چكيده به لاتين
Integral equations have amazingly far been reaching applications in the numerous areas, such as mathematics, physics, engineering and economics. So, solving this equations and designing appropriate numerical methods is considered a major branch in scientific research. The objective of this thesis is to solve one-dimensional and two-dimensional integral equations using direct and collocation methods. Due to strength points of the hybrid function of block-pulse functions and Legendre polynomials, the applied methods are Based on the use of these functions. Firstly, by introducing the hybrid functions, numerical solution of Fredholm integral equations of the first kind is expresed using these functions and direct method. Three-dimensional Fredholm integral equation is converted by the Legendre wavelet into an algebraic equation system. This approach is attractive and simple because of the Legendre wavelet orthogonality. The efficiency of the Legendre wavelet method is verified by a comparison of this method with the radial basis functions method. Then we develop the tension Spline approximation to obtain the numerical solution of Volterra-Fredholm integral equation. First, the tension Spline method was obtained, then the combination of this method with the quasi-linearization method has been used for solving the nonlinear Volterra-Fredholm integral equation. The error analysis of the method is given. Illustrative examples are included to verify the effectiveness and applicability of the presented approach. After that we present numerical solutions for Abel integral equations by hybrid block-pulse functions and Legendre polynomials. Hybrid functions give us the opportunity to attend a highly accurate solution by adjusting the orders of block-pulse functions and Legendre polynomials. Hybrid functions allow us to choose arbitrary polynomial basis functions and to adjust the order of polynomial basis functions and block-pulse functions to achieve an appropriate solution. At the end of each chapter, For showing the accuracy and efficiency of presented procedures numerical examples are given and compared with some other methods.
