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Title: Analytical Solutions of Boundary Value Problems
Authors: Ali, Liaqat
Keywords: Physical Sciences
Issue Date: 2017
Publisher: Abdul Wali Khan University, Mardan
Abstract: In this thesis, Optimal Homotopy Asymptotic Method (OHAM) and new version of Optimal Homotopy Asymptotic Method are used to solve nonlinear boundary value problems in finite and infinite intervals. The new version of OHAM is extended to solve systems of differntial equations forming in the model for Magnetohydrodynamics thin film fluid flow under the effect of thermophoresis and variable fluid properties as will as thin film flow over a permeable stretching surface with variable fluid properties and magnetic field. It consists of initial guess, auxiliary functions (containing unknown optimal convergence- control parameters) and a homotopy. The unknown optimal convergence-control parameters can be determined by minimizing the residual. Many ways explained in the coming section can be used to determined these parameters. Here we have used Galerkin’s method and least square method for this purpose. The results are compared with the already existing Runge-Kutta Method (RK-4) and Optimal Homotopy Asymptotic Method (OHAM). The new version gives efficient and accurate first-order approximate solution. The outcomes gained by this method are in excellent concurrence with the exact solution and hence proved that this method is effective and easy. Also Optimal Homotopy Perturbation Method and an efficient modification of the existing optimal homotopy perturbation method by using Daftardar-Jafari polynomials are applied to solve non linear problems of differnt orders occuring in real world. The new algorithm, called modified optimal homotopy perturbation method is also developed for an nth-order integro-differential equation and is then applied xx xxi to linear and nonlinear two point boundary value problems for higher order integrodifferential equations. The results illustrate excellent accuracy and efficiency of the proposed method. These methods are also used to solve integral equations as well as integro-differential equations. This method can solve all existing nonlinear problems very easily and giving more accurate results. The results obtained by using modified optimal homotopy perturbation method are compared with the results obtained by the application of Adomain decomposition method, homotopy perturbation method, variational iteration method, differential transform method and homotopy analysis method etc. An easy and efficient technique, called series method is applied to get a reliable analytic approximate solution of linear and nonlinear integral and integro- differential equations as well as their system also which are arising in the phenomena of everyday life. The new technique does not need to create a homotopy with an embedding parameter as in Homotopy Perturbation Method (HPM) and Optimal Homotopy Asymptotic Method (OHAM). It neither needs to find the Adomain Polynomials to overcome the nonlinear terms in Adomain Decomposition Method (ADM). Unlike VIM, nor Lagrange Multiplier. In this paper no restrictive assumptions are taken for nonlinear terms. The proposed technique consists of a series only in which the unknown constants are determined by the usual way mentioned in the paper. The obtained results by this technique are in good agreement with the exact solution and it is proved that this technique is effective and easy to apply. Some problems are solved to prove the above claims and also the results are compared with exact solutions as well as with the results obtained by already existing different techniques. Further more this thesis also explores Liquid Film Flow of Williamson Fluid over an Unstable Stretching Surface in a Porous Space . The Brownian motion and Thermophoresis effect of the liquid film flow on a stretching sheet have been observed. This research include, to focus on the variation in the thickness of the liquid film in a porous space. The self-similarity variables have been applied to convert the modelled equations into a set of non-linear coupled differential equations. These non-linear xxii differential equations have been treated through an analytical technique known as Homotopy Analysis Method (HAM). The effect of physical non-dimensional parameters like, Eckert Number, Prandtl Number, Porosity Parameter, Brownian Motion Parameter, Unsteadiness Parameter, Schmidt Number, Thermophoresis Parameter, Dimensionless Film Thickness, and Williamson Fluid Constant on the liquid film size are investigated and conferred in this endeavor. The obtained results through HAM are authenticated, from its comparison with numerical (ND-Solve Method). The graphical comparison of these two methods is elaborated. The numerical comparison with absolute errors are also been shown in the tables. The physical and numerical results using h curves for the residuals of the velocity, temperature and concentration profiles are obtained. Math type and mathematica are used for calculations and numerical simulations.
Gov't Doc #: 21095
Appears in Collections:PhD Thesis of All Public / Private Sector Universities / DAIs.

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