الفهرس 
FundamentalsOnly 14 pages are availabe for public view
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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.