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dc.contributor.authorAhsan, Muhammad-
dc.description.abstractDirect and inverse problems have been widely used in engineering fields and applied sciences in recent decades. Direct problems are well-posed and easy to solve but it is more challenging to deal with nonlinear direct problems. Inverse problems are more difficult because they are inherently ill-posed; that is, a small error perturbation will lead to a large amount of error in the solution reconstructed. Moreover, prediction of an unknown parameter in an inverse problem is not an easy task. A stable and accurate reliable solver in the form of Haar wavelet collocation methods (HWCMs) has been proposed for direct and inverse problems. The dissertation is categorized into six parts. The first and second parts include, the Burgers’ type equations and nonlinear Schro¨dinger equations. The Haar wavelets (HW) based numerical scheme along with the stability and convergence analysis has also been included with accurate numerical experiments. In the third part, the Inverse Heat Conduction Problems (IHCPs) with unknown time dependent heat source has been presented. A HW algorithm is developed for IHCP with unknown time dependent heat source. Furthermore, a suitable transformation is used to eliminate the unknown heat source to obtain a partial differential equation (PDE) without a heat source. After the elimination of the unknown non-homogeneous term, an implicit finite-difference approximation (FDA) is used to approximate the time derivative and HW are used for approximation of the space derivatives. The proposed method has small condition number of coefficient matrices for both one- and two-dimensional IHCPs. The stability analysis further strengthens the validity of the proposed HWCM. In the fourth part of the thesis, two different HW multi-resolution collocation pro cedures are proposed for IHCP with unknown source depending on space variables and unknown solution on the interior of the domain. A suitable transformation is used to eliminate the unknown heat source to obtain a homogeneous PDE without a heat source. In homogeneous PDE, the FDA is adopted to approximate the time derivative and HW are used for approximation of the space derivatives. Both the HWCMs produce a well conditioned coefficient matrix and no need to introduced the regularization procedure. ii Validity and applicability in the form of stability analysis of the proposed HWCM have also been discussed. In the fifth part, a new hybrid HWCM has been developed for the numerical solution of the IHCP with time-space dependent heat source. Due to the ill-posedness of the IHCP, capturing of the required time-space dependent source accurately is challenging. In the hybrid HWCM, a first order FDA is used for time dependent derivative and two different Haar series are used to approximate the space derivatives and the unknown source term. The present method is implemented on different types of heat sources, which are separable and non-separable with respect to space and time. The results of the proposed method are effective and stable under a quite large amount of noise levels. In the last part, a multi-resolution Haar wavelet-based collocation technique is proposed for nonlinear IHCP with a space dependent heat source. In this technique, we have used the FDA for time dependent derivative and two different Haar series for approximation of the space derivatives and unknown source term. Different types of test problems are considered to ensure accuracy and well-conditioned behavior of the present HWCM.en_US
dc.description.sponsorshipHigher Education Commission Pakistanen_US
dc.publisherUniversity of Engineering & Technology Peshawaren_US
dc.subjectPhysical Sciencesen_US
dc.titleHaar Wavelets Collocation Method for Solving Direct and Inverse Problemsen_US
Appears in Collections:PhD Thesis of All Public / Private Sector Universities / DAIs.

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