Radial Basis Functions Based Schemes for Fractional Differential Equations

Abstract

The primary objective of the present thesis is to explore on radial basis functions based numerical schemes for certain types of fractional differential equations. Unlike classical derivatives, non-local nature of the fractional derivatives makes extension of the existing schemes to fractional models complex and computationally expensive. In addition, compared to time fractional differential equations, attempts on RBF schemes for space and space-time fractional differential equations are less in the literature. This may be due to the difficulty in handling multidimensional space fractional derivatives because of the vector integral representation. Two approaches, namely, direct and integrated RBF collocation methods (DRBF and IRBF) are extended to approximate fractional order derivatives. In particular, we have proposed these schemes for nonlinear fractional models: fractional nonlinear ODEs (both initial and boundary value problems) and fractional Darboux problem. These nonlinear fractional DEs are appropriately approximated by a sequence of linear fractional DEs that converges to the solutions of the problem. The proposed sequences are generated via either generalised quasilinearisation or successive approximation techniques. In all these cases, existence and uniqueness of the solution and convergence of the proposed sequences are proved for continuous case. The numerical solutions thus obtained are extensively studied and analysed in terms of accuracy, convergence, time complexity as well as shape parameter dependency. While being capable to provide highly accurate approximations with exponential convergence rate, these characteristics of RBF based schemes are overwhelmed by infamous instability due to ill-conditioning of the governing system. Hence another important contribution to the thesis includes putting forth two algorithms based on Tikhonov regularisation and RBF-QR method to approximate fractional order derivatives. Using Chebyshev-Gauss quadrature, RBF-QR method is generalised to include all types of radial functions, wherein the algorithm was earlier restricted to Gaussian RBF. Then the proposed algorithms are validated using various fractional imodels by computing solutions for significantly small shape parameters. Also they are analysed to see the effect of increase in nodal points.