57 research outputs found
Numerical Solution of Some Nonlinear Volterra Integral Equations of the First Kind
In this paper, the solving of a class of the nonlinear Volterra integral equations (NVIE) of the first kind is investigated. Here, we convert NVIE of the first kind to a linear equation of the second kind. Then we apply the operational Tau method to the problem and prove convergence of the presented method. Finally, some numerical examples are given to show the accuracy of the method
On The Numerical Solution of Linear Fredholm-Volterra Ä°ntegro Differential Difference Equations With Piecewise Ä°ntervals
The numerical solution of a mixed linear integro delay differential-difference equation with piecewise interval is presented using the Chebyshev collocation method. The aim of this article is to present an efficient numerical procedure for solving a mixed linear integro delay differential difference equations. Our method depends mainly on a Chebyshev expansion approach. This method transforms a mixed linear integro delay differential-difference equations and the given conditions into a matrix equation which corresponds to a system of linear algebraic equation. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system Maple 10
Numerical solution of the higher-order linear Fredholm integro-differential-difference equation with variable coefficients
AbstractThe main aim of this paper is to apply the Legendre polynomials for the solution of the linear Fredholm integro-differential-difference equation of high order. This equation is usually difficult to solve analytically. Our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The operational matrices of delay and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the presented technique and a comparison is made with existing results
Numerical solution of fractional Fredholm integro-differential equations by spectral method with fractional basis functions
This paper presents an efficient spectral method for solving the fractional
Fredholm integro-differential equations. The non-smoothness of the solutions to
such problems leads to the performance of spectral methods based on the
classical polynomials such as Chebyshev, Legendre, Laguerre, etc, with a low
order of convergence. For this reason, the development of classic numerical
methods to solve such problems becomes a challenging issue. Since the
non-smooth solutions have the same asymptotic behavior with polynomials of
fractional powers, therefore, fractional basis functions are the best candidate
to overcome the drawbacks of the accuracy of the spectral methods. On the other
hand, the fractional integration of the fractional polynomials functions is in
the class of fractional polynomials and this is one of the main advantages of
using the fractional basis functions. In this paper, an implicit spectral
collocation method based on the fractional Chelyshkov basis functions is
introduced. The framework of the method is to reduce the problem into a
nonlinear system of equations utilizing the spectral collocation method along
with the fractional operational integration matrix. The obtained algebraic
system is solved using Newton's iterative method. Convergence analysis of the
method is studied. The numerical examples show the efficiency of the method on
the problems with smooth and non-smooth solutions in comparison with other
existing methods
Study on Solving Two-dimensional Linear and Nonlinear Volterra Partial Integro-differential Equations by Reduced Differential Transform Method
In this article, we study on the analytical and numerical solution of two-dimensional linear and nonlinear Volterra partial integro-differential equations with the appropriate initial condition by means of reduced differential transform method. The advantage of this method is its simplicity in using, it solves the problem directly without the need for linearization, perturbation, or any other transformation and gives the solution in the form of convergent power series with elegantly computed components. The validity and efficiency of this method are illustrated by considering five computational examples
An Iterative Scheme for Solving Systems of Nonlinear Fredholm Integrodifferential Equations
Using fixed-point techniques and Faber-Schauder systems in adequate Banach spaces, we approximate the solution of a system of nonlinear Fredholm integrodifferential equations of the second kind.This research is partially supported by Junta de AndalucĂa Grant FQM359 and the ETSIE of the University of Granada, Spain
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