Homotopy perturbation method to obtain positive solutions of nonlinear boundary value problems of fractional order. (English) Zbl 1474.65229

Summary: We use the homotopy perturbation method for solving the fractional nonlinear two-point boundary value problem. The obtained results by the homotopy perturbation method are then compared with the Adomian decomposition method. We solve the fractional Bratu-type problem as an illustrative example.


65L10 Numerical solution of boundary value problems involving ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
34A08 Fractional ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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[1] Jafari, H., An Introduction To Fractional Differential Equations (2013), Mazandaran University Press
[2] Jafari, H.; Sayevand, K.; Tajadodi, H.; Baleanu, D., Homotopy analysis method for solving Abel differential equation of fractional order, Central European Journal of Physics, 11, 10, 1523-1527 (2013)
[3] Kumar, S.; Singh, O. P., Numerical inversion of the abel integral equation using homotopy perturbation method, Zeitschrift Fur Naturforschung, 65, 677-682 (2010)
[4] Wazwaz, A.-M., Adomian decomposition method for a reliable treatment of the Bratu-type equations, Applied Mathematics and Computation, 166, 3, 652-663 (2005) · Zbl 1073.65068 · doi:10.1016/j.amc.2004.06.059
[5] Wazwaz, A. M., A reliable algorithm for obtaining positive solutions for nonlinear boundary value problems, Computers & Mathematics with Applications, 41, 10-11, 1237-1244 (2001) · Zbl 0983.65090 · doi:10.1016/S0898-1221(01)00094-3
[6] Adomian, G.; Rach, R., Analytic solution of nonlinear boundary value problems in several dimensions by decomposition, Journal of Mathematical Analysis and Applications, 174, 1, 118-137 (1993) · Zbl 0796.35017 · doi:10.1006/jmaa.1993.1105
[7] Adomian, G.; Elrod, M.; Rach, R., A new approach to boundary value equations and application to a generalization of Airy’s equation, Journal of Mathematical Analysis and Applications, 140, 2, 554-568 (1989) · Zbl 0678.65057 · doi:10.1016/0022-247X(89)90083-8
[8] Agrawal, O. P., Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynamics, 29, 1-4, 145-155 (2002) · Zbl 1009.65085 · doi:10.1023/A:1016539022492
[9] Metzler, R.; Klafter, J., Boundary value problems for fractional diffusion equations, Physica A. Statistical Mechanics and Its Applications, 278, 1-2, 107-125 (2000) · doi:10.1016/S0378-4371(99)00503-8
[10] Bai, Z.; Lü, H., Positive solutions for boundary value problem of nonlinear fractional differential equation, Journal of Mathematical Analysis and Applications, 311, 2, 495-505 (2005) · Zbl 1079.34048 · doi:10.1016/j.jmaa.2005.02.052
[11] Jafari, H.; Daftardar-Gejji, V., Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method, Applied Mathematics and Computation, 180, 2, 700-706 (2006) · Zbl 1102.65136 · doi:10.1016/j.amc.2006.01.007
[12] Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J. J., Fractional Calculus Models and Numerical Methods. Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, 3 (2012), World Scientific · Zbl 1248.26011 · doi:10.1142/9789814355216
[13] Podlubny, I., Fractional Differential Equations. Fractional Differential Equations, Mathematics in Science and Engineering, 198 (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0924.34008
[14] Shawagfeh, N. T., Analytical approximate solutions for nonlinear fractional differential equations, Applied Mathematics and Computation, 131, 2-3, 517-529 (2002) · Zbl 1029.34003 · doi:10.1016/S0096-3003(01)00167-9
[15] Jafari, H.; Momani, S., Solving fractional diffusion and wave equations by modified homotopy perturbation method, Physics Letters A, 370, 5-6, 388-396 (2007) · Zbl 1209.65111 · doi:10.1016/j.physleta.2007.05.118
[16] Jafari, H.; Ghasempoor, S.; Khalique, C. M., A comparison between Adomian’s polynomials and He’s polynomials for nonlinear functional equations, Mathematical Problems in Engineering, 2013 (2013) · Zbl 1299.34034 · doi:10.1155/2013/943232
[17] Odibat, Z. M., A new modification of the homotopy perturbation method for linear and nonlinear operators, Applied Mathematics and Computation, 189, 1, 746-753 (2007) · Zbl 1122.65092 · doi:10.1016/j.amc.2006.11.188
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