On the <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>R</mml:mi><mml:mi>F</mml:mi></mml:mrow></mml:math>-pair operations for the exact solution of some classes of nonlinear Volterra integral equations

Darania, P.; Hadizadeh, M.

doi:https://doi.org/10.1155/MPE/2006/97020

Mathematical Problems in Engineering

On this page

Abstract References Copyright Related Articles

Open Access

Volume 2006 | Article ID 097020 | https://doi.org/10.1155/MPE/2006/97020

On the RF-pair operations for the exact solution of some classes of nonlinear Volterra integral equations

P. Darania¹and M. Hadizadeh^2,3

Received01 May 2005

Revised17 Nov 2005

Accepted15 Jan 2006

Published24 May 2006

Abstract

We study the exact solution of some classes of nonlinear integral equations by series of some invertible transformations and RF-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.

References

M. B. Abd-el-Malek, “Application of the group-theoretical method to physical problems,” Journal of Nonlinear Mathematical Physics, vol. 5, no. 3, pp. 314–330, 1998.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
M. A. Abdou, “On the solution of linear and nonlinear integral equation,” Applied Mathematics and Computation, vol. 146, no. 2-3, pp. 857–871, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
L. M. Berkovich, “The generalized Emden-Fowler equation,” Symmetry in Nonlinear Mathematical Physics, vol. 1, pp. 155–163, 1997.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover, New York, 1967.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, Springer, New York, 1972.
View at: MathSciNet
H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover, New York, 1962.
View at: Zentralblatt MATH | MathSciNet
L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge, 1985.
View at: Zentralblatt MATH | MathSciNet
M. Ganesh and M. C. Joshi, “Numerical solvability of Hammerstein integral equations of mixed type,” IMA Journal of Numerical Analysis, vol. 11, no. 1, pp. 21–31, 1991.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
P. M. Lima, “Numerical methods and asymptotic error expansions for the Emden-Fowler equations,” Journal of Computational and Applied Mathematics, vol. 70, no. 2, pp. 245–266, 1996.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
C. D. Luning and W. L. Perry, “Positive solutions of negative exponent generalized Emden-Fowler boundary value problems,” SIAM Journal on Mathematical Analysis, vol. 12, no. 6, pp. 874–879, 1981.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
N. M. Madbouly, D. F. McGhee, and G. F. Roach, “Adomian's method for Hammerstein integral equations arising from chemical reactor theory,” Applied Mathematics and Computation, vol. 117, no. 2-3, pp. 241–249, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. K. Miller, Nonlinear Volterra Integral Equations, Mathematics Lecture Note Series, W. A. Benjamin, California, 1971.
View at: Zentralblatt MATH | MathSciNet
L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982.
View at: Zentralblatt MATH | MathSciNet
A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Florida, 1998.
View at: Zentralblatt MATH | MathSciNet
A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Chapman & Hall/CRC, Florida, 2nd edition, 2003.
View at: Zentralblatt MATH | MathSciNet
B. D. Vujanovic, A. M. Strauss, S. E. Jones, and P. P. Gillis, “Polynomial conservation laws of the generalized Emden-Fowler equation,” International Journal of Non-Linear Mechanics, vol. 33, no. 2, pp. 377–384, 1998.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A.-M. Wazwaz, “A new algorithm for solving differential equations of Lane-Emden type,” Applied Mathematics and Computation, vol. 118, no. 2-3, pp. 287–310, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. S. W. Wong, “On the generalized Emden-Fowler equation,” SIAM Review, vol. 17, no. 2, pp. 339–360, 1975.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
V. F. Zaitsev, “On discrete group analysis of ordinary differential equations,” Soviet Mathematics. Doklady, vol. 37, no. 2, pp. 403–406, 1988.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
V. F. Zaitsev and A. V. Flegontov, “Creation of new methods of mathematical physics, search of the exact solutions and first integrals of nonlinear differential equations,” Differential Equations and Control Process, no. 1, pp. 252–260, 1997.
View at: Google Scholar
V. F. Zaitsev and A. D. Polyanin, Discrete Group Methods for Integrating Equations of Nonlinear Mechanics, CRC Press, Florida, 1994.
View at: Zentralblatt MATH
http://eqworld.ipmnet.ru/ (The World of Mathematical Equations).

Copyright

Copyright © 2006 P. Darania and M. Hadizadeh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation Order printed copies

Views

169

Downloads

815

Citations