الفهرس 
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Abstract
the aim of the present thesis is to investigate the features of spectral collocation method for numerical solutions of different types of integral equations (IEs), integro-differential equations and fractional integro-differential equations subject to various kinds of non-local conditions. The speed of convergence is one of the great advantages of spectral collocation method. Moreover, it has exponential rates of convergence; it also has high level of accuracy.
In Chapter 1, we give a general introduction to IE, fractional calculus, and spectral methods with their advantages over the standard numerical methods. The orthogonal polynomials, their properties and expansion of functions in terms of them are introduced. Some general properties and important relations concerned with the Jacobi polynomials and shifted Jacobi polynomials are considered.
In Chapter 2, we propose two efficient algorithms for solving Abel’s and Volterra IEs. Also, we construct a numerical technique for solving two-dimensional fractional IEs with weakly singular. Using the novel methods, we can reduce such problems into those of a system of algebraic equations which greatly simplifies the problem. For testing the accuracy and validity of the proposed numerical technique, we apply it to solve several numerical examples. Thus the novel algorithm is more responsible for solving two-dimensional fractional IEs with weakly singular.
In Chapter 3, we propose shifted Legendre-Gauss-Lobatto collocation (SL-GL-C) method to solve the Fredholm and system of Volterra-Fredholm IEs. The collocation points are selected at the shifted Legendre-Gauss-Lobatto (SL-GL) interpolation nodes. The solution of such equation is approximated as a finite expansion of shifted Legendre polynomials for independent variables, and then we evaluate the residuals of the mentioned problem at the SL-GL quadrature points. Special attention is given to the comparison of the numerical results obtained by the new algorithms with those found by other known methods.
In Chapter 4, A numerical approach based on shifted Jacobi-Gauss collocation (SJ-G-C) method for solving one and two-dimensional mixed Volterra-Fredholm IEs (MV-F-IEs) is presented. The novel method together with the shifted Jacobi-Gauss (SJ-G) nodes is utilized to reduce the MV-F-IEs to a system of algebraic equations that can be easily solved. The present algorithm is extended to solve the two-dimensional MV-F-IEs. Convergence analysis for the present method is discussed and confirmed the exponential convergence of the spectral algorithm. Several numerical examples are presented to demonstrate the powerful and accuracy of the method.
Finally in Chapter 5, A new SJ-G-C algorithm is presented to numerical solve of several classes of fractional integro-differential equations (FI-DEs), namely Voltarra FI-DEs, Fredholm FI-DEs and system of Volterra FI-DEs subject to with initial and nonlocal boundary conditions. The present algorithm is also extended to solve mixed Voltarra-Fredholm FI-DEs. As a collocation nodes, we employ SJ-G points to reduce the aforementioned problem to a system of algebraic equations. The error analysis of the algorithm presented is analyzed. The Numerical comparisons listed here show that our numerical solutions produce high accurate results. Thus the novel algorithm is more responsible for solving FI-DEs.
The obtained numerical results are tabulated and displayed graphically whenever possible. These results show that the proposed algorithms of solutions are reliable accurate. Comparisons with previously obtained results by other researchers or exact known solutions are made throughout the context whenever available.
To the best of our knowledge, the formulae and algorithms stated and proved in Chapters 2 up to 5 are completely new. The Programs used in this thesis are performed using the PC machine, with Intel(R) Core i3 CPU 2.00 GHz, 2.00 GB of RAM, and the symbolic computation software Mathematica 9.0 has been also used.