AN OPERATIONAL MATRIX APPROACH TO SOLVE A 2D VARIABLE-ORDER REACTION ADVECTION DIFFUSION EQUATION WITH VIETA-FIBONACCI POLYNOMIALS

A reaction-advection-diffusion equation describes many physical phenomena, such as the transportation of particles, groundwater pollution, viscoelasticity, and many others. In this study, a well-known fractional operator of variable order is used to present the space-time variable-order reaction-advection-diffusion equation. The operational matrix of the variable order derivative is developed with the aid of shifted Vieta-Fibonacci polynomials. This operational matrix is used in the approximation of derivatives of variable order to construct residual associated with the considered problem, and then it is collocated at some points in the domain, which generates a system of non-linear algebraic equations. Newton's method is applied to solve the obtained system of non-algebraic equations. To validate the precision of the proposed scheme, some problems are solved by the proposed scheme, and its comparisons are made with the existing analytical solution, which clearly indicates the improved accuracy of the proposed method. The convergence of the scheme and error analysis are also discussed in this paper.