Quasi wavelet numerical approach of non-linear reaction diffusion and integro reaction-diffusion equation with Atangana–Baleanu time fractional derivative

2020 ◽  
Vol 130 ◽  
pp. 109456 ◽  
Author(s):  
Sachin Kumar ◽  
Prashant Pandey
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Reaction Diffusion Equation ◽  
Linear Reaction ◽  
Non Linear ◽  
Time Fractional Derivative

scholarly journals DOUBLE-QUASI-WAVELET NUMERICAL METHOD FOR THE VARIABLE-ORDER TIME FRACTIONAL AND RIESZ SPACE FRACTIONAL REACTION–DIFFUSION EQUATION INVOLVING DERIVATIVES IN CAPUTO–FABRIZIO SENSE

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040047
Author(s):  
SACHIN KUMAR ◽  
PRASHANT PANDEY ◽  
J. F. GÓMEZ-AGUILAR ◽  
D. BALEANU
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Numerical Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Variable Order ◽  
Nonlinear Reaction Diffusion Equation ◽  
Spatial Derivatives ◽  
Time Fractional Derivative ◽  
Fractional Type

Our motive in this scientific contribution is to deal with nonlinear reaction–diffusion equation having both space and time variable order. The fractional derivatives which are used are non-singular having exponential kernel. These derivatives are also known as Caputo–Fabrizio derivatives. In our model, time fractional derivative is Caputo type while spatial derivative is variable-order Riesz fractional type. To approximate the variable-order time fractional derivative, we used a difference scheme based upon the Taylor series formula. While approximating the variable order spatial derivatives, we apply the quasi-wavelet-based numerical method. Here, double-quasi-wavelet numerical method is used to investigate this type of model. The discretization of boundary conditions with the help of quasi-wavelet is discussed. We have depicted the efficiency and accuracy of this method by solving the some particular cases of our model. The error tables and graphs clearly show that our method has desired accuracy.

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.

scholarly journals Gradient Optimal Control of the Bilinear Reaction–Diffusion Equation

2021 ◽  
Author(s):  
El Hassan Zerrik ◽  
Abderrahman Ait Aadi
Keyword(s):  
Optimal Control ◽  
Diffusion Equation ◽  
Minimization Problem ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Time Dependent ◽  
Reaction Diffusion Equation ◽  
Energy Term ◽  
Bounded Controls ◽  
Special Cases

In this chapter, we study a problem of gradient optimal control for a bilinear reaction–diffusion equation evolving in a spatial domain Ω⊂Rn using distributed and bounded controls. Then, we minimize a functional constituted of the deviation between the desired gradient and the reached one and the energy term. We prove the existence of an optimal control solution of the minimization problem. Then this control is characterized as solution to an optimality system. Moreover, we discuss two special cases of controls: the ones are time dependent, and the others are space dependent. A numerical approach is given and successfully illustrated by simulations.

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