Radial Basis Function Meshless Method To Solve Fractional Differential Equations

Fractional partial diferential equations (FPDEs) are extensively applied in modelingphysical processes and material characteristics referring to memory properties, geneticcharacters and path dependence. In this thesis, we analyze radial basis function (RBF)meshless method for time fractional partial diferential equations (TFPDEs) with Caputodefinition and spatial factional diferential equations (SFPDEs) with Riemann-Liouvilledefinition.For TFPDEs with Caputo definition, an approximation scheme of the time fractionalderivative is obtained by finite diference method. Then, radial basis function approxi-mation is implemented on spatial terms. Combining with initial conditions and boundaryconditions, a full discrete scheme is constructed. The unconditional stability of this novelnumerical scheme is proved by using discrete Fourier transform and Gerschgorin theo-rem. The local truncation errors of this scheme are also obtained in this thesis by usingTaylor expansion and RBF theory, which depend on the time step and space step and theinvolved RBF.For SFPDEs with Riemann-Liouville definition, the RBF interpolation is imposedon spatial terms. We utilize Gaussian integration formula to approximate the relatedintegral. A full discrete scheme is obtained by imposing initial conditions and boundaryconditions. We prove that this novel scheme is stable under the condition that the shapeparameter in RBF and time step and space step satisfies some relation such that the spectraradius of the related matrix is less than1. The local truncation errors are also dependenton the involved RBF.Finally, several numerical examples for Fractional ordinary diferential equation,TFPDE with Dirichlet and Neumann boundary conditions and SFPDE with Dirichletboundary conditions are given to demonstrate the efciency of the proposed numericalschemes.