In this paper, we will present a new method for a Volterra integral equation with weakly singular KERNEL in the REPRODUCING KERNEL space. Firstly the equation is transformed into a new equivalent equation. Its exact solution is represented in the form of series in the REPRODUCING KERNEL space. In the mean time, the n-term approximation un(t) to the exact solution u (t) is obtained. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other.

Purpose: In this paper, we shall present an algorithm for solving more general singular second-order multi-point boundary value problems.Methods: The algorithm is based on the quasilinearization technique and the REPRODUCING KERNEL method for linear multi-point boundary value problems.Results: Three numerical examples are given to demonstrate the efficiency of the present method.Conclusions: Obtained results show that the present method is quite efficient.

In the beginning, we describe the fuzzy inner product space and the fuzzy Hilbert space. Our goal is to use the fuzzy REPRODUCING KERNEL method to solve the second-order fuzzy boundary value problem. The fuzzy convergence analysis of introduced method is discussed in detail. We present some examples in the end.

The aim of this paper is to use the REPRODUCING KERNEL Hilbert Space Method (RKHSM) to solve the linear and nonlinear fuzzy impulsive fractional differential equations. Finding the numerical solutions of this class of equations are a difficult topic to analyze. In this study, convergence analysis, estimations error and bounds errors are discussed in detail under some hypotheses which provide the theoretical basis of the proposed algorithm. Some numerical examples indicate that this method is an efficient one to solve the mentioned equations.

In this article, we offer an efficient method to find an approximate solution for quadratic optimal control problems. The approximate solution is offered in a finite series form in REPRODUCING KERNEL space. The convergence of proposed method is analyzed under some hypotheses which provide the theoretical basis of the proposed method for solving quadratic optimal control problems. Furthermore, in this study, we investigate the application of the proposed method to obtain the solution of equations that have formally been solved using Pontryagin’, s maximum principle. Moreover, many different types of quadratic optimal control problems are considered prototype examples. The obtained results demonstrate that the proposed method is truly effective and convenient to obtain the analytic and approximate solutions of quadratic optimal control problems.