چكيده به لاتين
In this thesis, first, we introduce some polynomials, basis functions, and wavelets, such as three-variable Bernstein polynomials, one- and two-variable Jacobi polynomials, one- and two-variable fractional Jacobi polynomials, hybrid of one- and two-dimensional block-pulse functions and Legendre polynomials, Müntz-Legendre wavelets, and the second kind Chebyshev wavelets. Also, we explain how to approximate functions by using them. Moreover, due to the advantages of using operational matrices based on orthogonal basis functions, we produce operational matrices and operational vectors of integration, fractional integration, and product based on the mentioned polynomials, basis functions, and wavelets. Next, we provide sufficient conditions for the local and global existence of solutions for two-dimensional nonlinear fractional Volterra and Fredholm integral equations, based on Schauder's and Tychonoff's fixed-point theorems. Then, we solve two-dimensional nonlinear fractional Volterra and Fredholm integral equations, two-dimensional nonlinear fractional integro-differential equations, distributed order fractional differential equations of the general form in the time domain using these operational matrices via collocation method. We also introduce formulas for the fractional Taylor series for two- and multi-variable functions. In addition, we solve three-dimensional Volterra-Fredholm integral equations of the first and second kinds using three-variable Bernstein polynomials. In each case, the error bound and convergence analysis of the proposed method are investigated. At the end of each chapter, numerical examples are given and compared with some other methods to show the accuracy and efficiency of the proposed methods.
