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Title: Local Kernel - Based Space - Time Methods for Initial and Boundary Value Problems
Authors: Ali, Hazrat
Keywords: Physical Sciences
Issue Date: 2020
Publisher: University of Engineering & Technology Peshawar
Abstract: The numerical approximation of initial and boundary value problems (IBVPs) is the most studied area of mathematics. These problems have significance in all physical sciences like the electromagnetic field, fluid flow, computer graphics and aircraft simulation. The present research constructs numerical schemes using a localized setting, which give rise to the small size of the matrices. The small-sized interpolation matrices would help us in approximating thousands of scattered data nodes in regularly and irregularly shaped domains. In the first part, an approximate scheme is constructed in the form of local setting using radial kernels for spatial approximation in the regular and irregular domains for wave type equations and non-linear biological population model and Runge-Kutta (RK-4) time-stepping procedure has been adopted. In the second part, a numerical scheme is used for the approximation of spatial as well as temporal operators (time is to be taken as a spatial variable) via radial kernels in the local setting. An advanced feature of the proposed meshless numerical scheme is to approximate the PDEs without any iterative scheme. This method is free of issues relating to stability due to time-stepping schemes. The global method gives dense matrix while local method results in sparse differentiation matrices, which would be very handy to solve large scale high dimensional problems with thousands of scattered interpolation nodes. The accuracy, stability and efficiency of the space-time method is demonstrated by solving examples related to Burgers’ equation, Black-Scholes model, vii backward heat conduction problem and KDV equations. The obtained results are in good comparison to the existing results in the literature. Keywords: Space-time meshless method, kernel function, complex-shaped domain, local meshless scheme, wave equation, non-linear biological population model, Burgers’ equation, Black-Scholes equation, non-linear PDEs.
Gov't Doc #: 20181
Appears in Collections:PhD Thesis of All Public / Private Sector Universities / DAIs.

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