×

Bäcklund transformation of fractional Riccati equation and infinite sequence solutions of nonlinear fractional PDEs. (English) Zbl 1474.35665

Summary: The Bäcklund transformation of fractional Riccati equation with nonlinear superposition principle of solutions is employed to establish the infinite sequence solutions of nonlinear fractional partial differential equations in the sense of modified Riemann-Liouville derivative. To illustrate the reliability of the method, some examples are provided.

MSC:

35R11 Fractional partial differential equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Miller, K. S.; Ross, B., An Introduction To the Fractional Calculus and Fractional Differential Equations, (1993), New York, NY, USA: Wiley, New York, NY, USA · Zbl 0789.26002
[2] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations, (2006), San Diego, Calif, USA: Elsevier, San Diego, Calif, USA · Zbl 1092.45003
[3] Podlubny, I., Fractional Differential Equations, (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0918.34010
[4] Duan, J. S.; Temuer, C.; Randolph, R., The Adomian decomposition method with convergence acceleration techniques for nonlinear fractional differential equations, Computers & Mathematics with Applications, 66, 5, 728-736, (2013) · Zbl 1348.34010
[5] El-Sayed, A. M. A.; Gaber, M., The Adomian decomposition method for solving partial differential equations of fractal order in finite domains, Physics Letters A, 359, 3, 175-182, (2006) · Zbl 1236.35003
[6] Odibat, Z.; Momani, S., The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics, Computers and Mathematics with Applications, 58, 11-12, 2199-2208, (2009) · Zbl 1189.65254
[7] Wu, G.-C.; Lee, E. W. M., Fractional variational iteration method and its application, Physics Letters A, 374, 25, 2506-2509, (2010) · Zbl 1237.34007
[8] Elbeleze, A. A.; Kilicman, A.; Taib, B. M., Homotopy perturbation method for fractional black-scholes European option pricing equations using sumudu transform, Mathematical Problems in Engineering, 2013, (2013) · Zbl 1299.91179
[9] El-Sayed, A. M. A.; Elsaid, A.; El-Kalla, I. L.; Hammad, D., A homotopy perturbation technique for solving partial differential equations of fractional order in finite domains, Applied Mathematics and Computation, 218, 17, 8329-8340, (2012) · Zbl 1245.65141
[10] Momani, S.; Odibat, Z.; Erturk, V. S., Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation, Physics Letters A, 370, 5-6, 379-387, (2007) · Zbl 1209.35066
[11] Odibat, Z.; Momani, S., A generalized differential transform method for linear partial differential equations of fractional order, Applied Mathematics Letters, 21, 2, 194-199, (2008) · Zbl 1132.35302
[12] Zhang, S.; Zhang, H.-Q., Fractional sub-equation method and its applications to nonlinear fractional PDEs, Physics Letters A, 375, 7, 1069-1073, (2011) · Zbl 1242.35217
[13] Guo, S.; Mei, L.; Li, Y.; Sun, Y., The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics, Physics Letters A, 376, 4, 407-411, (2012) · Zbl 1255.37022
[14] Lu, B., Bäcklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations, Physics Letters A, 376, 28-29, 2045-2048, (2012) · Zbl 1266.35139
[15] Jumarie, G., Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Computers and Mathematics with Applications, 51, 9-10, 1367-1376, (2006) · Zbl 1137.65001
[16] Lee, J.; Sakthivel, R., New exact travelling wave solutions of bidirectional wave equations, Journal of Physics, 76, 6, 819-829, (2011)
[17] Song, L.; Wang, Q.; Zhang, H., Rational approximation solution of the fractional Sharma-Tasso-Olever equation, Journal of Computational and Applied Mathematics, 224, 1, 210-218, (2009) · Zbl 1157.65074
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.