Lie point symmetries, conservation laws, and solutions of a space dependent reaction–diffusion equation

2014 ◽  
Vol 248 ◽  
pp. 386-398 ◽  
Author(s):  
Zhijie Cao ◽  
Yiping Lin
Keyword(s):  
Diffusion Equation ◽  
Conservation Laws ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Lie Point Symmetries

scholarly journals Conservation Laws and Exact Solutions for a Reaction-Diffusion Equation with a Variable Coefficient

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Zhijie Cao ◽  
Yiping Lin
Keyword(s):  
Diffusion Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Lie Groups ◽  
Variable Coefficient ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation

In this paper a variable-coefficient reaction-diffusion equation is studied. We classify the equation into three kinds by different restraints imposed on the variable coefficientb(x)in the process of solving the determining equations of Lie groups. Then, for each kind, the conservation laws corresponding to the symmetries obtained are considered. Finally, some exact solutions are constructed.

scholarly journals Numerical Analysis WSGD Scheme for One- and Two-Dimensional Distributed Order Fractional Reaction–Diffusion Equation with Collocation Method via Fractional B-Spline

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Mohammad Ramezani
Keyword(s):  
Diffusion Equation ◽  
Collocation Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
B Spline ◽  
Computational Work ◽  
Distributed Order ◽  
Fractional Reaction

AbstractThe main propose of this paper is presenting an efficient numerical scheme to solve WSGD scheme for one- and two-dimensional distributed order fractional reaction–diffusion equation. The proposed method is based on fractional B-spline basics in collocation method which involve Caputo-type fractional derivatives for $$0 < \alpha < 1$$ 0 < α < 1 . The most significant privilege of proposed method is efficient and quite accurate and it requires relatively less computational work. The solution of consideration problem is transmute to the solution of the linear system of algebraic equations which can be solved by a suitable numerical method. The finally, several numerical WSGD Scheme for one- and two-dimensional distributed order fractional reaction–diffusion equation.

Nonlinear solution of the reaction–diffusion equation using a two-step third–fourth-derivative block method

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Oluwaseun Adeyeye ◽  
Ali Aldalbahi ◽  
Jawad Raza ◽  
Zurni Omar ◽  
Mostafizur Rahaman ◽  
...  
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Block Method ◽  
Wide Range ◽  
Higher Derivatives ◽  
Fourth Derivative ◽  
Improved Accuracy

AbstractThe processes of diffusion and reaction play essential roles in numerous system dynamics. Consequently, the solutions of reaction–diffusion equations have gained much attention because of not only their occurrence in many fields of science but also the existence of important properties and information in the solutions. However, despite the wide range of numerical methods explored for approximating solutions, the adoption of block methods is yet to be investigated. Hence, this article introduces a new two-step third–fourth-derivative block method as a numerical approach to solve the reaction–diffusion equation. In order to ensure improved accuracy, the method introduces the concept of nonlinearity in the solution of the linear model through the presence of higher derivatives. The method obtained accurate solutions for the model at varying values of the dimensionless diffusion parameter and saturation parameter. Furthermore, the solutions are also in good agreement with previous solutions by existing authors.

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