Convergence of Discrete Adomian Method for Solving a Class of Nonlinear Fredholm Integral Equations

Affiliation(s)

Mathematics and Engineering Physics Department, Faculty of Engineering, Mansoura University, Mansoura, Egypt.

Mathematics and Engineering Physics Department, Faculty of Engineering, Mansoura University, Mansoura, Egypt.

ABSTRACT

In recent papers the solution of nonlinear Fredholm integral equations was discussed using Adomian decomposition method (ADM). For case in which the integrals are analytically impossible, ADM can not be applied. In this paper a discretized version of the ADM is introduced and the proposed version will be called discrete Adomian decomposition method (DADM). An accelerated formula of Adomian polynomials is used in calculations. Based on this formula, a new convergence approach of ADM is introduced. Convergence approach is reliable enough to obtain an explicit formula for the maximum absolute truncated error of the Adomian’s series solution. Also, we prove that the solution of nonlinear Fredholm integral equation by DADM converges to ADM solution. Finally, some numerical examples were introduced.

Cite this paper

I. Alkalla, R. Abd-Elmonem and A. Gomaa, "Convergence of Discrete Adomian Method for Solving a Class of Nonlinear Fredholm Integral Equations,"*Applied Mathematics*, Vol. 4 No. 1, 2013, pp. 217-222. doi: 10.4236/am.2013.41A033.

I. Alkalla, R. Abd-Elmonem and A. Gomaa, "Convergence of Discrete Adomian Method for Solving a Class of Nonlinear Fredholm Integral Equations,"

References

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[11] I. L. El-kalla, “Convergence of Adomian’s Method Applied to a Class of Volterra Type Integro-Differential Equations,” International Journal of Differential Equations and Applications, Vol. 10, No. 2, 2005, pp. 225-234.

[12] I. L. El-kalla, “Error Analysis of Adomian Series Solution to a Class of Nonlinear Differential Equations,” Applied Mathematics E-Notes, Vol. 7, 2007, pp. 214-221.

[13] I. L. El-Kalla, “Error Estimate of the Series Solution to a Class of Nonlinear Fractional Differential Equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 16, No. 3, 2011, pp. 1408-1413. doi:10.1016/j.cnsns.2010.05.030

[14] E. A. Az-Zo’bi and K. Al-Khaled, “A New Convergence Proof of the Adomian Decomposition Method for a Mixed Hyperbolic Elliptic System of Conservation Laws,” Applied Mathematics and Computation, Vol. 217, No. 8, 2010, pp. 4248-4256.

[15] E. Babolian and A. Davari, “Numerical Implementation of Adomain Decomposition Method,” Applied Mathematics and Computation, Vol. 153, No. 1, 2004, pp. 301-305. doi:10.1016/S0096-3003(03)00646-5

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[20] M. M. Hosseini and H. Nasabzadeh, “On the Convergence of Adomian Decomposition Method,” Applied Mathematics and Computation, Vol. 182, No. 1, 2006, pp. 536-543. doi:10.1016/j.amc.2006.04.015

[21] K. Abbaoui and Y. Cherruault, “Convergence of Adomian’s Method Applied to Differential Equations,” Computers & Mathematics with Applications, Vol. 28, No. 5, 1994, pp. 103-109. doi:10.1016/0898-1221(94)00144-8

[22] R. Rajaram and M. Najafi, “Analytical Treatment and Convergence of the Adomian Decomposition Method for a System of Coupled Damped Wave Equations,” Applied Mathematics and Computation, Vol. 212, No. 1, 2009, pp. 72-81. doi:10.1016/j.amc.2009.02.006

[23] E. Kreyszig, “Introductory Functional Analysis with Applications,” John Wiley & Sons Inc., New York, 1978.

[1] K. E. Atkinson, “A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind,” Society for Industrial and Applied Mathematics, Philadelphia, 1976, p. 237.

[2] F. G. Tricomi, “Integral Equations,” Dover Publications Inc., New York, 1985.

[3] L. M. Delves and J. L. Mohamed, “Computational Methods for Integral Equations,” Cambridge University Press, New York, 1985. doi:10.1017/CBO9780511569609

[4] M. A. Golbluerg, “Numerical Solution of Integral Equations,” Plenum Press, New York, 1990.

[5] G. Adomian, “Solving Frontier Problems of Physics: The Decomposition Method,” Kluwer Academic Publishers, Dordrecht, 1994.

[6] A. M. Wazwaz and S. A. Khuri, “Two Methods for Solving Integral Equations,” Applied Mathematics and Computation, Vol. 77, No. 1, 1996, pp. 79-89. doi:10.1016/0096-3003(95)00189-1

[7] A. M. Wazwaz, “A First Course in Integral Equations,” World Scientific Publishing Co., Singapore City, 1997. doi:10.1142/3444

[8] E. Babolian, J. Biazar and A. R. Vahidi, “The Decomposition Method Applied to Systems of Fredholm Integral Equations of the Second Kind,” Applied Mathematics and Computation, Vol. 148, No. 2, 2004, pp. 443-452. doi:10.1016/S0096-3003(02)00859-7

[9] A. M. Wazwaz, “A Comparison Study between the Modified Decomposition Method and the Traditional Methods for Solving Nonlinear Integral Equations,” Applied Mathematics and Computation, Vol. 181, No. 2, 2006, pp. 1703-1712. doi:10.1016/j.amc.2006.03.023

[10] Y. Cherruault, G. Adomian, K. Abbaoui and R. Rach, “Further Remarks on Convergence of Decomposition Method,” International Journal of Bio-Medical Computing, Vol. 38, No. 1, 1995, pp. 89-93. doi:10.1016/0020-7101(94)01042-Y

[11] I. L. El-kalla, “Convergence of Adomian’s Method Applied to a Class of Volterra Type Integro-Differential Equations,” International Journal of Differential Equations and Applications, Vol. 10, No. 2, 2005, pp. 225-234.

[12] I. L. El-kalla, “Error Analysis of Adomian Series Solution to a Class of Nonlinear Differential Equations,” Applied Mathematics E-Notes, Vol. 7, 2007, pp. 214-221.

[13] I. L. El-Kalla, “Error Estimate of the Series Solution to a Class of Nonlinear Fractional Differential Equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 16, No. 3, 2011, pp. 1408-1413. doi:10.1016/j.cnsns.2010.05.030

[14] E. A. Az-Zo’bi and K. Al-Khaled, “A New Convergence Proof of the Adomian Decomposition Method for a Mixed Hyperbolic Elliptic System of Conservation Laws,” Applied Mathematics and Computation, Vol. 217, No. 8, 2010, pp. 4248-4256.

[15] E. Babolian and A. Davari, “Numerical Implementation of Adomain Decomposition Method,” Applied Mathematics and Computation, Vol. 153, No. 1, 2004, pp. 301-305. doi:10.1016/S0096-3003(03)00646-5

[16] A. J. Jerri, “Introduction to Integral Equations with Applications,” John Wiley & Sons Inc., New York, 1999.

[17] J. Stoer and R. Bulirsch, “Introduction to Numerical Analysis,” 3rd Edition, Springer-Verlag, Berlin, Heidelberg, New York, 2002.

[18] K. E. Atkinson, “The Numerical Solution of Integral Equations of the Second Kind,” Cambridge University Press, New York, 1997. doi:10.1017/CBO9780511626340

[19] K. Abbaoui and Y. Cherruault, “New Ideas for Proving Convergence of Decomposition Methods,” Computers & Mathematics with Applications, Vol. 29, No. 7, 1995, pp. 103-108. doi:10.1016/0898-1221(95)00022-Q

[20] M. M. Hosseini and H. Nasabzadeh, “On the Convergence of Adomian Decomposition Method,” Applied Mathematics and Computation, Vol. 182, No. 1, 2006, pp. 536-543. doi:10.1016/j.amc.2006.04.015

[21] K. Abbaoui and Y. Cherruault, “Convergence of Adomian’s Method Applied to Differential Equations,” Computers & Mathematics with Applications, Vol. 28, No. 5, 1994, pp. 103-109. doi:10.1016/0898-1221(94)00144-8

[22] R. Rajaram and M. Najafi, “Analytical Treatment and Convergence of the Adomian Decomposition Method for a System of Coupled Damped Wave Equations,” Applied Mathematics and Computation, Vol. 212, No. 1, 2009, pp. 72-81. doi:10.1016/j.amc.2009.02.006

[23] E. Kreyszig, “Introductory Functional Analysis with Applications,” John Wiley & Sons Inc., New York, 1978.