A numerical technique based on operational matrices for solving nonlinear integro-differential equations

Document Type : Research Article

Author

• A. Golbabai

10.22067/ijnao.v4i1.34848

Abstract

This paper presents a computational method for solving two types of integro-differential equations, system of nonlinear high order Volterra-Fredholm integro-differential equation(VFIDEs) and nonlinear fractional order integro-differential equations.
Our tools for this aims is operational matrices of integration and fractional integration. By this method the given problems reduce to solve a system of algebraic equations. Illustrative examples are included to demonstrate the efficiency and high accuracy of the method.

Keywords

• Operational matrix of integration
• Volterra-Fredholm
• Nonlinear system of integro-differential equations
• Fractional order
• Legendre wavelet

20.1001.1.24236977.2014.4.1.3.1

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References

1.  Abbasbandy, S. and Taati, A. Numerical solution of the system of nonlinear Volterra integro-differential equations with nonlinear differential part by the operational Tau method and error estimation, journal of computational and Applied Mathematics, 231(2009)106-113.
2. Biazar, J., Ghazvini, H. and Eslami, M. He’s homotopy perturbation method for systems of integro-differential equations, Chaos Solition and Fractals, doi:10.1016/j.chaos.2007.06.001.
3. Caputo, M. Linear modles of dissipation whose Q is almost frequenty independent, Part II, J.Roy. Astr. Soc.,13:529-539.
4. El-shahed, M. Application of He,s homotopy perturbation method to Volterra ,s intgro-differential equation, International Journal of Nonlinear sciences and Numerical Simulation 8 (2) (2007) 223-228.
5. Elsayed, S. M., Kaya, D. and Zarea, S. The decomposition method applied to solve high order linear Volterra-Fredholm integro-differential equations, International Journal of Nonlinear sciences and Numerical Simulation 8 (2) (2007) 211-222.
6. Erjaee, G. H., Taghvafard, H. and Alnasr, M. Numerical solution of the high thermal loss problem presented by a fractional differential equation, Communication in Nonlinear sciences and Numerical Simulation 16 (2011) 1356–1362.
7. Gazanfari, B., Ghazanfari, A. G. and Veisif, F. Homotopy perturbation Method for the nonlinear fractional integro differentioal equations, Australian Journal of Basic an Applied Sciencec 4 (12) (2010) 5823-5829.
8. Golbabai, A. and Javidi, M. Application of He’s homotopy perturbation method for n-th order integro differential equations, Applied Mathematics and Computation, 190 (2007) 1409-1416.
9. Golbabai, A. and Javidi, M. A numerical solution for solving system of Fredholm integral equation by using perturbation method, Applied Math ematics and Computation, 189 (2007) 1921-1928.
10. Guf, J. S. and Jiang, W. S. The Haar wavelet operational m atrix of integration, International Journal of Systems Science, 27 (1996) 623-628.
11. Hu, X. B. and Wu, Y. T. Application of the Hirota bilinear formalism to a new integrable differential-difference equation, Physics Letters A, 246(1998) 523-529.
12. Kilicman, A. and Al Zhour, Z. A. A.Kronecker operational matrices for fractional calculus and some application, Applied Mathematic and Computation 187(2007)250-265.
13. Liu, N. and Ling, E. B. Legendre wavelets method for numerical solutions of partial differential equations, Numerical Methods for partial Differential Equations. 26 (2009) 81-94.
14. Podlubny, I. Fractional differential equations:an introduction to fractional derivatives, fractional differentional equations, to methods of their solution and some their application. New York: Academic Press; 1999.
15. Razzaghi, M. and Yousefi, S. Legendre wavelets method for constrained
optimal control problems, Mathematical method in Applied Sciences. 25 (2002)529-539.
16. Razzaghi, M. and Yousefi, S. Legendre wavelet method for the nonlinear Volterra-Fredholm integral equations, Mathematics and Computer in Simulation. 70 (2005) 1-8.
17. Razzaghi, M. and Yousefi, S. Legendre wavelets direct method for variational problems, Mathematics and Computer in Simulation. 53 (2000) 185-192.
18. Razzaghi, M. and Yousefi, S. The Legendre wavelet operational matrix of integration, International Journal of Systems Science, 32 (2001) 495-502.

19. Rehman, M. and Ali Khan, R. The Legendre wavelet methosd for solving fractional differentional equations, Communication in Nonlinear sciences and Numerical Simulation 16 (2011) 4163-4173.
20. Sun, F. Z., Gao, M., Lei, S. H., Zaho, Y. B., Wang, K., Shi, Y. T. and Wang, N. H. The fractal dimension of the fractal model of dropwise condensation and its experimental study, International Journal of Nonlinear sciences and Numerical Simulation 8 (2) (2007) 211-222.
21. Taghvafard, H. and Erjaee, G. H. On solving a system of singular Volterra integral equations of convolution type, Communication in Nonlinear sciences and Numerical Simulation 16 (2011) 3486–3492.
22. Tarasov, V. E. Fractional integro diffe rentional equations for electromagnetic waves in dielectric media, Theoritical and Mathematical Physics, 153 (3) (2009) 355-359.
23. Wang, H., Fu, H. M. , Zhang, H. F. and Hu, Z. Q. A practical thermo dynamic method to calculate the best glass-forming composition for bulk metallic glasses, International Journal of Nonlinear sciences and Numerical Simulation 8 (2) (2007) 171-178.
24. Xu, L., He, J. H. and Liu, Y. Electrospun nanoporous spheres whit Chinese drug, International Journal of Nonlinear sciences and Numerical Simulation 8 (2) (2007) 199-202.