LOCALIZED APPROXIMATION OF TIME DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS USING RADIAL KERNELS

Abstract

Time dependent partial differential equations (PDEs) model systems that experience change as a function of time. Time dependent PDEs have numerous applications such as diffusion, heat transfer, thermodynamics, population dynamics and wave phenomena. They are naturally parabolic or hyperbolic. Meshless methods have large advantages in accuracy over other methods, such as finite difference method (FDM), finite volume method (FVM), finite element method (FEM). The main features of the meshless methods are its simplicity, efficiency and invariance under euclidian transformation and can handle problems defined on complex shape domains. Meshless methods have some serious drawbacks as well. When the nodes are increased the method solve comparatively large system, and the ill-conditioning of the system matrix causes instability. Due to which it becomes difficult to achieve spectral convergence. This thesis is concerned with two issues that is to solve the ill- conditioning problem of the interpolation matrix by radial kernels in local setting and to replace the time marching scheme with the numerical inversion of Laplace transforms which eliminates temporal truncation errors and the need for many time integration steps. The method is applied to solve fractional and integer order time dependent PDEs. The method comprises of three steps. First the Laplace transform is applied to the partial differential equation and boundary conditions in a given differential system. Second, the kernel based method is employed to solve the transformed differential system. Third, the solution is represented as a contour integral evaluated to high accuracy by trapezoidal rule.