×

Two-dimensional differential transform method for solving linear and non-linear Schrödinger equations. (English) Zbl 1198.81089

Summary: We propose a reliable algorithm to develop exact and approximate solutions for the linear and nonlinear Schrödinger equations. The approach rest mainly on two-dimensional differential transform method which is one of the approximate methods. The method can easily be applied to many linear and nonlinear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. Several illustrative examples are given to demonstrate the effectiveness of the present method.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q55 NLS equations (nonlinear Schrödinger equations)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Khuri, S. A., A new approach to the cubic Schrödinger equation: an application of the decomposition technique, Appl Math Comput, 97, 251-254 (1988) · Zbl 0940.35187
[2] Wang, H., Numerical studies on the split-step finite difference method for nonlinear Schrödinger equations, Appl Math Comput, 170, 17-35 (2005) · Zbl 1082.65570
[3] Biazar, J.; Ghazvini, H., Exact solutions for Schrödinger equations by He’s homotopy perturbation method, Phys Lett A, 366, 79-84 (2007) · Zbl 1203.65207
[4] Wazwaz, A. M., A study on linear and nonlinear Schrödinger equations by the variational iteration method, Chaos Solitons & Fractals, 37, 1136-1142 (2008) · Zbl 1148.35353
[5] Sadighi, A.; Ganji, D. D., Analytic treatment of linear and nonlinear Schrödinger equations: a study with homotopy-perturbation and Adomian decomposition methods, Phys Lett A, 372, 465-469 (2008) · Zbl 1217.81069
[6] Zhou, J. K., Differential transform and its applications for electrical circuits (1986), Huazhong University Press: Huazhong University Press Wuhan, China
[7] Chen, C. K.; Ho, S. H., Solving partial differential equations by two-dimensional differential transform method, Appl Math Comput, 106, 171-179 (1999) · Zbl 1028.35008
[8] Jang, M. J.; Chen, C. L.; Liu, Y. C., Two-dimensional differential transform for partial differential equations, Appl Math Comput, 121, 261-270 (2001) · Zbl 1024.65093
[9] Abdel-Halim, Hassan, Different applications for the differential transformation in the differential equations, Appl Math Comput, 129, 183-201 (2002) · Zbl 1026.34010
[10] Ayaz, F., On two-dimensional differential transform method, Appl Math Comput, 143, 361-374 (2003) · Zbl 1023.35005
[11] Ayaz, F., Solution of the system of differential equations by differential transform method, Appl Math Comput, 147, 547-567 (2004) · Zbl 1032.35011
[12] Kurnaz, A.; Oturnaz, G.; Kiris, M. E., n-Dimensional differential transformation method for solving linear and nonlinear PDE’s, Int J Comput Math, 82, 369-380 (2005)
[13] Adbel-Halim, Hassan, Comparison differential transform technique with Adomian decomposition method for linear and nonlinear initial value problems, Chaos Solitons & Fractals, 36, 53-65 (2008) · Zbl 1152.65474
[14] Figen, Kangalgil; Fatma, Ayaz, Solitary wave solutions for the KdV and mKdV equations by differential transform method, Chaos Solitons & Fractals, 41, 1, 464-472 (2009) · Zbl 1198.35222
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.