An adaptive numerical approach for the solutions of fractional advection–diffusion and dispersion equations in singular case under Riesz's derivative operator

The fractional diffusion and dispersion equations are reinterpreted in determining the effect of fluid flow and displacement processes through certain compressible phenomena and then reconstructed by considering the flow conductivity, energy balance, flow chambers with the interconnected pores, and diffusion flow system. The adaptive reproducing kernel approach is formulated and analyzed to investigate numerical solutions of fractional advection-diffusion and dispersion equations in singular case on a finite domain with Riesz's fractional derivative. In such alternative representation, the reproducing kernel functions are obtained to provide analytic and approximate solutions in desired Hilbert spaces. To enable the utilized approach more, convergent analysis and error estimates are also given. To assure our results, some features with numerical experiments are presented to confirm the theoretical analysis and to illustrate the performance and effectiveness of the proposed scheme. Graphical and comparisons indicate the significant improvement of the algorithm in solving many singular fractional problems arising in physical issues.