On Discrete Adomian Decomposition Method with Chebyshev Abscissa for Nonlinear Integral Equations of Hammerstein Type
Affiliation(s)
Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University, Jeddah, Saudi Arabia.
Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University, Jeddah, Saudi Arabia.
ABSTRACT
We study approximate solutions of a nonlinear integral equation of Hammerstein type. We describe the principle of discrete Adomian decomposition method (DADM). DADM is considered in the case we evaluate numerical integration by using Chebyshev roots. This technique gives an accurate solutions as will shown by illustrate examples.
We study approximate solutions of a nonlinear integral equation of Hammerstein type. We describe the principle of discrete Adomian decomposition method (DADM). DADM is considered in the case we evaluate numerical integration by using Chebyshev roots. This technique gives an accurate solutions as will shown by illustrate examples.
Cite this paper
H. Bakodah and M. Darwish, "On Discrete Adomian Decomposition Method with Chebyshev Abscissa for Nonlinear Integral Equations of Hammerstein Type," Advances in Pure Mathematics, Vol. 2 No. 5, 2012, pp. 310-313. doi: 10.4236/apm.2012.25042.
H. Bakodah and M. Darwish, "On Discrete Adomian Decomposition Method with Chebyshev Abscissa for Nonlinear Integral Equations of Hammerstein Type," Advances in Pure Mathematics, Vol. 2 No. 5, 2012, pp. 310-313. doi: 10.4236/apm.2012.25042.
References
[1] J. Appell and C. Chen, “How to Solve Hammerstein Equations,” Journal of Integral Equations and Applications, Vol. 18, No. 3, 2006, pp. 287-296. doi:10.1216/jiea/1181075392 [2] K. Deimling, “Nonlinear Functional Analysis,” Springer-Verlag, Berlin, 1985. doi:10.1007/978-3-662-00547-7 [3] A.A. El-Bary, “Sobolev’s Method for Hammerstein Integral Equations,” Computers & Mathematics with Applications, Vol. 11, No. 2, 2006, pp. 91-94. [4] X. Liu, “On a Nonlinear Hammerstein Integral Equation with a Parameter,” Nonlinear Analysis, Vol. 70, No. 11, 2009, pp. 3887-3893. doi:10.1016/j.na.2008.07.038 [5] D. O’Regan and M. Meehan, “Existence Theory for Nonlinear Integral and Integrodifferential Equations,” Kluwer Academic Publishers, Dordrecht, 1998. doi:10.1007/978-94-011-4992-1 [6] G. Adomian, “A Review of the Decomposition Method in Applied Mathematics,” Journal of Mathematical Analysis and Applications, Vol. 135, No. 2, 1988, pp. 501-544. doi:10.1016/0022-247X(88)90170-9 [7] G. Adomian, “Solving Frontier Problems of Physics: The Decomposition Method,” Kluwer Academic Publishers, Dordrecht, 1994. [8] A. M. Wazwaz, “A Reliable Treatment for Mixed Volterra-Fredholm Integral Equations,” Applied Mathematics and Computation, Vol. 127, No. 2-3, 2002, pp. 405-414. doi:10.1016/S0096-3003(01)00020-0 [9] A. M. Wazwaz and S. M. El-Sayed, “A New Modification of the Adomian Decomposition Method for Linear and Nonlinear Operators,” Applied Mathematics and Computation, Vol. 122, No. 3, 2001, pp. 393-405. doi:10.1016/S0096-3003(00)00060-6 [10] H. Brunner, “Implicitly Linear Collocation Methods for Nonlinear Volterra Equations,” Applied Mathematics and Computation, Vol. 9, No. 3-5, 1992, pp. 235-247. [11] T. Tang, S. McKee and T. Diogo, “Product Integration Methods for an Integral Equation with Logarithmic Singular Kernel,” Applied Numerical Mathematics, Vol. 9, No. 3-5, 1992, pp. 259-266. doi:10.1016/0168-9274(92)90020-E [12] N. Bildik and M. Inc, “Modified Decomposition Method for Nonlinear Volterra-Fredholm Integral Equations,” Chaos Solitons Fractals, Vol. 33, No. 1, 2007, pp. 308- 313. doi:10.1016/j.chaos.2005.12.058 [13] H. L. Arora and F. I. Abdelwahid, “Solution of Non-Integer Order Differential Equations via the Adomian Decomposition Method,” Applied Mathematics Letters, Vol. 6, No. 1, 1993, pp. 21-23. doi:10.1016/0893-9659(93)90140-I [14] M. M. Hosseini and H. Nasabzadeh, “Modified Adomian Decomposition Method for Specific Second Order Ordinary Differential Equations,” Applied Mathematics and Computation, Vol. 186, No. 1, 2007, pp. 117-123. doi:10.1016/j.amc.2006.07.094 [15] D. Kaya, “The Use of Adomian Decomposition Method for Solving a Specific Nonlinear Partial Differential Equations,” Bulletin of the Belgian Mathematical Society Simon Stevin, Vol. 9, No. 3, 2002, pp. 343-349. [16] S. Shah, A. Shaikh and S. H. Sandilo, “Modified Decomposition Method for Nonlinear Volterra-Fredholm Integrodifferential Equation,” Journal of Basic & Applied Sciences, Vol. 6, No. 1, 2010, pp. 13-16. [17] S. H. Behiry, R. A. Abd-Elmonem and A. M. Gomaa, “Discrete Adomian Decomposition Solution of Nonlinear Fredholm Integral Equation,” Ain Shams Engineering Journal, Vol. 1, No. 1, 2010, pp. 97-101. doi:10.1016/j.asej.2010.09.009 [18] J. C. Mason and D. C. Handscomb, “Chebyshev Polynomials,” CRC Press, New York, 2003. [19] A. M. Wazwaz, “A First Course in Integral Equations,” World Scientific Publishing, Singapore City, 1997. [20] A. Jerri, “Introduction to Integral Equations with Applications,” John Wiley & Sons, New York, 1999.
[1] J. Appell and C. Chen, “How to Solve Hammerstein Equations,” Journal of Integral Equations and Applications, Vol. 18, No. 3, 2006, pp. 287-296. doi:10.1216/jiea/1181075392 [2] K. Deimling, “Nonlinear Functional Analysis,” Springer-Verlag, Berlin, 1985. doi:10.1007/978-3-662-00547-7 [3] A.A. El-Bary, “Sobolev’s Method for Hammerstein Integral Equations,” Computers & Mathematics with Applications, Vol. 11, No. 2, 2006, pp. 91-94. [4] X. Liu, “On a Nonlinear Hammerstein Integral Equation with a Parameter,” Nonlinear Analysis, Vol. 70, No. 11, 2009, pp. 3887-3893. doi:10.1016/j.na.2008.07.038 [5] D. O’Regan and M. Meehan, “Existence Theory for Nonlinear Integral and Integrodifferential Equations,” Kluwer Academic Publishers, Dordrecht, 1998. doi:10.1007/978-94-011-4992-1 [6] G. Adomian, “A Review of the Decomposition Method in Applied Mathematics,” Journal of Mathematical Analysis and Applications, Vol. 135, No. 2, 1988, pp. 501-544. doi:10.1016/0022-247X(88)90170-9 [7] G. Adomian, “Solving Frontier Problems of Physics: The Decomposition Method,” Kluwer Academic Publishers, Dordrecht, 1994. [8] A. M. Wazwaz, “A Reliable Treatment for Mixed Volterra-Fredholm Integral Equations,” Applied Mathematics and Computation, Vol. 127, No. 2-3, 2002, pp. 405-414. doi:10.1016/S0096-3003(01)00020-0 [9] A. M. Wazwaz and S. M. El-Sayed, “A New Modification of the Adomian Decomposition Method for Linear and Nonlinear Operators,” Applied Mathematics and Computation, Vol. 122, No. 3, 2001, pp. 393-405. doi:10.1016/S0096-3003(00)00060-6 [10] H. Brunner, “Implicitly Linear Collocation Methods for Nonlinear Volterra Equations,” Applied Mathematics and Computation, Vol. 9, No. 3-5, 1992, pp. 235-247. [11] T. Tang, S. McKee and T. Diogo, “Product Integration Methods for an Integral Equation with Logarithmic Singular Kernel,” Applied Numerical Mathematics, Vol. 9, No. 3-5, 1992, pp. 259-266. doi:10.1016/0168-9274(92)90020-E [12] N. Bildik and M. Inc, “Modified Decomposition Method for Nonlinear Volterra-Fredholm Integral Equations,” Chaos Solitons Fractals, Vol. 33, No. 1, 2007, pp. 308- 313. doi:10.1016/j.chaos.2005.12.058 [13] H. L. Arora and F. I. Abdelwahid, “Solution of Non-Integer Order Differential Equations via the Adomian Decomposition Method,” Applied Mathematics Letters, Vol. 6, No. 1, 1993, pp. 21-23. doi:10.1016/0893-9659(93)90140-I [14] M. M. Hosseini and H. Nasabzadeh, “Modified Adomian Decomposition Method for Specific Second Order Ordinary Differential Equations,” Applied Mathematics and Computation, Vol. 186, No. 1, 2007, pp. 117-123. doi:10.1016/j.amc.2006.07.094 [15] D. Kaya, “The Use of Adomian Decomposition Method for Solving a Specific Nonlinear Partial Differential Equations,” Bulletin of the Belgian Mathematical Society Simon Stevin, Vol. 9, No. 3, 2002, pp. 343-349. [16] S. Shah, A. Shaikh and S. H. Sandilo, “Modified Decomposition Method for Nonlinear Volterra-Fredholm Integrodifferential Equation,” Journal of Basic & Applied Sciences, Vol. 6, No. 1, 2010, pp. 13-16. [17] S. H. Behiry, R. A. Abd-Elmonem and A. M. Gomaa, “Discrete Adomian Decomposition Solution of Nonlinear Fredholm Integral Equation,” Ain Shams Engineering Journal, Vol. 1, No. 1, 2010, pp. 97-101. doi:10.1016/j.asej.2010.09.009 [18] J. C. Mason and D. C. Handscomb, “Chebyshev Polynomials,” CRC Press, New York, 2003. [19] A. M. Wazwaz, “A First Course in Integral Equations,” World Scientific Publishing, Singapore City, 1997. [20] A. Jerri, “Introduction to Integral Equations with Applications,” John Wiley & Sons, New York, 1999.