Numerical solution of the nonlinear diffusion equation by using non-standard/standard finite difference and Fibonacci collocation methods

In this article, the authors have introduced a new nonstandard/standard finite difference scheme with the Fibonacci polynomial. The considered nonlinear fractional diffusion equation is reduced into a system of linear ordinary differential equations which are solved by using nonstandard/standard finite difference method. The developed schemes are unconditionally stable and the effectiveness and efficiency of the method is confirmed by applying it in two linear and one nonlinear problems and through comparison of their numerical results with existing analytical results. After validation of the efficiency of the method, the proposed method is used to solve the highly nonlinear fractional order diffusion equation. The stability of the method is also discussed. The salient feature of the article is the graphical exhibitions of the effects on the solute profiles due to increasing in non-linearity of the model and of the order of the spatial derivative for different particular cases.