 8 0.0654
NSFD2 1.269362 × 10 8 1.923269 × 10 8 0.0677
EEFDM 1.207528 × 10 8 1.829585 × 10 8 0.0677
FIEFDM 1.207467 × 10 8 1.829491 × 10 8 0.0688
10NSFD1 1.269451 × 10 8 1.923411 × 10 8 0.2381
NSFD2 1.269451 × 10 8 1.923411 × 10 8 0.2354
EEFDM 1.207580 × 10 8 1.829667 × 10 8 0.2242
FIEFDM 1.207551 × 10 8 1.829624 × 10 8 0.2372
Table 18. A comparison between the exact and the numerical solutions at some values of x.
Table 18. A comparison between the exact and the numerical solutions at some values of x.
txExactNSFD1NSFD2EEFDMFIEFDM
1 0.1 2.636642 × 10 1 2.619319 × 10 1 2.619323 × 10 1 2.620016 × 10 1 2.620423 × 10 1
0.5 2.733691 × 10 1 2.684776 × 10 1 2.684788 × 10 1 2.686755 × 10 1 2.688005 × 10 1
0.9 2.830035 × 10 1 2.812149 × 10 1 2.812153 × 10 1 2.812875 × 10 1 2.813362 × 10 1
10 0.1 3.573732 × 10 1 3.559870 × 10 1 3.559874 × 10 1 3.560367 × 10 1 3.560550 × 10 1
0.5 3.651974 × 10 1 3.612915 × 10 1 3.612927 × 10 1 3.614329 × 10 1 3.614903 × 10 1
0.9 3.727452 × 10 1 3.713231 × 10 1 3.713236 × 10 1 3.713749 × 10 1 3.713974 × 10 1
Table 19. The absolute errors at some values of x for each of the numerical schemes.
Table 19. The absolute errors at some values of x for each of the numerical schemes.
txNSFD1NSFD2EEFDMFIEFDM
1 0.1 1.732294 × 10 3 1.731870 × 10 3 1.662559 × 10 3 1.621813 × 10 3
0.5 4.891452 × 10 3 4.890247 × 10 3 4.693599 × 10 3 4.568532 × 10 3
0.9 1.788565 × 10 3 1.788119 × 10 3 1.715994 × 10 3 1.667276 × 10 3
10 0.1 1.386199 × 10 3 1.385769 × 10 3 1.336481 × 10 3 1.318209 × 10 3
0.5 3.905879 × 10 3 3.904660 × 10 3 3.764468 × 10 3 3.707117 × 10 3
0.9 1.422126 × 10 3 1.421680 × 10 3 1.370378 × 10 3 1.347886 × 10 3
Table 20. The relative errors at some values of x for each of the numerical schemes.
Table 20. The relative errors at some values of x for each of the numerical schemes.
txNSFD1NSFD2EEFDMFIEFDM
1 0.1 6.570078 × 10 3 6.568471 × 10 3 6.305595 × 10 3 6.151057 × 10 3
0.5 1.789322 × 10 2 1.788881 × 10 2 1.716946 × 10 2 1.671196 × 10 2
0.9 6.319940 × 10 3 6.318365 × 10 3 6.063510 × 10 3 5.891364 × 10 3
10 0.1 3.878856 × 10 3 3.877653 × 10 3 3.739734 × 10 3 3.688607 × 10 3
0.5 1.069525 × 10 2 1.069191 × 10 2 1.030804 × 10 2 1.015099 × 10 2
0.9 3.815276 × 10 3 3.814081 × 10 3 3.676447 × 10 3 3.616104 × 10 3
Table 21. L 1 and L error norms with CPU times for the four numerical methods.
Table 21. L 1 and L error norms with CPU times for the four numerical methods.
tSchemes L 1 Error L Error CPU Time (Sec)
1NSFD1 3.228089 × 10 3 4.891452 × 10 3 0.0621
NSFD2 3.227293 × 10 3 4.890247 × 10 3 0.0637
EEFDM 3.097547 × 10 3 4.693599 × 10 3 0.0659
FIEFDM 3.014506 × 10 3 4.568532 × 10 3 0.0671
10NSFD1 2.576806 × 10 3 3.905879 × 10 3 0.2335
NSFD2 2.576002 × 10 3 3.904660 × 10 3 0.2281
EEFDM 2.483571 × 10 3 3.764468 × 10 3 0.2328
FIEFDM 2.445540 × 10 3 3.707117 × 10 3 0.2871
Table 22. A comparison between the exact and the numerical solutions at some values of x.
Table 22. A comparison between the exact and the numerical solutions at some values of x.
txExactNSFD1NSFD2EEFDMFIEFDM
1 0.1 3.197787 × 10 1 3.115395 × 10 1 3.115521 × 10 1 3.129688 × 10 1 3.131647 × 10 1
0.5 3.395893 × 10 1 3.164245 × 10 1 3.164602 × 10 1 3.204963 × 10 1 3.210964 × 10 1
0.9 3.581881 × 10 1 3.498705 × 10 1 3.498835 × 10 1 3.513357 × 10 1 3.515904 × 10 1
10 0.1 4.976668 × 10 1 4.974441 × 10 1 4.974445 × 10 1 4.974708 × 10 1 4.974707 × 10 1
0.5 4.979926 × 10 1 4.975407 × 10 1 4.975418 × 10 1 4.976175 × 10 1 4.976172 × 10 1
0.9 4.983164 × 10 1 4.981586 × 10 1 4.981589 × 10 1 4.981856 × 10 1 4.981855 × 10 1
Table 23. The absolute errors at some values of x for each of the numerical schemes.
Table 23. The absolute errors at some values of x for each of the numerical schemes.
txNSFD1NSFD2EEFDMFIEFDM
1 0.1 8.239162 × 10 3 8.226605 × 10 3 6.809932 × 10 3 6.613999 × 10 3
0.5 2.316489 × 10 2 2.312918 × 10 2 1.909310 × 10 2 1.849292 × 10 2
0.9 8.317574 × 10 3 8.304529 × 10 3 6.852408 × 10 3 6.597658 × 10 3
10 0.1 1.626796 × 10 4 1.622971 × 10 4 1.360407 × 10 4 1.361553 × 10 4
0.5 4.518804 × 10 4 4.507901 × 10 4 3.751243 × 10 4 3.754311 × 10 4
0.9 1.578207 × 10 4 1.574438 × 10 4 1.308204 × 10 4 1.309152 × 10 4
Table 24. The relative errors at some values of x for each of the numerical schemes.
Table 24. The relative errors at some values of x for each of the numerical schemes.
txNSFD1NSFD2EEFDMFIEFDM
1 0.1 2.576520 × 10 2 2.572593 × 10 2 2.129576 × 10 2 2.068305 × 10 2
0.5 6.821441 × 10 2 6.810926 × 10 2 5.622408 × 10 2 5.445674 × 10 2
0.9 2.322125 × 10 2 2.318483 × 10 2 1.913075 × 10 2 1.841954 × 10 2
10 0.1 3.269239 × 10 4 3.261553 × 10 4 2.733899 × 10 4