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.
Highlights
Spectral methods are powerful techniques that we use to numerically solve linear and nonlinear partial differential equations either in their strong or weak forms
Because of the pseudospectral method is an efficient and accurate numerical scheme for solving various problems in physical space, including variable coefficient and singularity, 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
We extend the application of the Legendre pseudospectral method to solve numerically the Fisher equation with variable coefficient, ut − b (t) uxx − cu (1 − u) = 0, (x, t) ∈ D × [0, T], (27)
Summary
Spectral methods (see, for instance, [1,2,3,4,5]) are powerful techniques that we use to numerically solve linear and nonlinear partial differential equations either in their strong or weak forms. We present an accurate numerical solution based on Legendre-Gauss-Lobatto collocation method for Fisher’s type equations. 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. There are no results on Legendre-Gauss-Lobatto collocation method for solving nonlinear Fisher-type equations subject to initial-boundary conditions.
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