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 study, a new approach based on the REPRODUCING KERNEL HILBERT SPACE METHOD is proposed to approximate the solution of the second kind fuzzy linear integral equations. For this purpose, at first by applying the concept of parametric form, the fuzzy integral equation is converted to a system of crisp integral equations. Then, this system is solved by using the REPRODUCING KERNEL METHOD free of the Gram-Schmidt orthogonalization process. Also, two numerical algorithms are proposed based on applying the Gram-Schmidt process and without using it. The general form of numerical solution accordingly the REPRODUCING KERNEL METHOD is introduced and the convergence theorem of solution of the proposed scheme to the exact solution is proved. Finally, a sample fuzzy integral equation is solved by means of both suggested algorithms and the results are compared for differents points and levels. Due to the difficulties in applying the Gram-Schmidt process, the obtained results of the new algorithm are satisfactory.

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.

In this study, we solve the nonlinear fractional differential equations with fractional integral boundary conditions. To solve the mentioned problems, we use an iterative METHOD based on the REPRODUCING KERNEL HILBERT SPACEs. In this METHOD, the REPRODUCING KERNEL of a finite-dimensional HILBERT SPACE is constructed using Fibonacci polynomials. With the help of the obtained positive definite KERNEL, we produce bases that exactly satisfy the given integral boundary conditions. Then using the obtained bases, we construct fractional derivative operational matrices and obtain an approximation of the problem with the help of a simple iteration METHOD. In fact, we construct an approximation of the solution in a finite-dimensional SPACE. We have also shown the convergence of the METHOD under certain conditions. To show the effectiveness of the proposed METHOD, we have solved some examples, and the obtained results are ‎presented.‎

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.

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.