الفهرس | Only 14 pages are availabe for public view |
Abstract The following is a summary of the main objectives of this thesis: • A survey study on orthogonal polynomials and in particular Jacobi poly- nomials and their classes. • A theoretical study of even-order differential equations subject to initial and boundary conditions. • A comprehensive study of spectral methods, particularly Galerkin, tau, and collocation methods. • Establishing operational matrices of derivatives for certain basis functions that satisfy the homogeneous initial and boundary conditions. • Application of some suitable algorithms for the numerical treatment of the linear and non-linear second-order boundary value problems. • Proposing a numerical algorithm based on the Jacobi Galerkin method to solve the linear second-order hyperbolic telegraph differential equations governed by initial and boundary conditions. • Comparing our algorithms with some other algorithms to demonstrate the applicability and accuracy of the proposed methods. The thesis consists of three chapters: Chapter 1: This chapter focuses on the following issues: • Presenting some fundamental properties and formulas concerned with Ja- cobi polynomials and their celebrated classes of orthogonal polynomials. • Highlighting the spectral methods and their advantages over other stan- dard methods. Chapter 2: The following points summarize the key objectives of this chapter: • Introducing kinds of orthogonal polynomials, namely, shifted Chebyshev polynomials of the third and fourth kinds. • Establishing two new operational matrices of derivatives of these shifted polynomials. • Developing two numerical algorithms for solving the linear and non-linear second-order BVPs in one dimension. • Extending the algorithm in one dimension to be capable of treating the second-order two-dimensional problems. • Presenting some numerical results to investigate the applicability and accuracy of the presented algorithms. The results of this chapter are published in: H. Ashry, W.M. Abd-Elhameed, G.M. Moatimid, and Y.H. Youssri. Spectral treatment of one and two dimensional second-order BVPs via certain modified shifted Chebyshev polynomials., Int. J. Appl. Comput. Math, 7(6):1–21, 2021. Chapter 3: The following points summarize the key objectives of this chapter: • Converting the linear hyperbolic telegraph type equation governed by its underlying conditions to a modified equation governed by boundary conditions only. • Implementing a numerical technique built on applying the shifted Jacobi Galerkin method for solving the one-dimensional linear second-order hy- perbolic telegraph differential equations. • Investigating the error analysis of the proposed Jacobi expansion. • Present some illustrative examples to investigate the applicability and accuracy of the method. The results of this chapter are published in: H. Ashry, W.M. Abd-Elhameed, G.M. Moatimid, and Y.H. Youssri. Robust shifted Jacobi-Galerkin method for solving linear hyperbolic telegraph type equation. PJM, 11(3): 504–518, 2022. |