A Series Solution for the Ginzburg-Landau Equation with a Time-Periodic Coefficient

ABSTRACT

The solution of the real Ginzburg-Landau (GL) equation with a time-periodic coefficient is obtained in the form of a series, with assured convergence, using the computer-assisted ‘Homotopy Analysis Method’ (HAM) propounded by Liao [1]. The formulation has been kept quite general to keep open the possibility of obtaining the solution of the GL equation for different continua as limiting cases of the present study. New ideas have been added and clear explanations are provided in the paper to the existing concepts in HAM. The method can easily be extended to solve complex GL equation, system of GL equations or even the GL equations with a diffusion term, each having a time-periodic coefficient. The necessary code in Mathematica that implements the HAM for the current problem is appended to the paper for use by the readers.

The solution of the real Ginzburg-Landau (GL) equation with a time-periodic coefficient is obtained in the form of a series, with assured convergence, using the computer-assisted ‘Homotopy Analysis Method’ (HAM) propounded by Liao [1]. The formulation has been kept quite general to keep open the possibility of obtaining the solution of the GL equation for different continua as limiting cases of the present study. New ideas have been added and clear explanations are provided in the paper to the existing concepts in HAM. The method can easily be extended to solve complex GL equation, system of GL equations or even the GL equations with a diffusion term, each having a time-periodic coefficient. The necessary code in Mathematica that implements the HAM for the current problem is appended to the paper for use by the readers.

Cite this paper

nullP. Siddheshwar, "A Series Solution for the Ginzburg-Landau Equation with a Time-Periodic Coefficient,"*Applied Mathematics*, Vol. 1 No. 6, 2010, pp. 542-554. doi: 10.4236/am.2010.16072.

nullP. Siddheshwar, "A Series Solution for the Ginzburg-Landau Equation with a Time-Periodic Coefficient,"

References

[1] S. J. Liao, “On the Proposed Homotopy Analysis Techniques for Nonlinear Problems and its Application,” Ph. D. Thesis, Shanghai Jiao Tong University, Shanghai, 1992.

[2] S. J. Liao, “Homotopy Analysis Method: A New Analytical Technique for Non-Linear Problems,” Communications in Non-linear Science & Numerical Simulation, Vol. 2, No. 2, 1997, pp. 95-100.

[3] S. J. Liao, “An Explicit, Totally Analytic Approximate Solution for Blasius’ Viscous Flow Problems,” International Journal of Non-Linear Mechanics, Vol. 34, No. 4, 1999, pp. 759-778.

[4] S. J. Liao, “Beyond Perturbation-Introduction to the Homotopy Analysis Method,” CRC Press, London, 2003.

[5] S. Liao and Y. Tan, “A General Approach to Obtain Series Solutions of Nonlinear Differential Equations,” Studies in Applied Mathematics, Vol. 119, No. 4, 2007, pp. 297-354.

[6] S. Abbasbandy, “The Application of the Homotopy Analysis Method to Nonlinear Equations Arising in Heat Transfer,” Physics Letters A, Vol. 360, No. 1, 2006, pp. 109-113.

[7] S. Abbasbandy, “The Application of Homotopy Analysis Method to Solve a Generalized Hirota-Satsuma Coupled KdV Equation,” Physics Letters A, Vol. 361, No. 6, 2007 pp. 478-483.

[8] T. Hayat and M. Sajid, “On Analytic Solution for Thin Film Flow of a Fourth Grade Fluid Down a Vertical Cylinder,” Physics Letter A, Vol. 361, No. 4-5, 2007, pp. 316-322.

[9] M. Sajid, T. Hayat and S. Asghar, “Comparison Between the HAM and HPM Solutions of Thin Film Flows of Non-Newtonian Fluids on a Moving Belt,” Nonlinear Dynamics, Vol. 50, No. 1-2, 2007, pp. 27-35.

[10] K. Yabushita, M. Yamashita and K. Tsuboi, “An Analytic Solution of Projectile Motion with the Quadratic Resistance Law Using the Homotopy Analysis Method,” Journal of Physics A: Mathematical and Theoretical, Vol. 40, 2007, pp. 8403-8416.

[11] F. M. Allan, “Derivation of the Adomian Decomposition Method Using the Homotopy Analysis Method,” Applied Mathematics and Computation, Vol. 190, No. 1, 2007, pp. 6-14.

[12] A. Alizadeh-Pahlavan, V. Aliakbar, F. Vakili-Farahani, and K. Sadeghy, “MHD Flows of UCM Fluids above Porous Stretching Sheets Using Two-Auxiliary-Parameter Homotopy Analysis Method,” Communication in Nonlinear Science and Numerical Simulation, Vol. 14, No. 2, 2009, pp. 473-488.

[13] V. Marinca, N. Herisanu, and L. Nemes, “Optimal Homotopy Asymptotic Method with Application to Thin Film Flow,” Central European Journal of Physics, Vol. 6, No. 3, 2008, pp. 648-653.

[1] S. J. Liao, “On the Proposed Homotopy Analysis Techniques for Nonlinear Problems and its Application,” Ph. D. Thesis, Shanghai Jiao Tong University, Shanghai, 1992.

[2] S. J. Liao, “Homotopy Analysis Method: A New Analytical Technique for Non-Linear Problems,” Communications in Non-linear Science & Numerical Simulation, Vol. 2, No. 2, 1997, pp. 95-100.

[3] S. J. Liao, “An Explicit, Totally Analytic Approximate Solution for Blasius’ Viscous Flow Problems,” International Journal of Non-Linear Mechanics, Vol. 34, No. 4, 1999, pp. 759-778.

[4] S. J. Liao, “Beyond Perturbation-Introduction to the Homotopy Analysis Method,” CRC Press, London, 2003.

[5] S. Liao and Y. Tan, “A General Approach to Obtain Series Solutions of Nonlinear Differential Equations,” Studies in Applied Mathematics, Vol. 119, No. 4, 2007, pp. 297-354.

[6] S. Abbasbandy, “The Application of the Homotopy Analysis Method to Nonlinear Equations Arising in Heat Transfer,” Physics Letters A, Vol. 360, No. 1, 2006, pp. 109-113.

[7] S. Abbasbandy, “The Application of Homotopy Analysis Method to Solve a Generalized Hirota-Satsuma Coupled KdV Equation,” Physics Letters A, Vol. 361, No. 6, 2007 pp. 478-483.

[8] T. Hayat and M. Sajid, “On Analytic Solution for Thin Film Flow of a Fourth Grade Fluid Down a Vertical Cylinder,” Physics Letter A, Vol. 361, No. 4-5, 2007, pp. 316-322.

[9] M. Sajid, T. Hayat and S. Asghar, “Comparison Between the HAM and HPM Solutions of Thin Film Flows of Non-Newtonian Fluids on a Moving Belt,” Nonlinear Dynamics, Vol. 50, No. 1-2, 2007, pp. 27-35.

[10] K. Yabushita, M. Yamashita and K. Tsuboi, “An Analytic Solution of Projectile Motion with the Quadratic Resistance Law Using the Homotopy Analysis Method,” Journal of Physics A: Mathematical and Theoretical, Vol. 40, 2007, pp. 8403-8416.

[11] F. M. Allan, “Derivation of the Adomian Decomposition Method Using the Homotopy Analysis Method,” Applied Mathematics and Computation, Vol. 190, No. 1, 2007, pp. 6-14.

[12] A. Alizadeh-Pahlavan, V. Aliakbar, F. Vakili-Farahani, and K. Sadeghy, “MHD Flows of UCM Fluids above Porous Stretching Sheets Using Two-Auxiliary-Parameter Homotopy Analysis Method,” Communication in Nonlinear Science and Numerical Simulation, Vol. 14, No. 2, 2009, pp. 473-488.

[13] V. Marinca, N. Herisanu, and L. Nemes, “Optimal Homotopy Asymptotic Method with Application to Thin Film Flow,” Central European Journal of Physics, Vol. 6, No. 3, 2008, pp. 648-653.