Numerical solutions of riesz fractional diffusion and advection-dispersion equations in porous media using iterative reproducing kernel algorithm

This paper presents an iterative reproducing kernel algorithm for obtaining the numerical solutions of Riesz fractional diffusion and advection-dispersion equations in porous media on a finite domain. The representation of the exact and the numerical solutions is given in the W (Ω) and H (Ω) inner product spaces. The computation of the required grid points relies on the R(y;s) (x; t) and r(y;s) (x; t) reproducing kernel functions. An efficient construction is given to obtain the numerical solution together with an existence proof of the exact solution based upon the reproducing kernel theory. Numerical solution of such Riesz fractional equations is acquired by interrupting the n-term of the exact solution. In this approach, numerical examples were analyzed to illustrate the design procedure and confirm the performance of the proposed algorithm in the form of tabulated data, numerical comparisons, and graphical results. Finally, the utilized results show the significant improvement of the algorithm while saving the convergence accuracy and time.