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Title:	Bernstein polynomials method and its modifications for solving differential equations
Authors:	Mohammed Hamed Turky Alshbool (P68841)
Supervisor:	Ishak Hashim, Prof. Dr.
Keywords:	Differential equations Integer Fractional Bernstein polynomials Universiti Kebangsaan Malaysia -- Dissertations
Issue Date:	12-Jan-2015
Description:	Mathematical models of real world phenomena are often given by differential equations. Many researchers have focused their effort on finding suitable solution methods to the equations. In this thesis, solution methods based on Bernstein polynomials are proposed for solving differential equation of integer and fractional orders, and fractional partial differential equations. Firstly, an approximate solution depending on Bernstein olynomials and the collocation method are used for the numerical solution of singular and nonsingular differential equations. The method is given with two different error estimates. A residual correction procedure is used to estimate the absolute error. Illustrative examples are included to demonstrate the validity and applicability of the presented technique. Further, a new operational matrix method based on Bernstein polynomials is introduced to approximate the analytical solutions of fractional differential equations. The fractional extension of the standard Bernstein polynomials are defined and then transformed into matrix form by using the Caputo fractional derivative. Each of the terms of the problem to obtain the matrix form is converted by means of fractional Bernstein matrices. The method is given with error estimates. By using the residual correction procedure, the absolute error can be estimated. Next, a new modification of the Bernstein polynomials method, named the multistage Bernstein polynomials (MB-polynomials) method, is applied to solve stiff systems of differential equations. MB-polynomials offer a simple reliable modification based on the adaptation of the standard Bernstein polynomials method. This method depends on the use of the multiple intervals, which allows a number of subintervals. MB-polynomials are highly suited to large-domain calculations. The results obtained with MB-polynomials are compared with those obtained using the standard Bernstein polynomials method and other methods able to solve a stiff system of ordinary differential equations. The results attest to the accuracy of the proposed method. Finally, we present a new modification of the Bernstein polynomials method and Chebyshev interpolation nodes to solve fractional partial differential equations. We examine our method to study the physical meaning for any fractional order in partial differential equations. The procedure of the method is explained and illustrative examples are included to demonstrate the validity of the method. In summary, it has been demonstrated that Bernstein polynomials based method is an efficient method for solving linear and nonlinear differential equations of integer and fractional orders.,Certification of Master's/Doctoral Thesis" is not available
Pages:	173
Publisher:	UKM, Bangi
Appears in Collections:	Faculty of Science and Technology / Fakulti Sains dan Teknologi

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