FIBONACCI COLLOCATION METHOD TO SOLVE TWO-DIMENSIONAL NONLINEAR FRACTIONAL ORDER ADVECTION-REACTION DIFFUSION EQUATION

In this article we used a spectral collocation method with the help of an operational matrix method of Fibonacci polynomial to solve two-dimensional non-linear fractional order advection reaction diffusion equation. The effects on solute concentration for various fractional order spatial and time derivatives and also for conservative and non-conservative cases are found computationally and the results are displayed graphically for different cases. With the spectral collocation method, the property of Kronecker product of matrices of Fibonacci polynomials to approximate the solution is used. Since the proposed method is based on spectral collocation method, the residue, initial and boundary conditions of the proposed mathematical model are collocated at a certain number of collocation points. As a result, a system of nonlinear algebraic equations are obtained which is solved with the help of Newton's method. The efficiency of the proposed method is validated by applying it on three existing problems to compare the numerical solution with analytical solutions of the problems through error and convergence analysis in the minimum computational time to show the simplicity, effectiveness, and higher accuracy of the concerned method.