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.
