Abstract
We present an efficient iterative method for solving a class of nonlinear second-order Fredholm integrodifferential equations associated with different boundary conditions. A simple algorithm is given to obtain the approximate solutions for this type of equations based on the reproducing kernel space method. The solution obtained by the method takes form of a convergent series with easily computable components. Furthermore, the error of the approximate solution is monotone decreasing with the increasing of nodal points. The reliability and efficiency of the proposed algorithm are demonstrated by some numerical experiments.
Highlights
The topic of integrodifferential equations (IDEs) which has attracted growing interest for some time has been recently developed in many applied fields, so a wide variety of problems in the physical sciences and engineering can be reduced to IDEs, in particular in relation to mathematical modeling of biological phenomena [1,2,3], aeroelasticity phenomena [4], population dynamics [5], neural networks [6], electrocardiology [7], electromagnetic [8], electrodynamics [9], and so on
It is important to study boundary value problems (BVPs) for especially the nonlinear IDEs, which can be classified into two types: Fredholm and Volterra IDEs, where the upper bound of the integral part of Fredholm type is a fixed number whilst it is a variable for Volterra type [10]
These types of IDEs arise in the theories of singular integral equations with degenerate symbol and BVPs for mixed type partial differential equations
Summary
The topic of integrodifferential equations (IDEs) which has attracted growing interest for some time has been recently developed in many applied fields, so a wide variety of problems in the physical sciences and engineering can be reduced to IDEs, in particular in relation to mathematical modeling of biological phenomena [1,2,3], aeroelasticity phenomena [4], population dynamics [5], neural networks [6], electrocardiology [7], electromagnetic [8], electrodynamics [9], and so on. The RKHS is a Hilbert space of functions in which pointwise evaluation is a continuous linear functional They are spaces that can be defined by reproducing kernels. We utilize the reproducing kernel concept to construct two reproducing kernel Hilbert spaces and to find out their representation of reproducing functions for solving the IDEs (1) and (2) via RKHS technique. It is worth mentioning that the reproducing kernel K of a Hilbert space H is unique, and the existence of K is due to the Riesz representation theorem, where K completely determines the space H. If δ ≤ |x2 −x1| ≤ ε/M1M, we can get |un(x2)−un(x1)| < ε
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