Solving Fredholm integro-differential equations using reproducing kernel Hilbert space method

In this study, the numerical solution of Fredholm integro–differential equation is discussed in a reproducing kernel Hilbert space. A reproducing kernel Hilbert space is constructed, in which the initial condition of the problem is satisfied. The exact solution u x ð Þ is represented in the form of series in the space W 2 2 ½a; bŠ. In the mean time, the n-term approxima te solution u n ðxÞ is obtained and is proved to converge to the exact solution uðxÞ. Furthermore, we present an iterative method for obtaining the solution in the space W 2 2 ½a; bŠ. Some examples are displayed to demonstrate the validity and applicability of the proposed method. The numerical result indicates that the proposed method is straightforward to implement, efficient, and accurate for solving linear and nonlinear Fredholm integro–dif-ferential equations. Integro–differential equation (IDE) has a great deal of application in different branches of sciences and engineering. It arises naturally in a variety of models from biologica l science, applied mathemati cs, physics, and other discipline s, such as theory of elasticity , biomechanics, electromagnet ic, electrodynamics , fluid dynamics, heat and mass transfer, oscillating magnetic field, etc. [1–4]. This class of equations is sometimes too complicated to be solved exactly because, generally, the solution cannot be exhibited in a closed form even when it exists. Therefore, finding either the analytical approximat ion or numerical solution of such equations are of great interest. In this paper, we are concerned with providing the numerical solution based on the use of reproducing kernel Hilbert space (RKHS) method for Fredholm IDEs of the general form