Numerical and theoretical treatment based on the compact finite difference and spectral collocation algorithms of the space fractional-order Fisher’s equation

2020 ◽  
Vol 31 (09) ◽  
pp. 2050122
Author(s):  
M. M. Khader ◽  
M. Adel
Keyword(s):  
Finite Difference ◽  
Fractional Order ◽  
Time Derivative ◽  
Theoretical Treatment ◽  
Algebraic Equations ◽  
Spectral Collocation ◽  
Compact Finite Difference ◽  
Fisher’S Equation ◽  
Fisher's Equation ◽  
Accurate Numerical Algorithm

This paper presents an accurate numerical algorithm to solve the space fractional-order Fisher’s equation where the derivative operator is described in the Caputo derivative sense. In the presented discretization process, first, we use the compact finite difference (CFD) for a semi-discrete occurrence in time derivative and implement the Chebyshev spectral collocation method (CSCM) of the third-kind to discretize the spatial fractional derivative. The presented method converts the problem understudy to be a system of algebraic equations which can be easily solved. To study the convergence and stability analysis, some theorems are given with their proofs. A numerical simulation is outputted to test the accuracy and applicability of our presented algorithm.

scholarly journals LIE SYMMETRY ANALYSIS TO FISHER'S EQUATION WITH TIME FRACTIONAL ORDER

2020 ◽  
Vol 10 (5) ◽  
pp. 2058-2067
Author(s):  
Zhenli Wang ◽  
◽  
Lihua Zhang ◽  
Hanze Liu ◽  
◽  
...  
Keyword(s):  
Fractional Order ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Lie Symmetry Analysis ◽  
Fisher’S Equation ◽  
Fisher's Equation

scholarly journals Some Dynamical Models Involving Fractional-Order Derivatives with the Mittag-Leffler Type Kernels and Their Applications Based upon the Legendre Spectral Collocation Method

2021 ◽  
Vol 5 (3) ◽  
pp. 131
Author(s):  
Hari M. Srivastava ◽  
Abedel-Karrem N. Alomari ◽  
Khaled M. Saad ◽  
Waleed M. Hamanah
Keyword(s):  
Collocation Method ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Spectral Collocation Method ◽  
Hybrid Technique ◽  
Algebraic Equations ◽  
Spectral Collocation ◽  
Finite Differences Method ◽  
Nonlinear Algebraic Equations ◽  
Raphson Method

Fractional derivative models involving generalized Mittag-Leffler kernels and opposing models are investigated. We first replace the classical derivative with the GMLK in order to obtain the new fractional-order models (GMLK) with the three parameters that are investigated. We utilize a spectral collocation method based on Legendre’s polynomials for evaluating the numerical solutions of the pr. We then construct a scheme for the fractional-order models by using the spectral method involving the Legendre polynomials. In the first model, we directly obtain a set of nonlinear algebraic equations, which can be approximated by the Newton-Raphson method. For the second model, we also need to use the finite differences method to obtain the set of nonlinear algebraic equations, which are also approximated as in the first model. The accuracy of the results is verified in the first model by comparing it with our analytical solution. In the second and third models, the residual error functions are calculated. In all cases, the results are found to be in agreement. The method is a powerful hybrid technique of numerical and analytical approach that is applicable for partial differential equations with multi-order of fractional derivatives involving GMLK with three parameters.

A NOTE ON THE LATTICE BOLTZMANN VERSUS FINITE-DIFFERENCE METHODS FOR THE NUMERICAL SOLUTION OF THE FISHER'S EQUATION

2013 ◽  
Vol 25 (01) ◽  
pp. 1340015 ◽  
Author(s):  
SAURO SUCCI
Keyword(s):  
Finite Difference ◽  
Lattice Boltzmann ◽  
Finite Difference Methods ◽  
Difference Schemes ◽  
Diffusion Reaction ◽  
Finite Difference Schemes ◽  
Fisher’S Equation ◽  
Advection Diffusion ◽  
Fisher's Equation ◽  
Better Than

We assess the Lattice Boltzmann (LB) method versus centered finite-difference schemes for the solution of the advection–diffusion–reaction (ADR) Fisher's equation. It is found that the LB method performs significantly better than centered finite-difference schemes, a property we attribute to the near absence of dispersion errors.

scholarly journals Numerical Solution of Fisher’s Equation Using Finite Difference

2015 ◽  
Vol 12 ◽  
pp. 27-34
Author(s):  
Sharefa Eisa Ali Alhazmi
Keyword(s):  
Nonlinear System ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Diffusion Equation ◽  
Numerical Solution ◽  
Difference Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Fisher’S Equation ◽  
Fisher's Equation

A numerical method is proposed to approximate the numeric solutions of nonlinear Fisher’s reaction diffusion equation with finite difference method. The method is based on replacing each terms in the Fisher’s equation using finite difference method. The proposed method has the advantage of reducing the problem to a nonlinear system, which will be derived and solved using Newton method. FTCS and CN method will be introduced, compared and tested.

A Fully Discrete Spectral Method for Fisher’s Equation on the Whole Line

2016 ◽  
Vol 6 (4) ◽  
pp. 400-415 ◽  
Author(s):  
Yu-Jian Jiao ◽  
Tian-Jun Wang ◽  
Qiong Zhang
Keyword(s):  
Finite Difference ◽  
Theoretical Analysis ◽  
Asymptotic Solution ◽  
Spectral Method ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Fisher’S Equation ◽  
Fully Discrete ◽  
Fisher's Equation ◽  
Solution Behaviour

AbstractA generalised Hermite spectral method for Fisher's equation in genetics with different asymptotic solution behaviour at infinities is proposed, involving a fully discrete scheme using a second order finite difference approximation in the time. The convergence and stability of the scheme are analysed, and some numerical results demonstrate its efficiency and substantiate our theoretical analysis.

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