Numerical solution of two-dimensional fractional-order reaction advection sub-diffusion equation with finite-difference Fibonacci collocation method

There are various methods to solve the physical life problem involving engineering, scientific and biological systems. It is found that numerical methods are approximate solutions. In this way, randomly generated finite difference grids achieve an approximation with fewer iterations. The idea of randomly generated grids in cartesian coordinates and polar form are compared with the exact, iterative method, uniform grids, and approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions. The most ideal and benchmarking method is the finite difference method over randomly generated grids on Cartesian coordinates, polar coordinates used for numerical solutions. This concept motivates the investigation of the effects of the randomly generated meshes. The two-dimensional equation is solved over randomly generated meshes to test randomly generated grids and the implementation. The feasibility of the numerical solution is analyzed by comparing simulation profiles.

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Numerical solution of highly non-linear fractional order reaction advection diffusion equation using the cubic B-spline collocation method

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2020-0112 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Kushal Dhar Dwivedi ◽

Subir Das ◽

Rajeev ◽

Dumitru Baleanu

Keyword(s):

Diffusion Equation ◽

Collocation Method ◽

Fractional Order ◽

Order Reaction ◽

Spline Collocation ◽

Diffusion Term ◽

B Spline ◽

Spline Collocation Method ◽

Advection Diffusion Equation ◽

Advection Diffusion

Abstract In this article, the approximate solution of the fractional-order reaction advection-diffusion equation with the prescribed initial and boundary conditions is found with the help of a cubic B-spline collocation method, which is unconditionally stable and convergent. The accuracy of the scheme is validated by applying the method on four existing problems having analytical solutions and through the evaluation of the absolute errors between numerical results and the exact solutions for different particular cases. Applying the proposed method on the last two numerical problems, it is shown that the method performs better than the existing methods even for very less number of spatial and temporal discretizations. The main contribution of the article is to develop an efficient method to solve the proposed fractional order nonlinear problem and to find the effect on solute concentration graphically due to increase in the non-linearity in the diffusion term for different particular values of parameters.

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

International Journal of Applied and Computational Mathematics ◽

10.1007/s40819-021-00969-9 ◽

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.

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The Best Perturbation Method for the Parameter Inversion of Two-Dimension Convection-Diffusion Equation

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.380-384.1143 ◽

2013 ◽

Vol 380-384 ◽

pp. 1143-1146

Author(s):

Xiang Guo Liu

Keyword(s):

Inverse Problem ◽

Diffusion Equation ◽

Perturbation Method ◽

Numerical Simulations ◽

Numerical Solution ◽

Two Dimensional ◽

Convection Diffusion ◽

Convection Diffusion Equation ◽

Parameter Inversion ◽

Parametric Inversion

The paper researches the parametric inversion of the two-dimensional convection-diffusion equation by means of best perturbation method, draw a Numerical Solution for such inverse problem. It is shown by numerical simulations that the method is feasible and effective.

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