Approximate Solutions of Fisher's Type Equations with Variable Coefficients

Abstract

The spectral collocation approximations based on Legendre polynomials are used to compute the numerical solution of time-dependent Fisher’s type problems. The spatial derivatives are collocated at a Legendre-Gauss-Lobatto interpolation nodes. The proposed method has the advantage of reducing the problem to a system of ordinary differential equations in time. The four-stage A-stable implicit Runge-Kutta scheme is applied to solve the resulted system of first order in time. Numerical results show that the Legendre-Gauss-Lobatto collocation method is of high accuracy and is efficient for solving the Fisher’s type equations. Also the results demonstrate that the proposed method is powerful algorithm for solving the nonlinear partial differential equations.

1. Introduction

Spectral methods (see, for instance, [1–5]) are powerful techniques that we use to numerically solve linear and nonlinear partial differential equations either in their strong or weak forms. What sets spectral methods apart from others like finite difference methods or finite element methods is that to get a spectral method we approximate the solutions by high order orthogonal polynomial expansions. The orthogonal polynomial approximations can have very high convergence rates, which allow us to use fewer degrees of freedom for a desired level of accuracy. The most common spectral method from the strong form of the equations is known as collocation. In collocation techniques, the partial differential equation must be satisfied at a set of grid, or more precisely, collocation points (see, for instance, [6–10]). Spectral methods also have become increasingly popular for solving fractional differential equations [11–21].

In this paper, we present an accurate numerical solution based on Legendre-Gauss-Lobatto collocation method for Fisher’s type equations. The Fisher equation in the form 𝑢_𝑡=𝐷𝑢_𝑥𝑥+𝜈𝑢(1−𝑢) (1) was firstly introduced by Fisher in [22] to describe the propagation of a mutant gene. Fisher equations have a wide application in a large number of the chemical kinetics [23], logistic population growth [24], flame propagation [25], population in one-dimensional habitual [26], neutron population in a nuclear reaction [27], neurophysiology [28], branching Brownian motion [23], autocatalytic chemical reactions [29], and nuclear reactor theory [30].

In recent years, many physicists and mathematicians have paid much attention to the Fisher equations due to their importance in mathematical physics. In [31], Öǧün and Kart utilized truncated Painlevè expansions for presenting some exact solutions of Fisher and generalized Fisher equations. Tan et al. [28] proposed the homotopy analysis method to find analytical solution of Fisher equations. Gunzburger et al. [32] applied the discrete finite element approximation for obtaining a numerical solution of the forced Fisher equation. Dag et al. [33] discussed and applied the B-spline Galerkin method for Fisher’s equation. Bastani and Salkuyeh [34] proposed the compact finite difference approach in combination with third-order Runge-Kutta scheme to solve Fisher’s equation. More recently, Mittal and Jain [35] investigated the cubic B-spline scheme for solving Fisher’s reaction-diffusion problem. However, the fisher equations have been studied in many other articles by numerous numerical methods such as pseudospectral method [36, 37], finite difference method [38–44], finite element method [45], B-spline algorithm [46], and Galerkin method [47, 48].

To increase the numerical solution accuracy, spectral collocation methods based on orthogonal polynomials are often chosen. Doha et al. [49] proposed and developed a new numerical algorithm for solving the initial-boundary system of nonlinear hyperbolic equations based on spectral collocation method; a Chebyshev-Gauss-Radau collocation method in combination with the implicit Runge-Kutta scheme are employed to obtain highly accurate approximations to this system of nonlinear hyperbolic equations. In [50], Bhrawy proposed an efficient Jacobi-Gauss-Lobatto collocation method for approximating the solution of the generalized Fitzhugh-Nagumo equation in which the Jacobi-Gauss-Lobatto points are used as collocation nodes for spatial derivatives. Moreover, the Jacobi spectral collocation methods are used to solve some problems in mathematical physics, (see, for instance, [51–53]).

Indeed, there are no results on Legendre-Gauss-Lobatto collocation method for solving nonlinear Fisher-type equations subject to initial-boundary conditions. Therefore, the objective of this work is to present a numerical algorithm for solving such equation based on Legendre-Gauss-Lobatto pseudospectral method. The spatial derivatives are approximated at these grid points by approximating the derivatives of Legendre polynomial that interpolates the solutions. Moreover, we set the boundary conditions in the collocation method. The problem is then reduced to system of first-order ordinary differential equations in time. The four-stage A-stable implicit Runge-Kutta scheme is proposed for treating the this system of equations. Finally, some illustrative examples are implemented to illustrate the efficiency and applicability of the proposed approach.

The rest of this paper is structured as follows. In the next section, some properties of Legendre polynomials, which are required for implementing our algorithm, are presented. Section 3 is devoted to the development of Gauss-Lobatto collocation technique for a general form of Fisher-type equations based on the Legendre polynomials, and in Section 4 the proposed method is implemented to obtain some numerical results for three problems of Fisher-type equations with known exact solutions. Finally, a brief conclusion is provided in Section 5.

2. Legendre Polynomials

The Legendre polynomials𝐿_𝑘(𝑥) (𝑘=0,1,…,) satisfy the following Rodrigues’ formula: 𝐿_𝑘(𝑥)=(−1)^𝑘2^𝑘𝐷𝑘!^𝑘1−𝑥^2𝑘; (2) we recall also that𝐿_𝑘(𝑥)is a polynomial of degree 𝑘, and therefore, the𝑞th derivative of𝐿_𝑘(𝑥) is given by 𝐿_𝑘^(𝑞)(𝑥)=^𝑘−𝑞_{𝑖=0(𝑘+𝑖=even)}𝐶_𝑞(𝑘,𝑖)𝐿_𝑖(𝑥), (3) where 𝐶_𝑞()=2𝑘,𝑖_𝑞−1(2𝑖+1)Γ𝑞+𝑘−𝑖/2Γ𝑞+𝑘+𝑖+1/2Γ𝑞ΓΓ.2−𝑞+𝑘−𝑖/23−𝑞+𝑘+𝑖/2 (4) The analytical form of Legendre polynomial is 𝐿_𝑛(𝑥)=^[𝑛/2]_𝑖=0𝑐_𝑘^(𝑛)𝑥^𝑛−2𝑘, (5) where𝑐_𝑘^(𝑛)=(−1)^𝑘(2𝑛−2𝑘)!/2^𝑛(𝑛−𝑘)!(𝑛−2𝑘)!𝑘!, and 𝑛2=𝑛2,even,𝑛−12,odd. (6) It is also generating from the following relation: 𝐿_𝑘+2(𝑥)=2𝑘+3𝑘+2𝑥𝐿_𝑘+1(𝑥)−𝑘+1𝐿𝑘+2^𝑘(𝑥), (7) with𝐿₀(𝑥)=1, 𝐿₁(𝑥)=𝑥, and satisfies the orthogonality condition 𝐿_𝑘(𝑥),𝐿_𝑙(𝑥)_𝑤=¹₋₁𝐿_𝑘(𝑥)𝐿_𝑙(𝑥)𝑤(𝑥)=ℎ_𝑘𝛿_𝑙𝑘. (8) where𝑤(𝑥)=1,ℎ_𝑘=2/(2𝑘+1). Let𝑆_𝑁be the space of all polynomials of degree ≤𝑁, then for any𝜙∈𝑆_2𝑁−1(0,𝐿), ¹₋₁𝑤(𝑥)𝜙(𝑥)𝑑𝑥=^𝑁_𝑗=0𝜛_𝑁,𝑗𝜙𝑥_𝑁,𝑗. (9) Let us define the following discrete inner product and norm: (𝑢,𝑣)_𝑤=^𝑁_𝑗=0𝑢𝑥_𝑁,𝑗𝑣𝑥_𝑁,𝑗𝜛_𝑁,𝑗, (10) where𝑥_𝑁,𝑗and𝜛_𝑁,𝑗are the nodes and the corresponding weights of the Legendre-Gauss-Lobatto quadrature formula on the interval (−1,1), respectively.

3. Legendre Spectral Collocation Method

Because of the pseudospectral method is an efficient and accurate numerical scheme for solving various problems in physical space, including variable coefficient and singularity (see, [54, 55]), we propose this method based on Legendre polynomials for approximating the solution of the nonlinear generalized Burger-Fisher model equation and Fisher model with variable coefficient.

3.1. (1+1)-Dimensional Generalized Burger-Fisher Equation

In this subsection, we derive a Legendre pseudospectral algorithm to solve numerically the generalized Burger-Fisher problem: 𝑢_𝑡+𝜈𝑢^𝛿𝑢_𝑥−𝑢_𝑥𝑥−𝛾𝑢1−𝑢^𝛿=0,(𝑥,𝑡)∈𝐷×[0,𝑇], (11) where𝐷={𝑥∶−1≤𝑥≤1}. Subject to 𝑢()(𝑡)𝑢()(𝑥)𝑥,𝑡=𝑔,𝑥=−1,1,𝑥,0=𝑓,𝑥∈𝐷. (12)(13)

In the following, we shall derive an efficient algorithm for the numerical solution of (11)–(13). Let the approximation of𝑢(𝑥,𝑡)be given in terms of the Legendre polynomials expansion: 𝑢(𝑥,𝑡)=^𝑁_𝑗=0𝑎_𝑗(𝑡)𝐿_𝑗(𝑥),𝐚=𝑎₀,𝑎₁,…,𝑎_𝑁^𝑇. (14) Making use of relations (8) and (10) gives 𝑢(𝑥,𝑡)=^𝑁_𝑗=01ℎ_𝑗^𝑁_𝑖=0𝐿_𝑗𝑥_𝑖𝐿_𝑗(𝑥)𝜛_𝑁,𝑖𝑢𝑥_𝑖,𝑡 (15) or equivalently 𝑢(𝑥,𝑡)=^𝑁_𝑖=0^𝑁_𝑗=01ℎ_𝑗𝐿_𝑗𝑥_𝑖𝐿_𝑗(𝑥)𝜛_𝑁,𝑖𝑢𝑥_𝑖,𝑡. (16)

The Gauss-Lobatto points were introduced by way of (9). We then saw that the polynomial approximation 𝑢(𝑥,𝑡)can be characterized by(𝑁+1)nodal values𝑢(𝑥_𝑖,𝑡). The approximation of the spatial partial derivatives of first-order for𝑢(𝑥,𝑡)can be computed at the Legendre Gauss-Lobatto interpolation nodes as 𝑢_𝑥𝑥_𝑛,𝑡=^𝑁_𝑖=0^𝑁_𝑗=01ℎ_𝑗𝐿_𝑗𝑥_𝑖𝐿_𝑗𝑥_𝑛^′𝜛_𝑁,𝑖𝑢𝑥_𝑖=,𝑡_𝑁𝑖=0𝐴_𝑛𝑖𝑢𝑥_𝑖=,𝑡_𝑁𝑖=0𝐴_𝑛𝑖𝑢_𝑖(𝑡),𝑛=0,1,…,𝑁, (17) where 𝐴_𝑛𝑖=^𝑁_𝑗=01ℎ_𝑗𝐿_𝑗𝑥_𝑖𝐿_𝑗𝑥_𝑛^′𝜛_𝑁,𝑖,𝑢_𝑖(𝑡)=𝑢𝑥_𝑖,𝑡. (18) Subsequently, the second-order spatial partial derivatives of𝑢(𝑥,𝑡)may be written at the same collocation nodes as 𝑢_𝑥𝑥𝑥_𝑛,𝑡=^𝑁_𝑖=0^𝑁_𝑗=01ℎ_𝑗𝐿_𝑗𝑥_𝑖𝐿_𝑗𝑥_𝑛^′′𝜛_𝑁,𝑖𝑢𝑥_𝑖=,𝑡_𝑁𝑖=0𝐵_𝑛𝑖𝑢𝑥_𝑖=,𝑡_𝑁𝑖=0𝐵_𝑛𝑖𝑢_𝑖, (19) where 𝐵_𝑛𝑖=^𝑁_𝑗=01ℎ_𝑗𝐿_𝑗𝑥_𝑖𝐿_𝑗(𝑥_𝑛)^′′𝜛_𝑁,𝑖. (20)

In collocation methods, one specifically seeks the approximate solution such that the problem (11) is satisfied exactly at the Legendre Gauss-Lobatto set of interpolation points 𝑥_𝑛; 𝑛=1,…,𝑁−1. The approximation is exact at the𝑁−1collocation points. Therefore, (11) after using relations (17)–(20), can be written as 𝑢^⋅_𝑛(𝑡)+𝜈𝑢^𝛿_𝑛(𝑡)^𝑁_𝑖=0𝐴_𝑛𝑖𝑢_𝑖(𝑡)−^𝑁_𝑖=0𝐵_𝑛𝑖𝑢_𝑖(𝑡)−𝛾𝑢_𝑛(𝑡)1−𝑢^𝛿_𝑛(𝑡)=0,𝑛=1,…,𝑁−1, (21) where𝑢_𝑛(𝑡)=𝑢(𝑥_𝑛,𝑡)and𝑢^⋅_𝑛(𝑡)=𝜕𝑢^⋅_𝑛(𝑡)/𝜕𝑡.

Now the two values𝑢₀(𝑡)and𝑢_𝑁(𝑡)can be determined from the boundary conditions (12), then (21) can be reformulated as 𝑢^⋅_𝑛(𝑡)+𝜈𝑢^𝛿_𝑛(𝑡)^𝑁−1_𝑖=1𝐴_𝑛𝑖𝑢_𝑖(𝑡)−^𝑁−1_𝑖=1𝐵_𝑛𝑖𝑢_𝑖(𝑡)+𝜈𝑢^𝛿_𝑛(𝑡)𝑑_𝑛(𝑡)−𝑑_𝑛(𝑡)−𝛾𝑢_𝑛(𝑡)1−𝑢^𝛿_𝑛(𝑡)=0,𝑛=1,…,𝑁−1, (22) where 𝑑_𝑛(𝑡)=𝐴_𝑛0𝑢₀(𝑡)+𝐴_𝑛𝑁𝑢_𝑁(𝑡),𝑑_𝑛(𝑡)=𝐵_𝑛0𝑢₀(𝑡)+𝐵_𝑛𝑁𝑢_𝑁(𝑡). (23)

Approximation (22) automatically satisfies the boundary conditions (12), but we need an initial condition for each of the 𝑢_𝑛(𝑡) to integrate (22) in time. The initial condition is usually taken to be the interpolant of the initial function𝑓(𝑥); that is𝑢_𝑛(0)=𝑓(𝑥_𝑛). Therefore, the approximation of (11)–(13) is reduced to the solution of system of ordinary differential equations in time. Consider 𝑢^⋅_𝑛(𝑡)+𝜈𝑢^𝛿_𝑛(𝑡)^𝑁−1_𝑖=1𝐴_𝑛𝑖𝑢_𝑖(𝑡)−^𝑁−1_𝑖=1𝐵_𝑛𝑖𝑢_𝑖(𝑡)+𝜈𝑢^𝛿_𝑛(𝑡)𝑑_𝑛(𝑡)−𝑑_𝑛(𝑡)−𝛾𝑢_𝑛(𝑡)1−𝑢^𝛿_𝑛(𝑡)𝑢=0,𝑛=1,…,𝑁−1,_𝑛(0)=𝑓𝑥_𝑛. (24)

Let us denote 𝑈^⋅(𝑡)=𝑢^⋅₁(𝑡),𝑢^⋅₂(𝑡),…,𝑢^⋅_𝑁−1(𝑡)^𝑇,𝑈(0)=𝑢₁(0),𝑢₂(0),…,𝑢_𝑁−1(0)^𝑇,𝑓=𝑓(𝑥₁),𝑓(𝑥₂),…,𝑓(𝑥_𝑁−1)^𝑇,𝐹(𝑡,𝑢(𝑡))=𝐹₁(𝑡,𝑢(𝑡)),𝐹₂(𝑡,𝑢(𝑡)),…,𝐹_𝑁−1(𝑡,𝑢(𝑡))^𝑇,𝐹_𝑛(𝑡,𝑢(𝑡))=−𝜈𝑢^𝛿_𝑛(𝑡)^𝑁−1_𝑖=1𝐴_𝑛𝑖𝑢_𝑖(𝑡)+^𝑁−1_𝑖=1𝐵_𝑛𝑖𝑢_𝑖(𝑡)−𝜈𝑢^𝛿_𝑛(𝑡)𝑑_𝑛(𝑡)+𝑑_𝑛(𝑡)+𝛾𝑢_𝑛(𝑡)1−𝑢^𝛿_𝑛(𝑡),𝑛=1,…,𝑁−1. (25) Then (24) can be written in the matrix form 𝑈^⋅(𝑡)((𝑡𝑈(0)=𝐹𝑡,𝑢))=𝑓. (26) This system of ordinary differential equations can be solved by using four-stage A-stable implicit Runge-Kutta scheme.

3.2. (1+1)-Dimensional Fisher Equation with Variable Coefficient

In this subsection, we extend the application of the Legendre pseudospectral method to solve numerically the Fisher equation with variable coefficient, 𝑢_𝑡−𝑏(𝑡)𝑢_𝑥𝑥−𝑐𝑢(1−𝑢)=0,(𝑥,𝑡)∈𝐷×[0,𝑇], (27) subject to the initial-boundary conditions 𝑢()(𝑥)𝑢()(𝑡).𝑥,0=𝑓,𝑥∈𝐷,𝑥,𝑡=𝑔 (28) Proceeding as in the previous subsection we can obtain𝑢_𝑥𝑥in the same form as (19), and then (27) can be collocated in the Legendre Gauss-Lobatto points as: 𝑢^⋅_𝑛(𝑡)−𝑏(𝑡)^𝑁−1_𝑖=1𝐵_𝑛𝑖𝑢_𝑖(𝑡)−𝑏(𝑡)𝑑_𝑛(𝑡)−𝑐𝑢_𝑛(𝑡)1−𝑢_𝑛(𝑡)=0,𝑛=1,…,𝑁−1, (29) which can be written in the matrix form 𝑈^⋅(𝑡)((𝑡,𝑈(0)=𝐹𝑡,𝑢))=𝑓, (30) where 𝐹_𝑛((𝑡(𝑡)𝑡,𝑢))=𝑏^𝑁−1_𝑖=1𝐵_𝑛𝑖𝑢_𝑖(𝑡)+𝑏(𝑡)𝑑_𝑛(𝑡)+𝑐𝑢_𝑛(𝑡)1−𝑢_𝑛(𝑡),𝑛=1,…,𝑁−1. (31)

4. Numerical Examples

In this section, three nonlinear time-dependent Fisher-type equations on finite interval are implemented to demonstrate the accuracy and capability of the proposed algorithm, and all of them were performed on the computer using a program written in Mathematica 8.0. The absolute errors in the given tables are 𝐸(𝑥,𝑡)=|𝑢(𝑥,𝑡)−𝑢(𝑥,𝑡)where 𝑢(𝑥,𝑡) and𝑢(𝑥,𝑡)are the exact and numerical solution at selected points(𝑥,𝑡).

Example 1. Consider the nonlinear time-dependent one-dimensional Fisher-type equations 𝑢_𝑡=𝑢_𝑥𝑥+𝑢(1−𝑢)𝑢−𝛾,(𝑥,𝑡)∈𝐷×[0,𝑇], (32) where𝐷={𝑥∶−1<𝑥<1}. Subject to 𝑢(1,𝑡)=1+𝛾2−𝛾−12tanh𝛾−12√21−1+𝛾√2𝑡,𝑢(−1,𝑡)=1+𝛾2+𝛾−12tanh𝛾−12√21+1+𝛾√2𝑡,𝑢(𝑥,0)=1+𝛾2−𝛾−12tanh𝛾−12√2(𝑥),𝑥∈𝐷. (33)

The exact solution is 𝑢(𝑥,𝑡)=1+𝛾2−𝛾−12tanh𝛾−12√2𝑥−1+𝛾√2𝑡. (34)

In Table 1, we introduce the absolute errors between the approximate and exact solutions for problem (32) using the proposed method for different values of𝑥and𝑡, with𝛾=10⁻²and𝑁=20.

Table 1 

Absolute errors for Example 1.

In case of𝛾=10⁻²and𝑁=20, the approximate solution and absolute errors of problem (32) are displayed in Figures 1(a) and 1(b), respectively. In Figure 2, we plotted the curves of approximate solutions and exact solutions of problem (32) for different values of 𝑡  (0.0,0.5 and 0.9) with𝛾=10⁻²and 𝑁=20. It is clear from this figure that approximate solutions and exact solutions completely coincide for the chosen values of𝑡.

(a)

(b)

Figure 1 

The result of the L-GL-C method at𝛾=10−2and𝑁=20. (a) The approximate solution. (b) The absolute error.

Figure 2 

The curves of approximate solutions and the exact solutions of problem (32) at𝑡=0.0, 𝑡=0.5, and 𝑡=0.9with𝛾=10−2and𝑁=20.

Example 2. Consider the nonlinear time-dependent one-dimensional generalized Burger-Fisher-type equations 𝑢_𝑡=𝑢_𝑥𝑥−𝜈𝑢^𝛿𝑢_𝑥+𝛾𝑢1−𝑢^𝛿,(𝑥,𝑡)∈𝐷×[0,𝑇], (35) where𝐷={𝑥∶−1<𝑥<1}. Subject to 𝑢()=11,𝑡2−12tanh𝜈𝛿2()×𝜈𝛿+11−+𝛾()𝛿+1𝛿+1𝜈𝑡^1/𝛿,𝑢()=1−1,𝑡2+12tanh𝜈𝛿2()×𝜈𝛿+11++𝛾()𝛿+1𝛿+1𝜈𝑡^1/𝛿,𝑢()=1𝑥,02−12tanh𝜈𝛿2()𝑥𝛿+1^1/𝛿,𝑥∈𝐷. (36)

The exact solution of (35) is 𝑢()=1𝑥,𝑡2−12tanh𝜈𝛿2()×𝜈𝛿+1𝑥−+𝛾()𝛿+1𝛿+1𝜈𝑡^1/𝛿. (37)

The absolute errors for problem (35) are listed in Table 2 using the L-GL-C method with𝜈=𝛾=10⁻², 𝑁=20, and various choices of𝛿.

Table 2 

Absolute errors for Example 2.

To illustrate the effectiveness of the Legendre pseudospectral method for problem (35), we displayed in Figures 3(a) and 3(b) the approximate solution and the absolute error with𝜈=𝛾=10⁻²,𝛿=1, and𝑁=20. The graph of curves of exact and approximate solutions with different values of 𝑡 (0.0, 0.5, and 0.9) is given in Figure 4. Moreover, the approximate solution and the absolute error with 𝜈=𝛾=10⁻², 𝛿=2, and 𝑁=20 are displayed in Figures 5(a) and 5(b), respectively. The curves of exact and approximate solutions of problem (35) with𝛿=2are displayed in Figure 6 with values of parameters listed in its caption.

(a)

(b)

Figure 3 

The result of the L-GL-C method at𝜈=𝛾=10−2,𝛿=1, and𝑁=20. (a) The approximate solution. (b)The absolute error.

Figure 4 

The curves of approximate solutions and the exact solutions of problem (35) at 𝑡=0.0, 𝑡=0.5, and 𝑡=0.9 with 𝜈=𝛾=10−2, 𝛿=1, and 𝑁=20.

(a)

(b)

Figure 5 

The result of the L-GL-C method at𝜈=𝛾=10−2,𝛿=2, and𝑁=20. (a) The approximate solution. (b)The absolute error.

Figure 6 

The curves of approximate solutions and the exact solutions of problem (35) at 𝑡=0.0, 𝑡=0.5, and 𝑡=0.9with 𝜈=𝛾=10−2, 𝛿=2, and𝑁=20.

Example 3. Consider the nonlinear time-dependent one-dimensional Fisher-type equations with variable coefficient 𝑢_𝑡𝑎=−6𝜇²𝑎coth6𝑡+𝑐𝑢_𝑥𝑥(),()[],+𝑎𝑢1−𝑢𝑥,𝑡∈𝐷×0,𝑇 (38) where𝐷={𝑥∶−1<𝑥<1}. Subject to 𝑢()=11,𝑡4𝑎coth6𝑡+𝑐sech²𝜇2+5𝑎𝑡+1122𝜇tanh2+5𝑎𝑡+1122,𝑢()=1−1,𝑡4𝑎coth6𝑡+𝑐sech²−𝜇2+5𝑎𝑡+1122−𝜇tanh2+5𝑎𝑡+1122,𝑢()=1𝑥,04coth(𝑐)sech²𝜇𝑥2+12tanh𝜇𝑥2+12,𝑥∈𝐷. (39)(40)

The exact solution of (38) is 𝑢()=1𝑥,𝑡4𝑎coth6𝑡+𝑐sech²𝜇𝑥2+5𝑎𝑡+1122tanh𝜇𝑥2+5𝑎𝑡+1122. (41)

Table 3 lists the absolute errors for problem (38) using the L-GL-C method. From numerical results of this table, it can be concluded that the numerical solutions are in excellent agreement with the exact solutions.

Table 3 

Absolute errors for Example 3.

5. Conclusion

In this paper, based on the Legendre-Gauss-Lobatto pseudospectral approximation we proposed an efficient numerical algorithm to solve nonlinear time-dependent Fisher-type equations with constant and variable coefficients. The method is based upon reducing the nonlinear partial differential equation into a system of first-order ordinary differential equations in the expansion coefficient of the spectral solution. Numerical examples were also provided to illustrate the effectiveness of the derived algorithms. The numerical experiments show that the Legendre pseudospectral approximation is simple and accurate with a limited number of collocation nodes.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.