Numerical Solutions of Integro-Differential Equations Using RKHSM

Numerical analysis: In this work, we make a discussion for the theory of reproducing Kernel Hilbert spaces associated with positive definite kernels and we illustrate their applications for a class of integro differential equations. We begin with the material that is contained in Aronszajnís classic paper on the theory of reproducing Kernel Hilbert spaces. In fact, we focus on their properties, generation of new spaces and relationships between their kernels and some theorems on extensions of functions and kernels. Moreover, we study the Sobolev space which is one of the most useful reproducing Kernel Hilbert spaces, construct a novel reproducing kernel space and give the way to express reproducing Kernel functions. Meanwhile, we employed a reproducing Kernel function and its conjugate operator to construct the complete orthonormal basis. This work investigates the solutions of a general form of first, second and fourth-order integro-differential equations using the reproducing Kernel Hilbert space method.