Tahereh Eftekhari

Bio: Tahereh Eftekhari is an academic researcher from Iran University of Science and Technology. The author has contributed to research in topics: Nonlinear system & Numerical analysis. The author has an hindex of 5, co-authored 11 publications receiving 60 citations.

Existence, uniqueness, and numerical solutions for two-dimensional nonlinear fractional Volterra and Fredholm integral equations in a Banach space

Abstract: The purpose of this research is to provide sufficient conditions for the local and global existence of solutions for two-dimensional nonlinear fractional Volterra and Fredholm integral equations, based on the Schauder’s and Tychonoff’s fixed-point theorems. Also, we provide sufficient conditions for the uniqueness of the solutions. Moreover, we use operational matrices of hybrid of two-dimensional block-pulse functions and two-variable shifted Legendre polynomials via collocation method to find approximate solutions of the mentioned equations. In addition, a discussion on error bound and convergence analysis of the proposed method is presented. Finally, the accuracy and efficiency of the presented method are confirmed by solving three illustrative examples and comparing the results of the proposed method with other existing numerical methods in the literature.

Control of accuracy on Taylor-collocation method to solve the weakly regular Volterra integral equations of the first kind by using the CESTAC method

Abstract: Finding the optimal parameters and functions of iterative methods is among the main problems of the Numerical Analysis. For this aim, a technique of the stochastic arithmetic (SA) is used to control of accuracy on Taylor-collocation method for solving first kind weakly regular integral equations (IEs). Thus, the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method is applied and instead of usual mathematical softwares the CADNA (Control of Accuracy and Debugging for Numerical Applications) library is used. Also, the convergence theorem of presented method is illustrated. In order to apply the CESTAC method we will prove a theorem that it will be our licence to use the new termination criterion instead of traditional absolute error. By using this theorem we can show that number of common significant digits (NCSDs) between two successive approximations are almost equal to NCSDs between exact and numerical solution. Finally, some examples are solved by using the Taylor-collocation method based on the CESTAC method. Several tables of numerical solutions based on the both arithmetics are presented. Comparison between number of iterations are demonstrated by using the floating point arithmetic (FPA) for different values of $\varepsilon$.