## Abstract and Applied Analysis

### The Combined RKM and ADM for Solving Nonlinear Weakly Singular Volterra Integrodifferential Equations

#### Abstract

The reproducing kernel method (RKM) and the Adomian decomposition method (ADM) are applied to solve $n$th-order nonlinear weakly singular Volterra integrodifferential equations. The numerical solutions of this class of equations have been a difficult topic to analyze. The aim of this paper is to use Taylor’s approximation and then transform the given $n$th-order nonlinear Volterra integrodifferential equation into an ordinary nonlinear differential equation. Using the RKM and ADM to solve ordinary nonlinear differential equation is an accurate and efficient method. Some examples indicate that this method is an efficient method to solve $n$th-order nonlinear Volterra integro-differential equations.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 258067, 10 pages.

Dates
First available in Project Euclid: 28 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364475967

Digital Object Identifier
doi:10.1155/2012/258067

Mathematical Reviews number (MathSciNet)
MR2999922

Zentralblatt MATH identifier
1256.65107

#### Citation

Lv, Xueqin; Shi, Sixing. The Combined RKM and ADM for Solving Nonlinear Weakly Singular Volterra Integrodifferential Equations. Abstr. Appl. Anal. 2012 (2012), Article ID 258067, 10 pages. doi:10.1155/2012/258067. https://projecteuclid.org/euclid.aaa/1364475967

#### References

• J. Morchało, “On two-point boundary value problem for an integro-differential equation of second order,” Polytechnica Posnaniensis, no. 9, pp. 51–66, 1975.
• R. P. Agarwal, “Boundary value problems for higher order integro-differential equations,” Nonlinear Analysis, vol. 7, no. 3, pp. 259–270, 1983.
• A.-M. Wazwaz, “A reliable algorithm for solving boundary value problems for higher-order integro-differentiable equations,” Applied Mathematics and Computation, vol. 118, no. 2-3, pp. 327–342, 2001.
• E. Babolian and A. S. Shamloo, “Numerical solution of Volterra integral and integro-differential equa-tions of convolution type by using operational matrices of piecewise constant orthogonal functions,” Journal of Computational and Applied Mathematics, vol. 214, no. 2, pp. 495–508, 2008.
• H. Brunner, “On the numerical solution of nonlinear Volterra-Fredholm integral equations by colloca-tion methods,” SIAM Journal on Numerical Analysis, vol. 27, no. 4, pp. 987–1000, 1990.
• D. Conte and I. D. Prete, “Fast collocation methods for Volterra integral equations of convolution type,” Journal of Computational and Applied Mathematics, vol. 196, no. 2, pp. 652–663, 2006.
• O. Lepik, “Haar wavelet method for nonlinear integro-differential equations,” Applied Mathematics and Computation, vol. 176, no. 1, pp. 324–333, 2006.
• M. Ghasemi, M. Tavassoli Kajani, and E. Babolian, “Application of He's homotopy perturbation method to nonlinear integro-differential equations,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 538–548, 2007.
• J. Saberi-Nadjafi and A. Ghorbani, “He's homotopy perturbation method: an effective tool for solving nonlinear integral and integro-differential equations,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2379–2390, 2009.
• Y. Mahmoudi, “Wavelet Galerkin method for numerical solution of nonlinear integral equation,” Applied Mathematics and Computation, vol. 167, no. 2, pp. 1119–1129, 2005.
• K. Maleknejad and Y. Mahmoudi, “Taylor polynomial solution of high-order nonlinear Volterra-Fredholm integro-differential equations,” Applied Mathematics and Computation, vol. 145, no. 2-3, pp. 641–653, 2003.
• G. Ebadi, M. Y. Rahimi-Ardabili, and S. Shahmorad, “Numerical solution of the nonlinear Volterra integro-differential equations by the tau method,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1580–1586, 2007.
• M. Zarebnia and Z. Nikpour, “Solution of linear Volterra integro-differential equations via Sinc func-tions,” International Journal of Applied Mathematics and Computer, vol. 2, pp. 1–10, 2010.
• A.-M. Wazwaz, “The combined Laplace transform-Adomian decomposition method for handling nonlinear Volterra integro-differential equations,” Applied Mathematics and Computation, vol. 216, no. 4, pp. 1304–1309, 2010.
• L. Bougoffa, R. C. Rach, and A. Mennouni, “An approximate method for solving a class of weakly-sin-gular Volterra integro-differential equations,” Applied Mathematics and Computation, vol. 217, no. 22, pp. 8907–8913, 2011.
• H. Yao, The reserch of algorithms for some singular differential equations of higher even-order [Ph.D. thesis], Department of Mathematics, Harbin Institute of Technology, 2008.
• M. Gui and Y. Lin, Nonlinear Numercial Analysis in the Reproducing Kernel Space, Nova Science Publish-er, New York, NY, USA, 2008.
• X. Lü and M. Cui, “Analytic solutions to a class of nonlinear infinite-delay-differential equations,” Journal of Mathematical Analysis and Applications, vol. 343, no. 2, pp. 724–732, 2008.
• F. Geng and M. Cui, “New method based on the HPM and RKHSM for solving forced Duffing equa-tions with integral boundary conditions,” Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp. 165–172, 2009.
• Z. Chen and Y. Lin, “The exact solution of a linear integral equation with weakly singular kernel,” Journal of Mathematical Analysis and Applications, vol. 344, no. 2, pp. 726–734, 2008.