Application of Reproducing Kernel Method for Solving Nonlinear Fredholm-Volterra Integrodifferential Equations

By Omar Abu Arqub, M. Al‐Smadi, Shaher Momani. 2012

91 citations recorded by the source corpus.

This paper investigates the numerical solution of nonlinear Fredholm-Volterra integro-differential equations using reproducing kernel Hilbert space method. The solution 𝑢(𝑥) is represented in the form of series in the reproducing kernel space. In the mean time, the n-term approximate solution 𝑢𝑛(𝑥) is obtained and it is proved to converge to the exact solution 𝑢(𝑥). Furthermore, the proposed method has an advantage that it is possible to pick any point in the interval of integration and as well the approximate solution and its derivative will be applicable. Numerical examples are included to demonstrate the accuracy and applicability of the presented technique. The results reveal that the method is very effective and simple.

Subfields: Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/1c828fe4dec6329fafaaa1467c2c4cb3ab6bf11a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Modeling and Simulation

In Numerical Analysis

Citations in the corpus

Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method (2015, background): “The reader is asked to refer to Abu Arqub (2015), Abu Arqub and Al-Smadi (2014), Abu Arqub et al. (2012, 2013a,b), Al-Smadi et al. (2013, 2014), Berlinet and Agnan (2004), Cui and Lin (2009), Daniel (2003), Geng (2009), Geng and Cui (2012), Geng and Qian (2013), Geng et al. (2014), Jiang and Chen…”
A Reliable Analytical Method for Solving Higher-Order Initial Value Problems (2013, background): “On the other aspect as well, the applications of other versions of series solutions to linear and nonlinear problems can be found in [13–18] and for numerical solvability of different categories of differential equations, one can consult the references [19, 20].”
An Optimization Algorithm for Solving Systems of Singular Boundary Value Problems (2014, background): “On the other hand, the numerical solvability of other version of differential equations and other related equations can be found in [34,35,36,37,38, 39,40] and references therein.”
A Novel Representation of the Exact Solution for Differential Algebraic Equations System Using Residual Power-Series Method (2015, unknown): “The applications of other versions for linear and nonlinear problems can be found in [19–26] and references therein.”
Approximate Analytical Solution by Residual Power Series Method for System of Fredholm Integral Equations (2016, background): “(24) Following the residual functions ( 24) and the fact dm dxm Res m 1 (0) = dm dxm Res m 2 (0) = 0 for m = 0,1,2, .”
A Representation of the Exact Solution of Generalized Lane-Emden Equations Using a New Analytical Method (2013, background): “However, the applications of other versions of series solutions to linear and nonlinear problems can be found in [20–25], and, to discern the numerical solvability of different categories of singular differential equations, one can consult [26].”