The spectral collocation approximations based on Legendre polynomials are used to compute the numerical solution of timedependent Fisher’s type problems. The spatial derivatives are collocated at a LegendreGaussLobatto interpolation nodes. The proposed method has the advantage of reducing the problem to a system of ordinary differential equations in time. The fourstage Astable implicit RungeKutta scheme is applied to solve the resulted system of first order in time. Numerical results show that the LegendreGaussLobatto 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.
Spectral methods (see, for instance, [
In this paper, we present an accurate numerical solution based on LegendreGaussLobatto collocation method for Fisher’s type equations. The Fisher equation in the form
In recent years, many physicists and mathematicians have paid much attention to the Fisher equations due to their importance in mathematical physics. In [
To increase the numerical solution accuracy, spectral collocation methods based on orthogonal polynomials are often chosen. Doha et al. [
Indeed, there are no results on LegendreGaussLobatto collocation method for solving nonlinear Fishertype equations subject to initialboundary conditions. Therefore, the objective of this work is to present a numerical algorithm for solving such equation based on LegendreGaussLobatto 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 firstorder ordinary differential equations in time. The fourstage Astable implicit RungeKutta 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
The Legendre polynomials
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, [
In this subsection, we derive a Legendre pseudospectral algorithm to solve numerically the generalized BurgerFisher problem:
In the following, we shall derive an efficient algorithm for the numerical solution of (
The GaussLobatto points were introduced by way of (
In collocation methods, one specifically seeks the approximate solution such that the problem (
Now the two values
Approximation (
Let us denote
In this subsection, we extend the application of the Legendre pseudospectral method to solve numerically the Fisher equation with variable coefficient,
In this section, three nonlinear timedependent Fishertype 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
Consider the nonlinear timedependent onedimensional Fishertype equations
The exact solution is
In Table
Absolute errors for Example







−1  0.1 

−1  0.2 

−0.5 

−0.5 


0 

0 


0.5 

0.5 


1 

1 

In case of
The result of the LGLC method at
The curves of approximate solutions and the exact solutions of problem (
Consider the nonlinear timedependent onedimensional generalized BurgerFishertype equations
The exact solution of (
The absolute errors for problem (
Absolute errors for Example














−1  0.1 

−1  0.1 

−1  0.1 

−0.5 

−0.5 

−0.5 


0 

0 

0 


0.5 

0.5 

0.5 


1 

1 

1 


−1  0.5 

−1  0.5 

−1  0.5 

−0.5 

−0.5 

−0.5 


0 

0 

0 


0.5 

0.5 

0.5 


1 

1 

1 

To illustrate the effectiveness of the Legendre pseudospectral method for problem (
The result of the LGLC method at
The curves of approximate solutions and the exact solutions of problem (
The result of the LGLC method at
The curves of approximate solutions and the exact solutions of problem (
Consider the nonlinear timedependent onedimensional Fishertype equations with variable coefficient
The exact solution of (
Table
Absolute errors for Example







−1  0.1 

−1  0.5 

−0.5 

−0.5 


0 

0 


0.5 

0.5 


1 

1 

In this paper, based on the LegendreGaussLobatto pseudospectral approximation we proposed an efficient numerical algorithm to solve nonlinear timedependent Fishertype equations with constant and variable coefficients. The method is based upon reducing the nonlinear partial differential equation into a system of firstorder 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.
The authors declare that there is no conflict of interests regarding the publication of this paper.