Variational Iterative Method Applied to Variational Problems with Moving Boundaries

ABSTRACT

In this paper, He’s variational iterative method has been applied to give exact solution of the Euler Lagrange equation which arises from the variational problems with moving boundaries and isoperimetric problems. In this method, general Lagrange multipliers are introduced to construct correction functional for the variational problems. The initial approximations can be freely chosen with possible unknown constant, which can be determined by imposing the boundary conditions. Illustrative examples have been presented to demonstrate the efficiency and applicability of the variational iterative method.

In this paper, He’s variational iterative method has been applied to give exact solution of the Euler Lagrange equation which arises from the variational problems with moving boundaries and isoperimetric problems. In this method, general Lagrange multipliers are introduced to construct correction functional for the variational problems. The initial approximations can be freely chosen with possible unknown constant, which can be determined by imposing the boundary conditions. Illustrative examples have been presented to demonstrate the efficiency and applicability of the variational iterative method.

KEYWORDS

Variational Iterative Method; Variational Problems; Moving Boundaries; Isoperimetric Problems

Variational Iterative Method; Variational Problems; Moving Boundaries; Isoperimetric Problems

Cite this paper

F. Ghomanjani and S. Ghaderi, "Variational Iterative Method Applied to Variational Problems with Moving Boundaries,"*Applied Mathematics*, Vol. 3 No. 5, 2012, pp. 395-402. doi: 10.4236/am.2012.35061.

F. Ghomanjani and S. Ghaderi, "Variational Iterative Method Applied to Variational Problems with Moving Boundaries,"

References

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[2] A. Saadatmandi and M. Dehghan, “He’s Variational Iteration Method for Solving a Partial Differential Equation Arising in Modeling of Water Waves,” Zeitschrift für Naturforschung, Vol. 64, 2009, pp. 783-787.

[3] J. H. He, “Variational Iteration Method a Kind of NonLinear Analytical Technique: Some Examples,” International Journal of Nonlinear Mechanics, Vol. 34, No. 4, 1999, pp. 699-708. doi:10.1016/S0020-7462(98)00048-1

[4] M. A. Abdou and A. A. Soliman, “Varitional Iteration Method for Solving Burgers’ and Coupled Burgers’ Equations,” Journal of Computational and Applied Mathmatics, Vol. 181, 2005, pp. 45-251.

[5] J. Biazar and H. Ghazvini, “He’s Variational Iteration Method for Solving Hyperbolic Differential Equations,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 8, 2007, pp. 311-314.

[6] M. Dehghan and M. Tatari, “The Use of He’s Variational Iteration Method for Solving a Fokker Planck Equation,” Physica Scripta, Vol. 74, No. 3, 2006, pp. 310-316. doi:10.1088/0031-8949/74/3/003

[7] M. Dehghan and A. Saadatmandi, “Variational Iteration Method for Solving the Wave Equation Subject to an Integral Conservation Condition,” Chaos, Solitons and Fractals, Vol. 41, No. 3, 2009, pp. 448-1453. doi:10.1016/j.chaos.2008.06.009

[8] M. Dehghan and F. Shakeri, “Approximate Solution of a Differential Equation Arising in Astrophysics Using the Variational Iteration Method,” New Astronomy, Vol. 13, No. 1, 2008, pp. 53-59. doi:10.1016/j.newast.2007.06.012

[9] M. Dehghan and F. Shakeri, “Application of He’s Variational Iteration Method for Solving the Cauchy Reaction-Diffusion Problem,” Journal of Computational and Applied Mathematics, Vol. 214, No. 2, 2008, pp. 435-446. doi:10.1016/j.cam.2007.03.006

[10] M. Dehghan and R. Salehi, “The Use of Variational Iteration Method and Adomian Decomposition Method to Solve the Eikonal Equation and Its Application in the Reconstruction Problem,” Communications in Numerical Methods in Engineering, in press.

[11] M. Dehghan and M. Tatari, “The Use of Adomian Decomposition Method for Solving Problems in Calculus of Variations,” Mathematical Problems in Engineering, Vol. 2006, 2006, pp. 1-12. doi:10.1155/MPE/2006/65379

[12] M. Dehghan and M. Tatari, “Identifying an Unknown Function in a Parabolic Equation with over Specified Data via He’s Variational Iteration Method,” Chaos, Solitons and Fractals, Vol. 36, 2008, pp. 57-166.

[13] J. H. He, “Variational Iteration Method for Autonomous Ordinary Differential Systems,” Applied Mathematics and Computation, Vol. 114, No. 2-3, 2000, pp. 115-123. doi:10.1016/S0096-3003(99)00104-6

[14] J. H. He and X. H. Wu, “Variational Iteration Method: New Development and Applications,” Computers and Mathematics with Applications, Vol. 54, No. 7-8, 2007, pp. 881-894. doi:10.1016/j.camwa.2006.12.083

[15] J. H. He, “Variational Iteration Method: Some Recent Results and New Interpretations,” Journal of Computational and Applied Mathematics, Vol. 207, No. 1, 2007, pp. 3-17. doi:10.1016/j.cam.2006.07.009

[16] J. H. He, Variational Iteration Method for Delay Differential Equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 2, No. 4, 1997, pp. 235236. doi:10.1016/S1007-5704(97)90008-3

[17] J. H. He, “Approximate Solution of Nonlinear Differential Equations with Convolution Product Nonlinearities,” Computer Methods in Applied Mechanics and Engineering, Vol. 167, No. 1-2, 1998, pp. 69-73. doi:10.1016/S0045-7825(98)00109-1

[18] J. H. He, “Approximate Analytical Solution for Seepage Flow with Fractional Derivatives in Porous Media,” Computer Methods in Applied Mechanics and Engineering, Vol. 167, No. 1-2, 1998, pp. 57-68. doi:10.1016/S0045-7825(98)00108-X

[19] M. Inc, “Numerical Simulation of KdV and mKdV Equations with Initial Conditions by the variational Iteration Method,” Chaos, Solitons and Fractals, Vol. 34, No. 4, 2007, pp. 1075-1081. doi:10.1016/j.chaos.2006.04.069

[20] S. Momani and S. Abuasad, “Application of He’s Variational Iteration Method to Helmholtz Equation,” Chaos, Solitons and Fractals, Vol. 27, No. 5, 2006, pp. 11191123. doi:10.1016/j.chaos.2005.04.113

[21] S. Momani and Z. Odibat, “Analytical Approach to Linear Fractional Partial Differential Equations Arising in Fluid Mechanics. Physics Letters A, Vol. 355, No. 4-5, 2006, pp. 271-279. doi:10.1016/j.physleta.2006.02.048

[22] H. Ozer, “Application of the Variational Iteration Method to the Boundary Value Problems with Jump Discontinuities Arising in Solid Mechanics,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 8, 2007, pp. 513-518.

[23] Z. M. Odibat and S. Momani, “Application of Variational Iteration Method to Nonlinear Differential Equations of Fractional Order,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 7, 2007, pp. 2734. doi:10.1515/IJNSNS.2006.7.1.27

[24] H. Sagan, “Introduction to the Calculus of Variations,” Courier Dover Publications, 1992.

[25] F. Shakeri and M. Dehghan, “Numerical Solution of the Klein-Gordon Equation via He’s Variational Iteration Method,” Nonlinear Dynamics, Vol. 51, No. 1-2, 2008, pp. 89-97. doi:10.1007/s11071-006-9194-x

[26] F. Shakeri and M. Dehghan, “Solution of a Model Describing Biological Species Living Together Using the Variational Iteration Method,” Mathematical and Computer Modeling, Vol. 48, No. 5-6, 2008, pp. 685-699. doi:10.1016/j.mcm.2007.11.012

[27] 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

[28] M. Tatari and M. Dehghan, “Improvement of He’s Variational Iteration Method for Solving Systems of Differential Equations,” Computers and Mathematics with Applications, Vol. 58, No. 11-12, 2009, pp. 2160-2166. doi:10.1016/j.camwa.2009.03.081

[29] M. Tatari and M. Dehghan, “On the Convergence of He's Variational Iteration Method,” Journal of Computational and Applied Mathematics, Vol. 207, No. 1, 2007, pp. 121128. doi:10.1016/j.cam.2006.07.017

[30] S. A. Youse and M. Dehghan, “The Use of He’s Variational Iteration Method for Solving Variational Problems,” International Journal of Computer Mathematics, Vol. 87, No. 6, 2010. doi:10.1080/00207160802283047

[31] A. M. Wazwaz, 2007. “A Comparison between the Variational Iteration Method and Adomian Decomposition Method,” Journal of Computational and Applied Mathematics, Vol. 207, No. 1, pp. 129-136. doi:10.1016/j.cam.2006.07.018c

[32] R. Memarbashi, “Variational Problems with Moving Boundaries Using Decomposition Method,” Mathematical Problems in Engineering, Vol. 2007, 2007, Article ID 10120.

[33] M. L. Krasnov, G. I. Makarenko and A. I. Kiselev, “Problems and Exercises in the Calculus of Variations,” Mir Publisher, Moscow, 1964.

[1] P. G. Engstrom and U. Brechtken-Manderschied, “Introduction to the Calculus of Variations,” Chapman and Hall/CRC, New York, 1991.

[2] A. Saadatmandi and M. Dehghan, “He’s Variational Iteration Method for Solving a Partial Differential Equation Arising in Modeling of Water Waves,” Zeitschrift für Naturforschung, Vol. 64, 2009, pp. 783-787.

[3] J. H. He, “Variational Iteration Method a Kind of NonLinear Analytical Technique: Some Examples,” International Journal of Nonlinear Mechanics, Vol. 34, No. 4, 1999, pp. 699-708. doi:10.1016/S0020-7462(98)00048-1

[4] M. A. Abdou and A. A. Soliman, “Varitional Iteration Method for Solving Burgers’ and Coupled Burgers’ Equations,” Journal of Computational and Applied Mathmatics, Vol. 181, 2005, pp. 45-251.

[5] J. Biazar and H. Ghazvini, “He’s Variational Iteration Method for Solving Hyperbolic Differential Equations,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 8, 2007, pp. 311-314.

[6] M. Dehghan and M. Tatari, “The Use of He’s Variational Iteration Method for Solving a Fokker Planck Equation,” Physica Scripta, Vol. 74, No. 3, 2006, pp. 310-316. doi:10.1088/0031-8949/74/3/003

[7] M. Dehghan and A. Saadatmandi, “Variational Iteration Method for Solving the Wave Equation Subject to an Integral Conservation Condition,” Chaos, Solitons and Fractals, Vol. 41, No. 3, 2009, pp. 448-1453. doi:10.1016/j.chaos.2008.06.009

[8] M. Dehghan and F. Shakeri, “Approximate Solution of a Differential Equation Arising in Astrophysics Using the Variational Iteration Method,” New Astronomy, Vol. 13, No. 1, 2008, pp. 53-59. doi:10.1016/j.newast.2007.06.012

[9] M. Dehghan and F. Shakeri, “Application of He’s Variational Iteration Method for Solving the Cauchy Reaction-Diffusion Problem,” Journal of Computational and Applied Mathematics, Vol. 214, No. 2, 2008, pp. 435-446. doi:10.1016/j.cam.2007.03.006

[10] M. Dehghan and R. Salehi, “The Use of Variational Iteration Method and Adomian Decomposition Method to Solve the Eikonal Equation and Its Application in the Reconstruction Problem,” Communications in Numerical Methods in Engineering, in press.

[11] M. Dehghan and M. Tatari, “The Use of Adomian Decomposition Method for Solving Problems in Calculus of Variations,” Mathematical Problems in Engineering, Vol. 2006, 2006, pp. 1-12. doi:10.1155/MPE/2006/65379

[12] M. Dehghan and M. Tatari, “Identifying an Unknown Function in a Parabolic Equation with over Specified Data via He’s Variational Iteration Method,” Chaos, Solitons and Fractals, Vol. 36, 2008, pp. 57-166.

[13] J. H. He, “Variational Iteration Method for Autonomous Ordinary Differential Systems,” Applied Mathematics and Computation, Vol. 114, No. 2-3, 2000, pp. 115-123. doi:10.1016/S0096-3003(99)00104-6

[14] J. H. He and X. H. Wu, “Variational Iteration Method: New Development and Applications,” Computers and Mathematics with Applications, Vol. 54, No. 7-8, 2007, pp. 881-894. doi:10.1016/j.camwa.2006.12.083

[15] J. H. He, “Variational Iteration Method: Some Recent Results and New Interpretations,” Journal of Computational and Applied Mathematics, Vol. 207, No. 1, 2007, pp. 3-17. doi:10.1016/j.cam.2006.07.009

[16] J. H. He, Variational Iteration Method for Delay Differential Equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 2, No. 4, 1997, pp. 235236. doi:10.1016/S1007-5704(97)90008-3

[17] J. H. He, “Approximate Solution of Nonlinear Differential Equations with Convolution Product Nonlinearities,” Computer Methods in Applied Mechanics and Engineering, Vol. 167, No. 1-2, 1998, pp. 69-73. doi:10.1016/S0045-7825(98)00109-1

[18] J. H. He, “Approximate Analytical Solution for Seepage Flow with Fractional Derivatives in Porous Media,” Computer Methods in Applied Mechanics and Engineering, Vol. 167, No. 1-2, 1998, pp. 57-68. doi:10.1016/S0045-7825(98)00108-X

[19] M. Inc, “Numerical Simulation of KdV and mKdV Equations with Initial Conditions by the variational Iteration Method,” Chaos, Solitons and Fractals, Vol. 34, No. 4, 2007, pp. 1075-1081. doi:10.1016/j.chaos.2006.04.069

[20] S. Momani and S. Abuasad, “Application of He’s Variational Iteration Method to Helmholtz Equation,” Chaos, Solitons and Fractals, Vol. 27, No. 5, 2006, pp. 11191123. doi:10.1016/j.chaos.2005.04.113

[21] S. Momani and Z. Odibat, “Analytical Approach to Linear Fractional Partial Differential Equations Arising in Fluid Mechanics. Physics Letters A, Vol. 355, No. 4-5, 2006, pp. 271-279. doi:10.1016/j.physleta.2006.02.048

[22] H. Ozer, “Application of the Variational Iteration Method to the Boundary Value Problems with Jump Discontinuities Arising in Solid Mechanics,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 8, 2007, pp. 513-518.

[23] Z. M. Odibat and S. Momani, “Application of Variational Iteration Method to Nonlinear Differential Equations of Fractional Order,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 7, 2007, pp. 2734. doi:10.1515/IJNSNS.2006.7.1.27

[24] H. Sagan, “Introduction to the Calculus of Variations,” Courier Dover Publications, 1992.

[25] F. Shakeri and M. Dehghan, “Numerical Solution of the Klein-Gordon Equation via He’s Variational Iteration Method,” Nonlinear Dynamics, Vol. 51, No. 1-2, 2008, pp. 89-97. doi:10.1007/s11071-006-9194-x

[26] F. Shakeri and M. Dehghan, “Solution of a Model Describing Biological Species Living Together Using the Variational Iteration Method,” Mathematical and Computer Modeling, Vol. 48, No. 5-6, 2008, pp. 685-699. doi:10.1016/j.mcm.2007.11.012

[27] 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

[28] M. Tatari and M. Dehghan, “Improvement of He’s Variational Iteration Method for Solving Systems of Differential Equations,” Computers and Mathematics with Applications, Vol. 58, No. 11-12, 2009, pp. 2160-2166. doi:10.1016/j.camwa.2009.03.081

[29] M. Tatari and M. Dehghan, “On the Convergence of He's Variational Iteration Method,” Journal of Computational and Applied Mathematics, Vol. 207, No. 1, 2007, pp. 121128. doi:10.1016/j.cam.2006.07.017

[30] S. A. Youse and M. Dehghan, “The Use of He’s Variational Iteration Method for Solving Variational Problems,” International Journal of Computer Mathematics, Vol. 87, No. 6, 2010. doi:10.1080/00207160802283047

[31] A. M. Wazwaz, 2007. “A Comparison between the Variational Iteration Method and Adomian Decomposition Method,” Journal of Computational and Applied Mathematics, Vol. 207, No. 1, pp. 129-136. doi:10.1016/j.cam.2006.07.018c

[32] R. Memarbashi, “Variational Problems with Moving Boundaries Using Decomposition Method,” Mathematical Problems in Engineering, Vol. 2007, 2007, Article ID 10120.

[33] M. L. Krasnov, G. I. Makarenko and A. I. Kiselev, “Problems and Exercises in the Calculus of Variations,” Mir Publisher, Moscow, 1964.