## On the solutions of nonlinear higher-order boundary value problems by using differential transformation method and Adomian decomposition method.(English)Zbl 1213.65148

Summary: We study higher-order boundary value problems (HOBVP) for higher-order nonlinear differential equation. We make comparison among differential transformation method (DTM), Adomian decomposition method (ADM), and exact solutions. We provide several examples in order to compare our results. We extend and prove a theorem for nonlinear differential equations by using the DTM. The numerical examples show that the DTM is a good method compared to the ADM since it is effective, uses less time in computation, easy to implement and achieve high accuracy. In addition, DTM has many advantages compared to ADM since the calculation of Adomian polynomial is tedious. From the numerical results, DTM is suitable to apply for nonlinear problems.

### MSC:

 65N99 Numerical methods for partial differential equations, boundary value problems
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### References:

 [1] A.-M. Wazwaz, “A reliable modification of Adomian decomposition method,” Applied Mathematics and Computation, vol. 102, no. 1, pp. 77-86, 1999. · Zbl 0928.65083 [2] A.-M. Wazwaz, “Approximate solutions to boundary value problems of higher order by the modified decomposition method,” Computers & Mathematics with Applications, vol. 40, no. 6-7, pp. 679-691, 2000. · Zbl 0959.65090 [3] A.-M. Wazwaz, “The numerical solution of fifth-order boundary value problems by the decomposition method,” Journal of Computational and Applied Mathematics, vol. 136, no. 1-2, pp. 259-270, 2001. · Zbl 0986.65072 [4] A.-M. Wazwaz, “The numerical solution of sixth-order boundary value problems by the modified decomposition method,” Applied Mathematics and Computation, vol. 118, no. 2-3, pp. 311-325, 2001. · Zbl 1023.65074 [5] M. Me, “The modified decomposition method for eighth-order boundary value problems,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1437-1444, 2007. · Zbl 1119.65069 [6] M. M. Hosseini and M. Jafari, “A note on the use of Adomian decomposition method for highorder and system of nonlinear differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, pp. 1952-1957, 2009. · Zbl 1221.65162 [7] M. Dehghan and F. Shakeri, “The numerical solution of the second Painlevé equation,” Numerical Methods for Partial Differential Equations, vol. 25, no. 5, pp. 1238-1259, 2009. · Zbl 1172.65037 [8] M. Dehghan and M. Tatari, “The use of Adomian decomposition method for solving problems in calculus of variations,” Mathematical Problems in Engineering, vol. 2006, Article ID 65379, 9 pages, 2006. · Zbl 1200.65050 [9] M. Dehghan and R. Salehi, “A seminumeric approach for solution of the eikonal partial differential equation and its applications,” Numerical Methods for Partial Differential Equations, vol. 26, no. 3, pp. 702-722, 2010. · Zbl 1189.65237 [10] M. Dehghan, J. M. Heris, and A. Saadatmandi, “Application of semi-analytic methods for the Fitzhugh-Nagumo equation, which models the transmission of nerve impulses,” Mathematical Methods in the Applied Sciences, vol. 33, no. 11, pp. 1384-1398, 2010. · Zbl 1196.35025 [11] M. Dehghan, M. Shakourifar, and A. Hamidi, “The solution of linear and nonlinear systems of Volterra functional equations using Adomian-Pade technique,” Chaos, Solitons and Fractals, vol. 39, no. 5, pp. 2509-2521, 2009. · Zbl 1197.65223 [12] M. Dehghan, A. Hamidi, and M. Shakourifar, “The solution of coupled Burgers’ equations using Adomian-Pade technique,” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1034-1047, 2007. · Zbl 1122.65388 [13] M. Dehghan and R. Salehi, “Solution of a nonlinear time-delay model in biology via semi-analytical approaches,” Computer Physics Communications, vol. 181, no. 7, pp. 1255-1265, 2010. · Zbl 1219.65062 [14] F. Shakeri and M. Dehghan, “Application of the decomposition method of adomian for solving the pantograph equation of order m,” Zeitschrift fur Naturforschung, vol. 65, no. 5, pp. 453-460, 2010. [15] M. Dehghan, “The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure,” International Journal of Computer Mathematics, vol. 81, no. 8, pp. 979-989, 2004. · Zbl 1056.65099 [16] S. S. Ray and R. K. Bera, “Analytical solution of a fractional diffusion equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 174, no. 1, pp. 329-336, 2006. · Zbl 1089.65108 [17] F. Ayaz, “On the two-dimensional differential transform method,” Applied Mathematics and Computation, vol. 143, no. 2-3, pp. 361-374, 2003. · Zbl 1023.35005 [18] F. Ayaz, “Solutions of the system of differential equations by differential transform method,” Applied Mathematics and Computation, vol. 147, no. 2, pp. 547-567, 2004. · Zbl 1032.35011 [19] V. S. Ertürk and S. Momani, “Comparing numerical methods for solving fourth-order boundary value problems,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1963-1968, 2007. · Zbl 1119.65066 [20] A. Arikoglu and I. Ozkol, “Solution of fractional differential equations by using differential transform method,” Chaos, Solitons and Fractals, vol. 34, no. 5, pp. 1473-1481, 2007. · Zbl 1152.34306 [21] I. H. Abdel-Halim Hassan and V. S. Ertürk, “Solutions of different types of the linear and non-linear higher-order boundary value problems by differential transformation method,” European Journal of Pure and Applied Mathematics, vol. 2, no. 3, pp. 426-447, 2009. · Zbl 1213.35027
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