A Strong Method for Solving Systems of Integro-Differential Equations

ABSTRACT

The introduced method in this paper consists of reducing a system of integro-differential equations into a system of algebraic equations, by expanding the unknown functions, as a series in terms of Chebyshev wavelets with unknown coefficients. Extension of Chebyshev wavelets method for solving these systems is the novelty of this paper. Some examples to illustrate the simplicity and the effectiveness of the proposed method have been presented.

The introduced method in this paper consists of reducing a system of integro-differential equations into a system of algebraic equations, by expanding the unknown functions, as a series in terms of Chebyshev wavelets with unknown coefficients. Extension of Chebyshev wavelets method for solving these systems is the novelty of this paper. Some examples to illustrate the simplicity and the effectiveness of the proposed method have been presented.

KEYWORDS

Systems of Integro-Differential Equations, Chebyshev Wavelets Method, Mother Wavelet, Operational Matrix

Systems of Integro-Differential Equations, Chebyshev Wavelets Method, Mother Wavelet, Operational Matrix

Cite this paper

nullJ. Biazar and H. Ebrahimi, "A Strong Method for Solving Systems of Integro-Differential Equations,"*Applied Mathematics*, Vol. 2 No. 9, 2011, pp. 1105-1113. doi: 10.4236/am.2011.29152.

nullJ. Biazar and H. Ebrahimi, "A Strong Method for Solving Systems of Integro-Differential Equations,"

References

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[2] E. Babolian and F. Fattahzadeh, “Numerical Solution of Differential Equations by Using Chebyshev Wavelet Operational Matrix of Integration,” Applied Mathematics and Computations, Vol. 188, No. 1, 2007, pp. 417-426.

[3] Y. Li, “Solving a Nonlinear Fractional Differential Equation Using Chebyshev Wavelets,” Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 9, 2010, pp. 2284-2292. doi:10.1016/j.cnsns.2009.09.020

[4] J. Biazar, “Solution of Systems of Integral-Differential Equations by Adomian Decomposition Method,” Applied Mathematics and Computation, Vol. 168, No. 2, 2005, pp. 1232-1238. doi:10.1016/j.amc.2004.10.015

[5] J. Biazar, H. Ghazvini and M. Eslami, “He’s Homotopy Perturbation Method for Systems of Integro-Differential Equations,” Chaos, Solitions and Fractals, Vol. 39, No. 3, 2009, pp. 1253-1258. doi:10.1016/j.chaos.2007.06.001

[6] E. Yusufoglu (Agadjanov), “An E?cient Algorithm for Solving Integro-Di?erential Equations System,” Applied Mathematics and Computation, Vol. 192, No. 1, 2007, pp. 51-55. doi:10.1016/j.amc.2007.02.134

[7] J. Biazar and H. Aminikhah, “A New Technique for SolvIng Integro-Differential Equations,” Computers and Mathematics with Applications, Vol. 58, No. 11-12, 2009, pp. 2084- 2090. doi:10.1016/j.camwa.2009.03.042

[8] J. Pour-Mahmoud and M. Y. Rahimi-Ardabili and S. Shamoad, “Numerical Solution of the System of Fredholm Integro-Differential Equations by the Tau Method,” Applied Mathematics and Computation, Vol. 168, 2005, pp. 465-478.

[9] S. Abbasbandy and A. Taati, “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 Ma-thematics, Vol. 231, No. 1, 2009, pp. 106-113. doi:10.1016/j.cam.2009.02.014

[10] A. Arikoglu and I. Ozkol, “Solutions of Integral and Integro-Differential Equation Systems by Using Differential Transform Method,” Computers and Mathematics with Applications, Vol. 56, No. 9, 2008, pp. 2411-2417. doi:10.1016/j.camwa.2008.05.017

[11] M. Gachpazan, “Numerical Scheme to Solve Integro- Di?erential Equations System,” Journal of Advanced Research in Scientific Computing, Vol. 1, No. 1, 2009, pp. 11-21.

[12] K. Maleknejad, F. Mirzaee and S. Abbasbandy, “Solving Linear Integro-Differential Equations System by Using Rationalized Haar Functions Method,” Applied Mathematics and Computation, Vol. 155, No. 2, 2004, pp. 317-328. doi:10.1016/S0096-3003(03)00778-1

[13] K. Maleknejad and M. Tavassoli Kajani, “Solving Linear Integro-Differential Equation System by Galerkin Methods with Hybrid Functions,” Applied Mathematics and Computation, Vol. 159, No. 3, 2004, pp. 603-612. doi:10.1016/j.amc.2003.10.046

[14] I. Daubeches, “Ten Lectures on Wavelets,” SIAM, Philadelphia, 1992.

[15] Ole Christensen and K. L. Christensen, “Approximation Theory: From Taylor Polynomial to Wavelets,” Birkhauser, Boston, 2004.

[1] E. Babolian and F. Fattahzadeh, “Numerical Computation Method in Solving Integral Equations by Using Chebyshev Wavelet Operational Matrix of Integration,” Applied Mathematics and Computations, Vol. 188, No. 1, 2007, pp. 1016-1022. doi:10.1016/j.amc.2006.10.073

[2] E. Babolian and F. Fattahzadeh, “Numerical Solution of Differential Equations by Using Chebyshev Wavelet Operational Matrix of Integration,” Applied Mathematics and Computations, Vol. 188, No. 1, 2007, pp. 417-426.

[3] Y. Li, “Solving a Nonlinear Fractional Differential Equation Using Chebyshev Wavelets,” Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 9, 2010, pp. 2284-2292. doi:10.1016/j.cnsns.2009.09.020

[4] J. Biazar, “Solution of Systems of Integral-Differential Equations by Adomian Decomposition Method,” Applied Mathematics and Computation, Vol. 168, No. 2, 2005, pp. 1232-1238. doi:10.1016/j.amc.2004.10.015

[5] J. Biazar, H. Ghazvini and M. Eslami, “He’s Homotopy Perturbation Method for Systems of Integro-Differential Equations,” Chaos, Solitions and Fractals, Vol. 39, No. 3, 2009, pp. 1253-1258. doi:10.1016/j.chaos.2007.06.001

[6] E. Yusufoglu (Agadjanov), “An E?cient Algorithm for Solving Integro-Di?erential Equations System,” Applied Mathematics and Computation, Vol. 192, No. 1, 2007, pp. 51-55. doi:10.1016/j.amc.2007.02.134

[7] J. Biazar and H. Aminikhah, “A New Technique for SolvIng Integro-Differential Equations,” Computers and Mathematics with Applications, Vol. 58, No. 11-12, 2009, pp. 2084- 2090. doi:10.1016/j.camwa.2009.03.042

[8] J. Pour-Mahmoud and M. Y. Rahimi-Ardabili and S. Shamoad, “Numerical Solution of the System of Fredholm Integro-Differential Equations by the Tau Method,” Applied Mathematics and Computation, Vol. 168, 2005, pp. 465-478.

[9] S. Abbasbandy and A. Taati, “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 Ma-thematics, Vol. 231, No. 1, 2009, pp. 106-113. doi:10.1016/j.cam.2009.02.014

[10] A. Arikoglu and I. Ozkol, “Solutions of Integral and Integro-Differential Equation Systems by Using Differential Transform Method,” Computers and Mathematics with Applications, Vol. 56, No. 9, 2008, pp. 2411-2417. doi:10.1016/j.camwa.2008.05.017

[11] M. Gachpazan, “Numerical Scheme to Solve Integro- Di?erential Equations System,” Journal of Advanced Research in Scientific Computing, Vol. 1, No. 1, 2009, pp. 11-21.

[12] K. Maleknejad, F. Mirzaee and S. Abbasbandy, “Solving Linear Integro-Differential Equations System by Using Rationalized Haar Functions Method,” Applied Mathematics and Computation, Vol. 155, No. 2, 2004, pp. 317-328. doi:10.1016/S0096-3003(03)00778-1

[13] K. Maleknejad and M. Tavassoli Kajani, “Solving Linear Integro-Differential Equation System by Galerkin Methods with Hybrid Functions,” Applied Mathematics and Computation, Vol. 159, No. 3, 2004, pp. 603-612. doi:10.1016/j.amc.2003.10.046

[14] I. Daubeches, “Ten Lectures on Wavelets,” SIAM, Philadelphia, 1992.

[15] Ole Christensen and K. L. Christensen, “Approximation Theory: From Taylor Polynomial to Wavelets,” Birkhauser, Boston, 2004.