Numerical Solution of Linear Volterra Integro-Differential Equation using Runge-Kutta-Fehlberg Method

Ali Filiz

doi:doi:10.11648/j.acm.20140301.12

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Numerical Solution of Linear Volterra Integro-Differential Equation using Runge-Kutta-Fehlberg Method

Ali Filiz

Published in Applied and Computational Mathematics (Volume 3, Issue 1)

Received: 18 August 2013 Published: 20 February 2014

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Abstract

In this paper a new fourth and fifth-order numerical solution of linear Volterra integro-differential equation is discussed. One popular technique that uses here for error control is called the Runge-Kutta-Fehlberg method for Ordinary Differential Equation (ODE) part and Newton-Cotes formulae for integral parts.

Published in	Applied and Computational Mathematics (Volume 3, Issue 1)
DOI	10.11648/j.acm.20140301.12
Page(s)	9-14
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), 2014. Published by Science Publishing Group

Keywords

A fourth and Fifth-Order Accuracy, Lagrange Polynomial Interpolating, Newton-Cotes Formulas, Runge-Kutta Methods, Linear Volterra Integro-Differential Equation

References

[1]	M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs and Mathematical Tables, New York: Dover, 1972, pp. 885–887.
[2]	A. Asanov, Uniqueness of the solution of systems of convolution-type Volterra integral equations of the first kind, In: Inverse problems for differential equations of the mathematical physics (Russian), Novasibirsk: Akad. Nauk SSSR Sibirsk. Otdel. Vychil. Tsentr, 1978, Vol 155, pp. 2–34.
[3]	R. L. Burden and J. D. Faires, Numerical Analysis, New York: Brooks/Cole Publishing Company, USA, 1997, ch.5.
[4]	C. T. H. Baker, The Numerical Treatment of Integral Equations, Clarendon Press; Oxford University Press, 1977.
[5]	C. T. H. Baker, G. A. Bochorov, A. Filiz, N. J. Ford, C. A. H. Paul, F. A. Rihan, A. Tang, R. M. Thomas, H. Tian, D. R. Wille "Numerical Modelling by Retarded Functional Differential Equations," Numerical Analysis Report, Manchester Center for Computational Mathematics, No:335, ISS 130-1725,1998.
[6]	C. T. H. Baker, G. A. Bochorov, A. Filiz, N. J. Ford, C. A. H. Paul, F. A. Rihan, A. Tang, R. M. Thomas, H. Tian, D. R. Wille "Numerical Modelling by Delay and Volterra Functional Differential Equations," Numerical Analysis Report, In: Computer Mathematics and its Aplications-Advances & Developments (1994-2005), Elias A. Lipitakis (Editor), LEA Publishers, Athens, Greece, 2006, pp. 233-256.
[7]	R. Bellman, A Survey of the Theory of the Boundedness Stability and Asymptotic Behaviour of Solutions of Linear and Non-linear differential and difference equations, Washington, D. C., 1949.
[8]	K. L. Cooke, "Functional differential equations close to cifferential equation," Amer. Math. Soc., 1966, Vol.72, pp. 285-288.
[9]	A. Filiz, "On the solution of Volterra and Lotka-Volterra Type Equations," LMS supported One Day Meeting in Delayed Differential equation (Liverpool, UK), 12th March 2000.
[10]	A. Filiz, "Numerical Solution of Some Volterra Integral Equations," PhD Thesis, The University of Manchester, 2000.
[11]	A. Filiz, "Fourth-order robust numerical method for integro-differential equations," Asian Journal of Fuzzy and Applied Mathematics, 2013, Vol. 1 I, pp. 28-33.
[12]	P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, 1985.
[13]	C. W. Ueberhuber, Numerical Computation 2: Methods, Software and analysis, Berlin: Springer-Verlag, 1997.
[14]	V. Volterra, Leçons Sur la Theorie Mathematique de la Lutte Pour La Vie, Gauthier-villars, Paris, 1931.
[15]	V. Volterra, Theory of Functional and of Integro-Differential Equations. Dover, New York, 1959.
[16]	V. Volterra, "Sulle Equazioni Integro-differenziali Della Teoria Dell’elastica," Atti Della Reale Accademia dei Lincei 18 (1909), Reprinted in Vito Volterra, Opera Mathematiche; Memorie e Note, Vol. 3, Accademia dei Lincei Rome, 1957.
[17]	Wolfram MathWorld, Newton-Cotes Formulas, http://mathworld.wolfram.com/Newton-CotesFormulas.html

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Ali Filiz. (2014). Numerical Solution of Linear Volterra Integro-Differential Equation using Runge-Kutta-Fehlberg Method. Applied and Computational Mathematics, 3(1), 9-14. https://doi.org/10.11648/j.acm.20140301.12

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ACS Style

Ali Filiz. Numerical Solution of Linear Volterra Integro-Differential Equation using Runge-Kutta-Fehlberg Method. Appl. Comput. Math. 2014, 3(1), 9-14. doi: 10.11648/j.acm.20140301.12

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AMA Style

Ali Filiz. Numerical Solution of Linear Volterra Integro-Differential Equation using Runge-Kutta-Fehlberg Method. Appl Comput Math. 2014;3(1):9-14. doi: 10.11648/j.acm.20140301.12

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@article{10.11648/j.acm.20140301.12,
  author = {Ali Filiz},
  title = {Numerical Solution of Linear Volterra Integro-Differential Equation using Runge-Kutta-Fehlberg Method},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {1},
  pages = {9-14},
  doi = {10.11648/j.acm.20140301.12},
  url = {https://doi.org/10.11648/j.acm.20140301.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140301.12},
  abstract = {In this paper a new fourth and fifth-order numerical solution of linear Volterra integro-differential equation is discussed. One popular technique that uses here for error control is called the Runge-Kutta-Fehlberg method for Ordinary Differential Equation (ODE) part and Newton-Cotes formulae for integral parts.},
 year = {2014}
}

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TY  - JOUR
T1  - Numerical Solution of Linear Volterra Integro-Differential Equation using Runge-Kutta-Fehlberg Method
AU  - Ali Filiz
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AB  - In this paper a new fourth and fifth-order numerical solution of linear Volterra integro-differential equation is discussed. One popular technique that uses here for error control is called the Runge-Kutta-Fehlberg method for Ordinary Differential Equation (ODE) part and Newton-Cotes formulae for integral parts.
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Author Information

Ali Filiz

Department of Mathematics, Adnan Menderes University, 09010 AYDIN-TURKEY

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APA Style

Ali Filiz. (2014). Numerical Solution of Linear Volterra Integro-Differential Equation using Runge-Kutta-Fehlberg Method. Applied and Computational Mathematics, 3(1), 9-14. https://doi.org/10.11648/j.acm.20140301.12

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ACS Style

Ali Filiz. Numerical Solution of Linear Volterra Integro-Differential Equation using Runge-Kutta-Fehlberg Method. Appl. Comput. Math. 2014, 3(1), 9-14. doi: 10.11648/j.acm.20140301.12

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AMA Style

Ali Filiz. Numerical Solution of Linear Volterra Integro-Differential Equation using Runge-Kutta-Fehlberg Method. Appl Comput Math. 2014;3(1):9-14. doi: 10.11648/j.acm.20140301.12

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@article{10.11648/j.acm.20140301.12,
  author = {Ali Filiz},
  title = {Numerical Solution of Linear Volterra Integro-Differential Equation using Runge-Kutta-Fehlberg Method},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {1},
  pages = {9-14},
  doi = {10.11648/j.acm.20140301.12},
  url = {https://doi.org/10.11648/j.acm.20140301.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140301.12},
  abstract = {In this paper a new fourth and fifth-order numerical solution of linear Volterra integro-differential equation is discussed. One popular technique that uses here for error control is called the Runge-Kutta-Fehlberg method for Ordinary Differential Equation (ODE) part and Newton-Cotes formulae for integral parts.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - Numerical Solution of Linear Volterra Integro-Differential Equation using Runge-Kutta-Fehlberg Method
AU  - Ali Filiz
Y1  - 2014/02/20
PY  - 2014
N1  - https://doi.org/10.11648/j.acm.20140301.12
DO  - 10.11648/j.acm.20140301.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 9
EP  - 14
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20140301.12
AB  - In this paper a new fourth and fifth-order numerical solution of linear Volterra integro-differential equation is discussed. One popular technique that uses here for error control is called the Runge-Kutta-Fehlberg method for Ordinary Differential Equation (ODE) part and Newton-Cotes formulae for integral parts.
VL  - 3
IS  - 1
ER  -

Copy | Download