Mathematics

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Browsing Mathematics by Author "Malak Tawfiq Abdelfattah Meqbil"

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    Optimal Homotopy Asymptotic Method for Solving Multidimensional First Order Systems of Partial Differential Equations
    (Al-Quds University, 2024-07-15) Malak Tawfiq Abdelfattah Meqbil; ملاك توفيق عبدالفتاح مقبل
    In this thesis, a semi-analytic approximating method, namely Optimal Homotopy Asymptotic Method (OHAM), which is developed from the Homotopy Analysis Method (HAM), is used to find continuous approximate solutions for linear and non-linear first-order systems of partial differential equations. Within this work, the geometrical topological homotopy concept is used to construct the algorithm for solving such systems. A homotopy equation that depends on an embedding parameter belonging to interval [0, 1] is assumed. As the parameter varies from 0 to 1 the solution of the homotopy equation (which is assumed to be a power series of the embedding parameter) varies continuously from a solution, which is easy to find, to the exact solution. The approximate continuous solution is obtained by truncating the series and using a finite number of its terms. Least Squares Method is used to determine the so-called control-convergence parameters that appear in the approximate solution. The derived algorithm is applied to solve some examples and the obtained solutions are compared with exact solutions. The results confirm the validity of OHAM and reveal that OHAM is effective, simple, and explicit. Moreover, it is independent of the small parameters required in perturbation methods. Furthermore, the convergence domain of OHAM is easily modifiable, depending on the convergence-control parameters that appear in the approximated solutions, enhancing its versatility.