A Novel Analytical Implementation of Nonlinear Volterra Integral Equations

2012 ◽  
Vol 67 (12) ◽  
pp. 674-678 ◽  
Author(s):  
Majid Khan ◽  
Muhammad Asif Gondal ◽  
Syeda Iram Batool
Keyword(s):  
Exact Solutions ◽  
Integral Equations ◽  
Volterra Integral Equations ◽  
Volterra Equations ◽  
Volterra Type ◽  
Transform Method ◽  
Novel Technique ◽  
Homotopy Perturbation ◽  
Type Integral ◽  
Nonlinear Volterra Equations

This article aims at preferring a new and viable algorithm, specifically a two-step homotopy perturbation transform algorithm (TSHPTA). This novel technique is a feasible way in finding exact solutions with a small amount of calculations. As a simple but typical example, it demonstrates the strength and the great potential of the two-step homotopy perturbation transform method to solve nonlinear Volterra-type integral equations efficiently. The results reveal that the proposed scheme is suitable for the nonlinear Volterra equations.

scholarly journals Data dependence of solutions for Fredholm-Volterra integral equations in L2[a, b]

2013 ◽  
Vol 5 (1) ◽  
pp. 5-19
Author(s):  
Szilárd András
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Continuous Dependence ◽  
Main Tool ◽  
Volterra Integral Equations ◽  
Data Dependence ◽  
Solution Operator ◽  
Volterra Type ◽  
Type Integral

Abstract In this paper we study the continuous dependence and the differentiability with respect to the parameter λ ∈ [λ1, λ2] of the solution operator S : [λ1, λ2] → L2[a, b] for a mixed Fredholm-Volterra type integral equation. The main tool is the fiber Picard operators theorem (see [9], [8], [11], [3] and [2]).

scholarly journals Extended partial b-metric spaces and some fixed point results

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 2837-2850 ◽  
Author(s):  
V. Parvaneh ◽  
Z. Kadelburg
Keyword(s):  
Fixed Point ◽  
Integral Equations ◽  
Metric Space ◽  
Metric Spaces ◽  
Volterra Type ◽  
Fundamental Lemma ◽  
Contractive Mappings ◽  
Type Integral ◽  
Convergence Of Sequences ◽  
Weakly Contractive

In this paper, we introduce the concept of extended partial b-metric space. We demonstrate a fundamental lemma for the convergence of sequences in such spaces. Then we prove some fixed point results for weakly contractive mappings in the setup of ordered extended partial b-metric spaces. An example is given to verify the effectiveness and applicability of our main results. An application of these results to Volterra-type integral equations is provided at the end.

scholarly journals Approximate Solutions of the Generalized Abel’s Integral Equations Using the Extension Khan’s Homotopy Analysis Transformation Method

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Mohamed S. Mohamed ◽  
Khaled A. Gepreel ◽  
Faisal A. Al-Malki ◽  
Maha Al-Humyani
Keyword(s):  
Exact Solutions ◽  
Integral Equations ◽  
Method Comparison ◽  
Approximate Solutions ◽  
Transformation Method ◽  
Theory Of Elasticity ◽  
Transform Method ◽  
Numerical Examples ◽  
Homotopy Analysis ◽  
User Friendly

User friendly algorithm based on the optimal homotopy analysis transform method (OHATM) is proposed to find the approximate solutions to generalized Abel’s integral equations. The classical theory of elasticity of material is modeled by the system of Abel integral equations. It is observed that the approximate solutions converge rapidly to the exact solutions. Illustrative numerical examples are given to demonstrate the efficiency and simplicity of the proposed method. Finally, several numerical examples are given to illustrate the accuracy and stability of this method. Comparison of the approximate solution with the exact solutions shows that the proposed method is very efficient and computationally attractive. We can use this method for solving more complicated integral equations in mathematical physical.

Numerical Algorithms of Optimal Complexity for Weakly Singular Volterra Integral Equations

2006 ◽  
Vol 6 (4) ◽  
pp. 436-442 ◽  
Author(s):  
A.N. Tynda
Keyword(s):  
Numerical Methods ◽  
Integral Equations ◽  
Numerical Algorithms ◽  
Volterra Integral Equations ◽  
Volterra Equations ◽  
Fredholm Integral Equations ◽  
Optimal Complexity ◽  
Weakly Singular ◽  
Different Types ◽  
Singular Kernels

AbstractIn this paper we construct complexity order optimal numerical methods for Volterra integral equations with different types of weakly singular kernels. We show that for Volterra equations (in contrast to Fredholm integral equations) using the ”block-by-block” technique it is not necessary to employ the additional iterations to construct complexity optimal methods.

scholarly journals Convergence Analysis of Legendre Pseudospectral Scheme for Solving Nonlinear Systems of Volterra Integral Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Emran Tohidi ◽  
O. R. Navid Samadi ◽  
S. Shateyi
Keyword(s):  
Nonlinear Systems ◽  
Theoretical Prediction ◽  
Numerical Solution ◽  
Integral Equations ◽  
Volterra Integral Equations ◽  
Volterra Equations ◽  
Exponential Rate ◽  
Exponential Rate Of Convergence ◽  
Legendre Spectral Method ◽  
Pseudospectral Scheme

We are concerned with the extension of a Legendre spectral method to the numerical solution of nonlinear systems of Volterra integral equations of the second kind. It is proved theoretically that the proposed method converges exponentially provided that the solution is sufficiently smooth. Also, three biological systems which are known as the systems of Lotka-Volterra equations are approximately solved by the presented method. Numerical results confirm the theoretical prediction of the exponential rate of convergence.

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