Approximate Analytical Solution of a Class of Singular Differential Equations with Dirichlet Boundary Conditions by the Modified Adomian Decomposition Method
Keywords:
Modified Adomian Decomposition Method, Lane-Emden Type Equations, Singularity, Dirichlet Boundary Conditions.
Abstract
In this paper, the difficulty associated with the numerical solution of a class of singular differential equation with dirichlet-boundary conditions is considered and solved by the Modified Adomian Decomposition Method (MADM), based on a new operator propose to remove its singularity . The new scheme is tested for some examples and the obtained results present. These results reveal the suitability and efficiency of the propose method for this class of problem especially when comparisons is made with the exact solution and other techniques in the literature.
References
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[35] Adomian, G., Rach, R., Shaiwagfeh, N. T. On the analytic solution of the Lane-Emden equations. Phys. letter, 2, 161-181, (1995).
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[37] Wazwaz, A. M. Necessary condition for the appearance of noise term in decomposition series, Appl. Math. Comput. 81, 265-274, (1997).
[2] Junfeng, L. U. Variation iteration method for solving two point boundary value problems. J. Comput. Appl. Math, 207, 92-95, (2007).
[3] Ravi Kanth, A. S. V., Reddy, Y. N. Cubic spline for a class of singular two point boundary value problems. Appl. Math. Comput, 170, 733-740, (2005).
[4] Al-Khaled, K. Theory and computation in singular boundary value problems. Chaos Solitons Fractals, 33(2):678-684, (2007).
[5] Caglar, N., Caglar, H. B-spline solution of singular boundary value problems. Appl. Math. Comput, 182, 1509-1513, (2006).
[6] Sanni Bataineh, A., Noorani, M. S. M., Hashim, I. Approximate solutions of singular two point boundary value problems by modified homotopy analysis method, Physics Letters. A, 372, 4062-4066, (2008).
[7] Inc, M., Ergut, E., Cherrault, Y. A different approach for solving singular two point boundary value problems: analytical and numerical treatment. Kybernetes, 34, 934-940, (2005).
[8] Jang, B. Two point boundary value problems by the extended Adomian decomposition method. J. Comput. Appl. Math, 219, 253-263, (2008).
[9] Aly, E. H., Ebaid, A., and Rach, R. Advances in the Adomian decomposition method for solving two-point nonlinear boundary value problems with neumann boundary conditions. Comput. Math. Aplic, 63, 1056-1065, (2012).
[10] Chun, C., Ebaid, A., Lec, Mi, Aly, Emad, An approach for solving singular two point boundary value problems: analytical and numerical treatment. Anzian Journal 53 (2011).
[11] Shawagfeh, N. T. Non perturbative approximate solution of Lane-Emden equations, J. Math. Phys, 34, 4364, (1993).
[12] Wazwaz, A. M. A new analytical algorithm of Lane -Emden type equation. Appl. Math. Comput, , 118: 287-310 ,(2001).
[13] Parand, K., Dehghan, M., Riezaeia, A. R., Ghaderi, S. M. An approximate algorithm for the solution of the nonlinear Lane-Emden type eqautions arising in astophysics using Hermite function collocation method, comput. phys. comm, 181, 1096, (2010).
[14] Rarikanth, A. S. V., Reddy, Y. N. Solving singular two point boundary value problems using continuous crenetic algorithm. Appl. Math. Comput, 155, 249, (2004).
[15] Ratib Anakira, N., Alomart, A. K., Hashim, I. Numerical Scheme for solving singular two point boundary value problems. Journal of Applied Mathematics, , (2013). doi.org/10.1155/2013/468909.
[16] Wazwaz, A. M. The modied decomposition method for analytical treatment of dierential equations. Appl. Math. Compt, 173, 165-176, (2006).
[17] El-Sayed, S. M. Integral methods for computing solutions of a class of singular two-point boundary value problems. Appl. Math. Comput, 130, 235-241, (2002).
[18] Ebaid, A. Exact solutions for a class of nonlinear singular two-point boundary value problems. The decomposition method. Z. Naturfarsch. A, 65, 145-150, (2010).
[19] Adomian, G. A review of the decomposition method in applied mathematics. J. Math. Anal. Appl, 135, 501, (1988).
[20] Adomian, G. Solving frontier problems of physics: The decomposition method. Kluwer, Boston, MA, 1994., (1994).
[21] Adomian, G. A review of the decomposition method and some recent results for non linear equations. Math. Comput. modelling, 3, 17-43., (1992).
[22] Wazwaz, A. M. A reliable modication of Adomian decomposition method. Apple. Math.Comput, 102(1), 77-86, (1999).
[23] Wazwaz, A. M. A new method for solving singular initial value problems in the second order ordinary dierential equation. Appl. Math. Comput, 128, 45-57, (2002).
[24] Wazwaz, A. M. A new algorithm for calculating Adomian polynomials fof nonlinear operators. Appl. Math. Comput, 111, 53, (2000).
[25] Pourdarvish, A. A. Reliable symbolic implementation of algorithm for calculating Adomian polynomials. Appl. Math. Comput, 172, 545-550, (2006).
[26] Babolian, E., Biazar, Z. An alternate algorithm for computing Adomian polynomials in special case. Appl. Math. Comput, 132:167-172, (2002).
[27] Biazar, J., Shaof, S. M. Int. J. Contemp. Math. Sciences, 2(20): 975-982, (2007).
[28] Cherruault, Y. Convergence of Adomian method. Kyberneters, 8(2), 31, (1998).
[29] Cherruault, Y. Modeles et methods et methodes, mathematiques pour les sciences du vivant, presses, universitaires, de france, (1998).
[30] Cherruault, Y. Optimization methodes locales et Globales, presses universitaries de france, (1999).
[31] Himoun, N., Abbaoui, K., Cherruault,Y. New results of convergence of Adomian's method, 28(4), 423-429, (1999).
[32] Hosseini, M. M., Nasabzadeh, H. On the convergence of Adomian decomposition method. Apple. Math.Comput. 182, 536-543, (2006).
[33] Cherruault, Y., Adomian, G., Abbaoui, K. and Rach, R. Further remarks on convergence of decomposition method. Bio medical comput, 38, 89-93, (1995).
[34] Abbaoui, K., Cherrault, Y. New idea for proving convergence of decomposition method. Comput. Math. Appl, 29(7); 103-108, (1996).
[35] Adomian, G., Rach, R., Shaiwagfeh, N. T. On the analytic solution of the Lane-Emden equations. Phys. letter, 2, 161-181, (1995).
[36] Adomian, G. and Rach, R. Noise term in decomposition series solution. Comput. Math. Appl, 24 (11), 61-64, (1992).
[37] Wazwaz, A. M. Necessary condition for the appearance of noise term in decomposition series, Appl. Math. Comput. 81, 265-274, (1997).
Published
2019-02-17
How to Cite
Yahya, Q. H., Osilagun, J. A., & Adegbindin, A. O. (2019). Approximate Analytical Solution of a Class of Singular Differential Equations with Dirichlet Boundary Conditions by the Modified Adomian Decomposition Method. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 2019(1), 433 - 442. Retrieved from http://ijmso.unilag.edu.ng/article/view/312
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