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Explicit series solutions of some linear and nonlinear Schrödinger equations via the homotopy analysis method. (English) Zbl 1221.35389
Summary: By means of the homotopy analysis method (HAM), the solutions of some Schrodinger equations are exactly obtained in the form of convergent Taylor series. The HAM contains the auxiliary parameter \(\hbar\), that provides a convenient way of controlling the convergent region of series solutions. This analytical method is employed to solve linear and nonlinear examples to obtain the exact solutions. HAM is a powerful and easy-to-use analytic tool for nonlinear problems.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35C10 Series solutions to PDEs
35J10 Schrödinger operator, Schrödinger equation
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
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[1] Sadighi A, Ganji DD. Analytic treatment of linear and nonlinear Schrdinger equations: a study with homotopy-perturbation and Adomian decomposition methods. Phys Lett A. doi:10.1016/j.physleta.2007.07.065. · Zbl 1217.81069
[2] Batiha K. Approximate analytical solution for the ZakharovKuznetsov equations with fully nonlinear dispersion. J Comput Appl Math. doi:10.1016/j.cam.2007.04.020. · Zbl 1138.65092
[3] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Boston · Zbl 0802.65122
[4] Adomian, G.; Rach, R., Noise terms in decomposition solution series, J comput math appl, 24, 11, 61-64, (1992) · Zbl 0777.35018
[5] Ruan, J.; Zhengyi, L., A modified algorithm for the Adomian decomposition method with applications to lotkavolterra systems, Math comp model, 46, 9-10, 1214-1224, (2007) · Zbl 1133.65046
[6] Bahnasawi, A.A.; El-Tawil, M.A.; Abdel-Naby, A., Solving Riccati differential equation using adomians decomposition method, Appl math comput, 157, 503-514, (2004) · Zbl 1054.65071
[7] Wazwaz A. A study on linear and nonlinear Schrodinger equations by the variational iteration method. Chaos Soliton Fract. doi:10.1016/j.chaos.2006.10.009. · Zbl 1148.35353
[8] He, J.-H.; Wu, X.-H., Construction of solitary solution and compacton-like solution by variational iteration method, Chaos soliton fract, 29, 108-113, (2006) · Zbl 1147.35338
[9] Batiha, B.; Noorani, M.S.M.; Hashim, I., Application of variational iteration method to heat and wave-like equations, Phys lett A, 369, 1-2, 55-61, (2007) · Zbl 1209.80040
[10] Odibat, Z.; Momani, S., Approximate solutions for boundary value problems of time-fractional wave equation, Appl math comput, 181, 767-774, (2006) · Zbl 1148.65100
[11] Wang, H., Numerical studies on the split-step finite difference method for nonlinear schrodinger equations, Appl math comput, 170, 17-35, (2005) · Zbl 1082.65570
[12] Biazar, J.; Ghazvini, H., Exact solutions for non-linear schrodinger equations by hes homotopy perturbation method, Phys lett A, 366, 79-84, (2007) · Zbl 1203.65207
[13] Khuri, S.A., A new approach to the cubic schrodinger equation: an application of the decomposition technique, Appl math comput, 97, 251-254, (1998) · Zbl 0940.35187
[14] Liao SJ. The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. thesis, Shanghai Jiao Tong University; 1992.
[15] Liao, S.J., Beyond perturbation: introduction to the homotopy analysis method, (2003), CRC Press, Chapman and Hall Boca Raton
[16] Liao, S.J., An approximate solution technique which does not depend upon small parameters (part 2): an application in fluid mechanics, Int J nonlinear mech, 32, 815-822, (1997) · Zbl 1031.76542
[17] Liao, S.J., An approximate solution technique which does not depend upon small parameters: a special example, Int J nonlinear mech, 30, 371-380, (1995) · Zbl 0837.76073
[18] Liao, S.J., Comparison between the homotopy analysis method and homotopy perturbation method, Appl math comput, 169, 1186-1194, (2005) · Zbl 1082.65534
[19] Liao, S.J., A new branch of solutions of boundary-layer flows over an impermeable stretched plate, Int J heat mass transf, 48, 2529-3259, (2005) · Zbl 1189.76142
[20] Liao, S.J.; Pop, I., Explicit analytic solution for similarity boundary layer equations, Int J heat mass transf, 47, 75-78, (2004) · Zbl 1045.76008
[21] Liao, S.J., On the homotopy analysis method for nonlinear problems, Appl math comput, 147, 499-513, (2004) · Zbl 1086.35005
[22] Ayub, M.; Rasheed, A.; Hayat, T., Exact flow of a third grade fluid past a porous plate using homotopy analysis method, Int J eng sci, 41, 2091-2103, (2003) · Zbl 1211.76076
[23] Hayat, T.; Khan, M.; Asghar, S., Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid, Acta mech, 167, 213-232, (2004) · Zbl 1063.76108
[24] Hayat, T.; Khan, M., Homotopy solutions for a generalized second-grade fluid past a porous plate, Nonlinear dyn, 42, 395-405, (2005) · Zbl 1094.76005
[25] Tan, Y.; Abbasbandy, S., Homotopy analysis method for quadratic Riccati differential equation, Comm nonlinear sci numer simul, 13, 3, 539-546, (2008) · Zbl 1132.34305
[26] Abbasbandy, S., The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys lett A, 109-113, (2006) · Zbl 1236.80010
[27] Abbasbandy, S., The application of homotopy analysis method to solve a generalized hirotasatsuma coupled KdV equation, Phys lett A, 15, 1-6, (2006)
[28] Hayat, T.; Sajid, M., On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder, Phys lett A, 361, 316-322, (2007) · Zbl 1170.76307
[29] Wang, C.; Wu, Y.; Wu, W., Solving the nonlinear periodic wave problems with the homotopy analysis method, Wave motion, 41, 329-337, (2004)
[30] Wu, Y.; Wang, C.; Liao, S.J., Solving the one-loop soliton solution of the Vakhnenko equation by means of the homotopy analysis method, Chaos soliton fract, 23, 1733-1740, (2005) · Zbl 1069.35060
[31] Sami Bataineh A, Noorani MSM, Hashim I. Solving systems of ODEs by homotopy analysis method. Comm Nonlinear Sci Numer Simul. doi:10.1016/j.cnsns.2007.05.026. · Zbl 1221.65194
[32] Yabushita, K.; Yamashita, M.; Tsuboi, K., An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method, J phys A: math ther, 40, 8403-8416, (2007) · Zbl 1331.70041
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