Volterra Integral Equations and Some Nonlinear Integral Equations with Variable Limit of Integration as Generalized Moment Problems

We study the exact solution of some classes of nonlinear integral equations by series of some invertible transformations andRF-pair operations. We show that this method applies to several classes of nonlinear Volterra integral equations as well and give some useful invertible transformations for converting these equations into differential equations of Emden-Fowler type. As a consequence, we analyze the effect of the proposed operations on the exact solution of the transformed equation in order to find the exact solution of the original equation. Some applications of the method are also given. This approach is effective to find a great number of new integrable equations, which thus far, could not be integrated using the classical methods.

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Numerical Treatment of Nonlinear Stochastic Itô–Volterra Integral Equations by Piecewise Spectral-Collocation Method

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4042440 ◽

2019 ◽

Vol 14 (3) ◽

Cited By ~ 1

Author(s):

Fakhrodin Mohammadi

Keyword(s):

Approximate Solution ◽

Finite Number ◽

Convergence Analysis ◽

Integral Equations ◽

Lagrange Interpolation ◽

Interpolation Method ◽

Approximate Solutions ◽

Volterra Integral Equations ◽

Numerical Treatment ◽

Nonlinear Integral Equations

This paper deals with the approximate solution of nonlinear stochastic Itô–Volterra integral equations (NSIVIE). First, the solution domain of these nonlinear integral equations is divided into a finite number of subintervals. Then, the Chebyshev–Gauss–Radau points along with the Lagrange interpolation method are employed to get approximate solution of NSIVIE in each subinterval. The method enjoys the advantage of providing the approximate solutions in the entire domain accurately. The convergence analysis of the numerical method is also provided. Some illustrative examples are given to elucidate the efficiency and applicability of the proposed method.

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A Comparative Study on Modifications of Decomposition Method

International Journal of Advances in Scientific Research ◽

10.7439/ijasr.v2i8.3484 ◽

2016 ◽

Vol 2 (8) ◽

pp. 157

Author(s):

Jamshad Ahmad ◽

Madiha Tahir ◽

Liaqat Tahir ◽

Muhammad Naeem

Keyword(s):

Comparative Study ◽

Laplace Transform ◽

Integral Equations ◽

Decomposition Method ◽

Mathematical Physics ◽

Volterra Integral Equations ◽

Nonlinear Integral Equations ◽

Recursive Relations

Many mathematical physics models are contributed to give rise to of nonlinear integral equations. In this paper, we study the performance of two recently developed modifications of well known so called Adomians decomposition method applied using Laplace transform to nonlinear Volterra integral equations. Three nonlinear Volterra integral equations are solved analytically by implementing these modifications. From the obtained results, it may be concluded that that the modified techniques are reliable, efficient and easy to use through recursive relations that involve simple integrals. Moreover, these particular examples show the reliability and the performance of proposed modifications.

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Numerical Study of Two-Dimensional Volterra Integral Equations by RDTM and Comparison with DTM

Abstract and Applied Analysis ◽

10.1155/2013/929478 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10 ◽

Cited By ~ 2

Author(s):

Reza Abazari ◽

Adem Kılıçman

Keyword(s):

Integral Equations ◽

Numerical Study ◽

Volterra Integral Equations ◽

Differential Transform Method ◽

The Other ◽

Two Dimensional ◽

Nonlinear Integral Equations ◽

Transform Method ◽

Engineering Sciences ◽

Differential Transform

The two-dimensional Volterra integral equations are solved using more recent semianalytic method, the reduced differential transform method (the so-called RDTM), and compared with the differential transform method (DTM). The concepts of DTM and RDTM are briefly explained, and their application to the two-dimensional Volterra integral equations is studied. The results obtained by DTM and RDTM together are compared with exact solution. As an important result, it is depicted that the RDTM results are more accurate in comparison with those obtained by DTM applied to the same Volterra integral equations. The numerical results reveal that the RDTM is very effective, convenient, and quite accurate compared to the other kind of nonlinear integral equations. It is predicted that the RDTM can be found widely applicable in engineering sciences.

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