Gegenbauer wavelet operational matrix method for solving variable-order non-linear reaction-diffusion and Galilei invariant advection-diffusion equations

In this article, a variable-order operational matrix of Gegenbauer wavelet method based on Gegenbauer wavelet is applied to solve a space-time fractional variable-order non-linear reaction-diffusion equation and non-linear Galilei invariant advection diffusion equation for different particular cases. Operational matrices for integer-order differentiation and variable-order differentiation have been derived. Applying collocation method and using the said matrices, fractional-order non-linear partial differential equation is reduced to a system of non-linear algebraic equations, which have been solved using Newton iteration method. The salient feature of the article is the stability analysis of the proposed method. The efficiency, accuracy and reliability of the proposed method have been validated through a comparison between the numerical results of six illustrative examples with their existing analytical results obtained from literature. The beauty of the article is the physical interpretation of the numerical solution of the concerned variable-order reaction-diffusion equation for different particular cases to show the effect of reaction term on the pollution concentration profile.