Abstract
The reproducing kernel method and Taylor series to determine a solution for nonlinear Abel’s integral equations are combined. In this technique, we first convert it to a nonlinear differential equation by using Taylor series. The approximate solution in the form of series in the reproducing kernel space is presented. The advantages of this method are as follows: First, it is possible to pick any point in the interval of integration and as well the approximate solution. Second, numerical results compared with the existing method show that fewer nodes are required to obtain numerical solutions. Furthermore, the present method is a reliable method to solve nonlinear Abel’s integral equations with weakly singular kernel. Some numerical examples are given in two different spaces.
Highlights
Abel’s integral equation, linear or nonlinear, arises in many branches of scientific fields (Singh, Pandey, & Singh, 2009), such as seismology, microscopy, radio astronomy, atomic scattering, electron emission, radar ranging, X-ray radiography, plasma diagnostics, and optical fiber evaluation
The aim of this paper is to introduce the reproducing kernel method to solve nonlinear Abel’s integral equation
The orthonormal system {ψi(x)}∞i=1 of W2m[0, 1] is constructed from { i(x)}∞i=1 by using the Gram– Schmidt algorithm, and the approximate solution will be obtained by calculating a truncated series based on these functions, such that
Summary
Abel’s integral equation, linear or nonlinear, arises in many branches of scientific fields (Singh, Pandey, & Singh, 2009), such as seismology, microscopy, radio astronomy, atomic scattering, electron emission, radar ranging, X-ray radiography, plasma diagnostics, and optical fiber evaluation. A variety of numerical and analytic methods for solving these equations are presented. We use a reproducing kernel Hilbert space approach that allows us to formulate the estimation problem as an unconstrained numeric maximization problem easy to solve. The aim of this paper is to introduce the reproducing kernel method to solve nonlinear Abel’s integral equation. We introduce construction of the method in the reproducing kernel space for solving Equation (1.1). 2. Construction of the method we construct the space W2m[0, 1] and formulate the reproducing kernel function Rx(y) in the space W2m[0, 1]. A transformation of the Equation (1.1) Using modified Taylor series, the nonlinear Abel’s integral equations with weakly singular kernel transform into nonlinear differential equations that can be solved .
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
