Extended Haar Wavelet Quasilinearization method for solving boundary value problems / Nor Artisham Che Ghani

Nor Artisham , Che Ghani (2018) Extended Haar Wavelet Quasilinearization method for solving boundary value problems / Nor Artisham Che Ghani. PhD thesis, University of Malaya.

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Abstract

Several computational methods have been proposed to solve single nonlinear ordinary differential equations. In spite of the enormous numerical effort, however yet numerically accurate and robust algorithm is still missing. Moreover, to the best of our knowledge, only a few works are dedicated to the numerical solution of coupled nonlinear ordinary differential equations. Hence, a robust algorithm based on Haar wavelets and the quasilinearization process is provided in this study for solving both numerical solutions; single nonlinear ordinary differential equations and systems of coupled nonlinear ordinary differential equations, including two of them are the new problems with some additional related parameters. In this research, the generation of Haar wavelets function, its series expansion and one-dimensional matrix for a chosen interval 0, B is introduced in detail. We expand the usual defined interval 0, 1 to 0, B because the actual problem does not necessarily involve only limit B to one, especially in the case of coupled nonlinear ordinary differential equations. To achieve the target, quasilinearization technique is used to linearize the nonlinear ordinary differential equations, and then the Haar wavelet method is applied in the linearized problems. Quasilinearization technique provides a sequence of function which monotonic quadratically converges to the solution of the original equations. The highest derivatives appearing in the differential equations are first expanded into Haar series. The lower order derivatives and the solutions can then be obtained quite easily by using multiple integration of Haar wavelet. All the values of Haar wavelet functions are substituted into the quasilinearized problem. The wavelet coefficient can be calculated easily by using MATLAB software. The universal subprogram is introduced to calculate the integrals of Haar wavelets. This will provide small computational time. The initial approximation can be determined from mathematical or physical consideration. In the demonstration problem, the performance of Haar wavelet quasilinearization method (HWQM) is compared with the existing numerical solutions that showed the same basis found in the literature. For the beginning, the computation was carried out for lower resolution. As expected, the more accurate results can be obtained by increasing the resolution and the convergence are faster at collocation points. For systems of coupled nonlinear ordinary differential equations, the equations are obtained through the similarity transformations. The transformed equations are then solved numerically. This is contrary to Runge-Kutta method, where the boundary value problems of HWQM need not to be reduced into a system of first order ordinary differential equations. Besides in terms of accuracy, efficiency and applicability in solving nonlinear ordinary differential equations for a variety of boundary conditions, this method also allow simplicity, fast and small computation cost since most elements of the matrices of Haar wavelet and its integration are zeros, it were contributed to the speeding up of the computation. This method can therefore serve as very useful tool in many physical applications.

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