Open Access Articles Version 1 https://doi.org/10.12345/hvycys24

Approximate solution of a class of two-dimensional non-linear variable order fractional reaction-diffusion equations in porous media

Vol. 1 No. 1 (2026): Volume 1, Issue 1, (January - June) 2026 Prashant Pandey1 Suraj Prasad2 Priya Kumari3
  1. K.S. Saket P.G. College, Ayodhya, India , IN
  2. K.S. Saket P.G. College, Ayodhya, India , IN
  3. Jhunjhunwala Group of Institution, Faizabad, India , IN
Published: June 30, 2026

Abstract

In the present scientific article, an efficient operational matrix based on the famous Bernstein polynomials is applied for the numerical solution of two-dimensional non-linear variable  order reaction-diffusion equation in porous media with given initial and boundary conditions. An operational matrix is constructed for fractional variable order differentiation w.r.to space variable x, y and time t, so that our proposed model is converted into a system of non-linear algebraic equations with the help of collocation method, which can be solved employing the Newton-Iteration method.  The salient features of the article are finding the stability analysis and error bounds of the proposed method and also the validation and the effectiveness of the method through the RMS,  L_{\infty}  and L 2 errors.  The physical presentation of the these errors for considered two-dimensional non-linear variable order reaction-diffusion with their exact solutions shows the method is too good for finding the solution of these kind of problems. 

Keywords:
Fractional calculus Variable order derivative Sub-diffusion Porous Media Stability and Convergence Analysis

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