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|Title:||Haar Wavelet Approach for Numerical Solution of Three-Dimensional Patial Differential Equations|
|Publisher:||University of Peshawar, Peshawar.|
|Abstract:||In this dissertation new algorithms have been presented for the numerical solution of three-dimensional elliptic and parabolic partial differential equa- tions subject to Dirichlet boundary conditions. A numerical method based on Haar wavelet collocation technique(HWCT) is being formulated for numer- ical solution of 3D elliptic partial di erential equations(PDEs) and systems involving such equations. The newly developed numerical technique is ap- plied to both linear and nonlinear 3D elliptic PDEs and systems involving such PDEs. The proposed numerical is also applied to the time-invariant fully nonlinear Navier-Stoke's model equations. In case of solving linear el- liptic PDEs, the resulting algebraic system of equations is solved by using Gauss elimination method. Whereas for nonlinear elliptic PDEs we used Newton's or Broyden's method. A hybrid numerical method based on nite di erence method and 3D HWCT is developed for the numerical solution of 3D parabolic PDEs and systems involving such PDEs. This hybrid numerical technique is applied to both linear and nonlinear parabolic PDEs including systems. In case of non- linear parabolic PDEs quasilinearization technique was applied to linearize the nonlinear terms. The time derivatives involved in parabolic PDEs is ap- proximated by the finite difference method. The numerical simulations of all newly developed numerical techniques are performed using MATLAB. The efficiency and accuracy of the proposed numerical methods is vali- dated via various linear and nonlinear test problems including systems from the literature. The numerical results of these test problems are compared with the exact solutions as well as with the existent methods in literature. The maximum absolute errors and experimental rate of convergence have been calculated for different number of collocation points. The numerical re- sults shows better accuracy, efficiency and simple applicability of the newly developed numerical methods.|
|Gov't Doc #:||16619|
|Appears in Collections:||PhD Thesis of All Public / Private Sector Universities / DAIs.|
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