The Method of Successive Approximations (Neumann’s Series) of Volterra Integral Equation of the Second Kind

Teshome Bayleyegn Matebie

doi:doi:10.11648/j.pamj.20160506.16

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The Method of Successive Approximations (Neumann’s Series) of Volterra Integral Equation of the Second Kind

Teshome Bayleyegn Matebie

Published in Pure and Applied Mathematics Journal (Volume 5, Issue 6)

Received: 12 October 2016 Accepted: 28 October 2016 Published: 23 December 2016

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Abstract

In this paper, the solving of a class of both linear and nonlinear Volterra integral equations of the first kind is investigated. Here, by converting integral equation of the first kind to a linear equation of the second kind and the ordinary differential equation to integral equation we are going to solve the equation easily. The method of successive approximations (Neumann’s series) is applied to solve linear and nonlinear Volterra integral equation of the second kind. Some examples are presented to illustrate methods.

Published in	Pure and Applied Mathematics Journal (Volume 5, Issue 6)
DOI	10.11648/j.pamj.20160506.16
Page(s)	211-219
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Volterra Integral Equation, First Kind, Second Kind, Kernel, Method of Successive Approximations

References

[1]	W. V. Lovitt. (1950). Linear Integral Equations, Dover Represented.
[2]	S.S.SASTRY. (2003). Introductory Methods of Numerical Analysis, Third Edition.
[3]	M. Rahman. (2007). Integral Equations and their Applications, 1st ed; WIT Press.
[4]	Peter Linz. (1985). Analytical and Numerical Methods for Volterra Equations.
[5]	Porter, David. (1990). Integral equations A practical treatment, from spectral theory to applications, first edition.
[6]	F. Smithies (1958). Cambridge Tracts in Mathematics and Mathematical Physics.
[7]	Peter J. Collins. (2006). Differential and Integral equations, WIT Press.
[8]	William Squire. (1970). Modern Analytic and Computational Methods in Science and Mathematics.
[9]	W. Pogorzelski, Integral Equations and their Applications, volume I.
[10]	Abdul-Majid Wazwaz. A First Course in Integral Equations - Solutions Manual, 2nd ed; World Scientific Publishing Co. Pte. Ltd.
[11]	T. A. Burton Eds. (2005). Volterra Integral and Differential Equations, 2nd ed; Academic Press, Elsevier.

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Teshome Bayleyegn Matebie. (2016). The Method of Successive Approximations (Neumann’s Series) of Volterra Integral Equation of the Second Kind. Pure and Applied Mathematics Journal, 5(6), 211-219. https://doi.org/10.11648/j.pamj.20160506.16

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ACS Style

Teshome Bayleyegn Matebie. The Method of Successive Approximations (Neumann’s Series) of Volterra Integral Equation of the Second Kind. Pure Appl. Math. J. 2016, 5(6), 211-219. doi: 10.11648/j.pamj.20160506.16

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AMA Style

Teshome Bayleyegn Matebie. The Method of Successive Approximations (Neumann’s Series) of Volterra Integral Equation of the Second Kind. Pure Appl Math J. 2016;5(6):211-219. doi: 10.11648/j.pamj.20160506.16

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@article{10.11648/j.pamj.20160506.16,
  author = {Teshome Bayleyegn Matebie},
  title = {The Method of Successive Approximations (Neumann’s Series) of Volterra Integral Equation of the Second Kind},
  journal = {Pure and Applied Mathematics Journal},
  volume = {5},
  number = {6},
  pages = {211-219},
  doi = {10.11648/j.pamj.20160506.16},
  url = {https://doi.org/10.11648/j.pamj.20160506.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20160506.16},
  abstract = {In this paper, the solving of a class of both linear and nonlinear Volterra integral equations of the first kind is investigated. Here, by converting integral equation of the first kind to a linear equation of the second kind and the ordinary differential equation to integral equation we are going to solve the equation easily. The method of successive approximations (Neumann’s series) is applied to solve linear and nonlinear Volterra integral equation of the second kind. Some examples are presented to illustrate methods.},
 year = {2016}
}

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TY  - JOUR
T1  - The Method of Successive Approximations (Neumann’s Series) of Volterra Integral Equation of the Second Kind
AU  - Teshome Bayleyegn Matebie
Y1  - 2016/12/23
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PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20160506.16
AB  - In this paper, the solving of a class of both linear and nonlinear Volterra integral equations of the first kind is investigated. Here, by converting integral equation of the first kind to a linear equation of the second kind and the ordinary differential equation to integral equation we are going to solve the equation easily. The method of successive approximations (Neumann’s series) is applied to solve linear and nonlinear Volterra integral equation of the second kind. Some examples are presented to illustrate methods.
VL  - 5
IS  - 6
ER  -

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Author Information

Teshome Bayleyegn Matebie

College of Natural Science, Department of Mathematics, Arba Minch University, Arba Minch, Ethiopia

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APA Style

Teshome Bayleyegn Matebie. (2016). The Method of Successive Approximations (Neumann’s Series) of Volterra Integral Equation of the Second Kind. Pure and Applied Mathematics Journal, 5(6), 211-219. https://doi.org/10.11648/j.pamj.20160506.16

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ACS Style

Teshome Bayleyegn Matebie. The Method of Successive Approximations (Neumann’s Series) of Volterra Integral Equation of the Second Kind. Pure Appl. Math. J. 2016, 5(6), 211-219. doi: 10.11648/j.pamj.20160506.16

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AMA Style

Teshome Bayleyegn Matebie. The Method of Successive Approximations (Neumann’s Series) of Volterra Integral Equation of the Second Kind. Pure Appl Math J. 2016;5(6):211-219. doi: 10.11648/j.pamj.20160506.16

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@article{10.11648/j.pamj.20160506.16,
  author = {Teshome Bayleyegn Matebie},
  title = {The Method of Successive Approximations (Neumann’s Series) of Volterra Integral Equation of the Second Kind},
  journal = {Pure and Applied Mathematics Journal},
  volume = {5},
  number = {6},
  pages = {211-219},
  doi = {10.11648/j.pamj.20160506.16},
  url = {https://doi.org/10.11648/j.pamj.20160506.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20160506.16},
  abstract = {In this paper, the solving of a class of both linear and nonlinear Volterra integral equations of the first kind is investigated. Here, by converting integral equation of the first kind to a linear equation of the second kind and the ordinary differential equation to integral equation we are going to solve the equation easily. The method of successive approximations (Neumann’s series) is applied to solve linear and nonlinear Volterra integral equation of the second kind. Some examples are presented to illustrate methods.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - The Method of Successive Approximations (Neumann’s Series) of Volterra Integral Equation of the Second Kind
AU  - Teshome Bayleyegn Matebie
Y1  - 2016/12/23
PY  - 2016
N1  - https://doi.org/10.11648/j.pamj.20160506.16
DO  - 10.11648/j.pamj.20160506.16
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 211
EP  - 219
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20160506.16
AB  - In this paper, the solving of a class of both linear and nonlinear Volterra integral equations of the first kind is investigated. Here, by converting integral equation of the first kind to a linear equation of the second kind and the ordinary differential equation to integral equation we are going to solve the equation easily. The method of successive approximations (Neumann’s series) is applied to solve linear and nonlinear Volterra integral equation of the second kind. Some examples are presented to illustrate methods.
VL  - 5
IS  - 6
ER  -

Copy | Download