Meshless Computational Techniques for Solving Fractional Partial Differential Equations Arising in Certain Physical Systems

Abstract

Starting from fluids flow in nature, from waves and winds to blood and lava to the diffusion in a tokamak, partial differential equations (PDEs) describe almost every nonlinear phenomenon in the real world. However, the physical behavior of many scientific processes, such as anomalous diffusion, fluid dynamics, plasma physics, mechanics, and chemical kinetics, is better characterized using a non-integer order dynamic model based on fractional calculus. Solving fractional PDEs (FPDEs) is crucial to enhance the understanding of these processes. In this dissertation, meshless computational techniques are developed to solve the FPDEs with applications in shallow water waves, plasma physics, nonlinear optics, fluid mechanics, and chemotaxis. Methods based on radial basis functions (RBFs) are emphasized due to their higher accuracy, simpler extension to higher dimensions, and ability to deal with complex geometries. This dissertation mainly implements Kansa’s global RBF and local RBF partition of unity (LRBF-PU) techniques to simulate FPDEs. In some cases, analytical approaches, such as the Kudryashov method and the tanh method, have been used to obtain explicit solutions for comparison with the numerical results. The Kansa RBF approach has been applied to solve the fractional Gilson Pickering equation and fractional Schamel–KdV equation with applications in plasma physics and the fractional Newell–Whitehead–Segel equation with applications to binary fluid mixtures. Also, the fractional Dullin–Gottwald–Holm equation, fractional coupled KdV–mKdV system, fractional Oskolkov–Benjamin–Bona–Mahony–Burgers equation, and fractional Ito equation describing surface waves in shallow water have been numerically solved. Higher dimensional FPDEs in mathematical physics: fractional Nizhnik–Novikov–Veselov equations and fractional modified Konopelchenko Dubrovsky equations are also investigated numerically. Moreover, the LRBF-PU method has been employed to solve the fractional Benjamin–Ono equation describing long internal waves in deep stratified fluids and the fractional Keller–Segel model for chemotaxis. The stability and convergence of the numerical schemes are theoretically established, and numerical experiments are performed. Numerical simulations and graphical comparisons manifest that the proposed methods are promising in handling FPDEs. Also, the findings are helpful in understanding various physical systems occurring in the real world.