On the Homotopy Analysis Method and Optimal Value of the Convergence Control Parameter: Solution of Euler-Lagrange Equation

Affiliation(s)

Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.

Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.

ABSTRACT

This paper presents, an efficient approach for solving Euler-Lagrange Equation which arises from calculus of variations. Homotopy analysis method to find an approximate solution of variational problems is proposed. An optimal value of the convergence control parameter is given through the square residual error. By minimizing the the square residual error, the optimal convergence-control parameters can be obtained. It is showed that the homotopy analysis method was valid and feasible to the study of variational problems.

This paper presents, an efficient approach for solving Euler-Lagrange Equation which arises from calculus of variations. Homotopy analysis method to find an approximate solution of variational problems is proposed. An optimal value of the convergence control parameter is given through the square residual error. By minimizing the the square residual error, the optimal convergence-control parameters can be obtained. It is showed that the homotopy analysis method was valid and feasible to the study of variational problems.

Cite this paper

J. Saberi-Nadjafi, R. Buzhabadi and H. Nik, "On the Homotopy Analysis Method and Optimal Value of the Convergence Control Parameter: Solution of Euler-Lagrange Equation,"*Applied Mathematics*, Vol. 3 No. 8, 2012, pp. 873-881. doi: 10.4236/am.2012.38129.

J. Saberi-Nadjafi, R. Buzhabadi and H. Nik, "On the Homotopy Analysis Method and Optimal Value of the Convergence Control Parameter: Solution of Euler-Lagrange Equation,"

References

[1] I. M. Gelfand and S. V. Fomin, “Calculus of Variations,” Prentice Hall, New Jersey, 1963.

[2] L. Elsgolts, “Differential Equations and the Calculus of Variations,” Mir Publisher Moscow, 1977.

[3] L. Elsgolts, “Calculus of Variations,” Pergamon Press, Oxford, 1962.

[4] C. F. Chen and C. H. Hsiao, “A Walsh Series Direct Method for Solving Variational Problems,” Journal of the Franklin Institute, Vol. 300, No. 4, 1975, pp. 265-280. doi:10.1016/0016-0032(75)90199-4

[5] M. Razzaghi and M. Razzaghi, “Fourier Series Direct Method for Variational Problems,” International Journal of Control, Vol. 48, No. 3, 1988, pp. 887-895. doi:10.1080/00207178808906224

[6] M. Dehghan and M. Tatari, “The Use of Adomian Decomposition Method for Solving Problems in Calculus of Variations,” Mathematical Problems in Engineering, 2006, pp. 653-679.

[7] M. Tatari and M. Dehghan, “Solution of Problems in Calculus of Variations via He’s Variational Iteration Method,” Physics Letters A, Vol. 362, No. 5-6, 2007, pp. 401-406. doi:10.1016/j.physleta.2006.09.101

[8] O. Abdulaziz, I. Hashim and M. S. H. Chowdhury, “Solving Variational Problems by Homotopy Perturbation Method,” International Journal for Numerical Methods in Engineering, Vol. 75, No. 6, 2008, pp. 709-721. doi:10.1002/nme.2279

[9] S. J. Liao, “The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems,” Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, 1992.

[10] S. J. Liao, “Beyond Perturbation: Introduction to the Homotopy Analysis Method,” CRC Press, Chapman & Hall, Boca Raton, 2003.

[11] S. J. Liao, “On the Homotopy Anaylsis Method for Nonlinear Problems,” Applied Mathematics and Computation, Vol. 147, No. 2, 2004, pp. 499-513. doi:10.1016/S0096-3003(02)00790-7

[12] S. J. Liao, “Comparison between the Homotopy Analysis Method and Homotopy Perturbation Method,” Applied Mathematics and Computation, Vol. 169, No. 2, 2005, pp. 618-634. doi:10.1016/j.amc.2004.10.058

[13] S. J. Liao, “Homotopy Analysis Method: A New Analytical Technique for Nonlinear Problems,” Communications in Nonlinear Science and Numerical Simulation, Vol. 2, No. 2, 1997, pp. 95-100. doi:10.1016/S1007-5704(97)90047-2

[14] T. Hayat, T. Javed and M. Sajid, “Analytic Solution for Rotating Flow and Heat Transfer Analysis of a Third-Grade Fluid,” Acta Mechanica, Vol. 191, No. 3-4, 2007, pp. 219-229. doi:10.1007/s00707-007-0451-y

[15] S. Abbasbandy, “Soliton Solutions for the 5th-Order KdV Equation with the Homotopy Analysis Method,” Nonlinear Dynamics, Vol. 51, No. 1-2, 2008, pp. 83-87. doi:10.1007/s11071-006-9193-y

[16] S. P. Zhu, “An Exact and Explicit Solution for the Valuation of American Put Options,” Quantitative Finance, Vol. 6, No. 3, 2006, pp. 229-242. doi:10.1080/14697680600699811

[17] S. Effati and H. Saberi Nik, “Analytic-Approximate Solution for a Class of Nonlinear Optimal Control Problems by Homotopy Analysis Method,” Asian-European Journal of Mathematics, in press.

[18] J. H. He, “Homotopy Perturbation Technique,” Computer Methods in Applied Mechanics and Engineering, Vol. 178, No. 3-4, 1999, pp. 257-262. doi:10.1016/S0045-7825(99)00018-3

[19] 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, Vol. 40, No. 29, 2007, pp. 8403-8416. doi:10.1088/1751-8113/40/29/015

[20] L. N. Zhang and J. H. He, “Homotopy Perturbation Method for the Solution of the Electrostatic Potential Differential Equation,” Mathematical Problems in Engineering, 2006, pp. 838-848.

[1] I. M. Gelfand and S. V. Fomin, “Calculus of Variations,” Prentice Hall, New Jersey, 1963.

[2] L. Elsgolts, “Differential Equations and the Calculus of Variations,” Mir Publisher Moscow, 1977.

[3] L. Elsgolts, “Calculus of Variations,” Pergamon Press, Oxford, 1962.

[4] C. F. Chen and C. H. Hsiao, “A Walsh Series Direct Method for Solving Variational Problems,” Journal of the Franklin Institute, Vol. 300, No. 4, 1975, pp. 265-280. doi:10.1016/0016-0032(75)90199-4

[5] M. Razzaghi and M. Razzaghi, “Fourier Series Direct Method for Variational Problems,” International Journal of Control, Vol. 48, No. 3, 1988, pp. 887-895. doi:10.1080/00207178808906224

[6] M. Dehghan and M. Tatari, “The Use of Adomian Decomposition Method for Solving Problems in Calculus of Variations,” Mathematical Problems in Engineering, 2006, pp. 653-679.

[7] M. Tatari and M. Dehghan, “Solution of Problems in Calculus of Variations via He’s Variational Iteration Method,” Physics Letters A, Vol. 362, No. 5-6, 2007, pp. 401-406. doi:10.1016/j.physleta.2006.09.101

[8] O. Abdulaziz, I. Hashim and M. S. H. Chowdhury, “Solving Variational Problems by Homotopy Perturbation Method,” International Journal for Numerical Methods in Engineering, Vol. 75, No. 6, 2008, pp. 709-721. doi:10.1002/nme.2279

[9] S. J. Liao, “The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems,” Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, 1992.

[10] S. J. Liao, “Beyond Perturbation: Introduction to the Homotopy Analysis Method,” CRC Press, Chapman & Hall, Boca Raton, 2003.

[11] S. J. Liao, “On the Homotopy Anaylsis Method for Nonlinear Problems,” Applied Mathematics and Computation, Vol. 147, No. 2, 2004, pp. 499-513. doi:10.1016/S0096-3003(02)00790-7

[12] S. J. Liao, “Comparison between the Homotopy Analysis Method and Homotopy Perturbation Method,” Applied Mathematics and Computation, Vol. 169, No. 2, 2005, pp. 618-634. doi:10.1016/j.amc.2004.10.058

[13] S. J. Liao, “Homotopy Analysis Method: A New Analytical Technique for Nonlinear Problems,” Communications in Nonlinear Science and Numerical Simulation, Vol. 2, No. 2, 1997, pp. 95-100. doi:10.1016/S1007-5704(97)90047-2

[14] T. Hayat, T. Javed and M. Sajid, “Analytic Solution for Rotating Flow and Heat Transfer Analysis of a Third-Grade Fluid,” Acta Mechanica, Vol. 191, No. 3-4, 2007, pp. 219-229. doi:10.1007/s00707-007-0451-y

[15] S. Abbasbandy, “Soliton Solutions for the 5th-Order KdV Equation with the Homotopy Analysis Method,” Nonlinear Dynamics, Vol. 51, No. 1-2, 2008, pp. 83-87. doi:10.1007/s11071-006-9193-y

[16] S. P. Zhu, “An Exact and Explicit Solution for the Valuation of American Put Options,” Quantitative Finance, Vol. 6, No. 3, 2006, pp. 229-242. doi:10.1080/14697680600699811

[17] S. Effati and H. Saberi Nik, “Analytic-Approximate Solution for a Class of Nonlinear Optimal Control Problems by Homotopy Analysis Method,” Asian-European Journal of Mathematics, in press.

[18] J. H. He, “Homotopy Perturbation Technique,” Computer Methods in Applied Mechanics and Engineering, Vol. 178, No. 3-4, 1999, pp. 257-262. doi:10.1016/S0045-7825(99)00018-3

[19] 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, Vol. 40, No. 29, 2007, pp. 8403-8416. doi:10.1088/1751-8113/40/29/015

[20] L. N. Zhang and J. H. He, “Homotopy Perturbation Method for the Solution of the Electrostatic Potential Differential Equation,” Mathematical Problems in Engineering, 2006, pp. 838-848.