Homotopy Perturbation Method for Solving Moving Boundary and Isoperimetric Problems

ABSTRACT

In this paper, homotopy perturbation method is applied to solve moving boundary and isoperimetric problems. This method does not depend upon a small parameter in the equation. homotopy is constructed with an imbedding parameter p, which is considered as a “small parameter”. Finally, we use combined homotopy perturbation method and Green’s function method for solving second order problems. Some examples are given to illustrate the effectiveness of methods. The results show that these methods provides a powerful mathematical tools for solving problems.

In this paper, homotopy perturbation method is applied to solve moving boundary and isoperimetric problems. This method does not depend upon a small parameter in the equation. homotopy is constructed with an imbedding parameter p, which is considered as a “small parameter”. Finally, we use combined homotopy perturbation method and Green’s function method for solving second order problems. Some examples are given to illustrate the effectiveness of methods. The results show that these methods provides a powerful mathematical tools for solving problems.

Cite this paper

S. Ghaderi, "Homotopy Perturbation Method for Solving Moving Boundary and Isoperimetric Problems,"*Applied Mathematics*, Vol. 3 No. 5, 2012, pp. 403-409. doi: 10.4236/am.2012.35062.

S. Ghaderi, "Homotopy Perturbation Method for Solving Moving Boundary and Isoperimetric Problems,"

References

[1] U. Brechtken-Manderschied, “Introduction to the Calculus of Variations,” Chapman Hall, London, 1991.

[2] H. Sagan, “Introduction to the Calculus of Variations,” Dover Publications, New York, 1992.

[3] J.-H. He, “A Coupling Method for a Homotopy Technique and a Perturbation Technique for Nonlinear Problems,” International Journal of Non-Linear Mechanics, Vol. 35, 2000, pp. 37-43.

[4] U. Ascher, M. Robert, M. M. Mattheij and R. D. Russell, “Numerical Solution of Boundary Value Problems for Ordinary Differential Equations,” SIAM, Philadelphia, 1995.

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

[6] D. D. Ganji, H. Tari and M. Bakhshi, “Variational Iteration Method and Homotopy Perturbation Method for Nonlinear Evalution Equations,” Computers & Mathematics with Applications, Vol. 54, No. 7-8, 2007, pp. 1018-1024.

[7] J. H. He, “Perturbation Method: A New Nonlinear Analytical Technique,” Applied Mathematics and Computation, Vol. 135, No. 1, 2003, pp. 73-79. doi:10.1016/S0096-3003(01)00312-5

[8] M. Inokuti, H. Sekine and T. Mura, “General Use of the Lagrange Multiplier in Non-Linear Mathematical Physics,” In: S. Nemat-Nassed, Ed., Variational Methods in the Mechanics of Solids, Pergamon Press, New York, 1978, pp. 156-162.

[9] S. J. Liao, “An Approximate Solution Technique Not Depending on Small Parameters: A Special Example,” International Journal of Non-Linear Mechanics, Vol. 30, No. 3, 1995, pp. 371-380. doi:10.1016/0020-7462(94)00054-E

[10] S. J. Liao, “Boundary Element Method for General Nonlinear Differential Operators,” Engineering Analysis with Boundary Element, Vol. 20, No. 2, 1997, pp. 91-99. doi:10.1016/S0955-7997(97)00043-X

[11] J. H. He, “Asymptotology by Homotopy Perturbation Method,” Applied Mathematics and Computation, Vol. 156, No. 3, 2004, pp. 591-596. doi:10.1016/j.amc.2003.08.011

[12] J. H. He, “Homotopy Perturbation Technique,” Applied Mechanics and Engineering, Vol. 178, 1997, pp. 257262.

[13] R. Memarbashi, “Variational Problems with Moving Boundaries Using Decomposition Method,” Mathematical Problems in Engineering, Vol. 2007, 2007, Article ID 10120. doi:10.1155/2007/10120

[14] J. H. He, “Homotopy Perturbation Method for Solving Boundary Value Problems,” Physics Letters A, Vol. 350, No. 1-2, 2006, pp. 87-88.

[15] M. L. krasnov, G. I. Makarenko and A. L. Kiselev, “Problems and Exercises in the Calculus of Variations,” George Yankovsky, Moscow, 1975.

[16] Y.-G. Wang and H.-F. Song and D. Li, “Solving TwoPoint Boundary Value Problems Using Combined Homotopy Perturbation Method and Green’s Function Method,” Applied Mathematics and Computation, Vol. 212, No. 2, 2009, pp. 366-376.

[1] U. Brechtken-Manderschied, “Introduction to the Calculus of Variations,” Chapman Hall, London, 1991.

[2] H. Sagan, “Introduction to the Calculus of Variations,” Dover Publications, New York, 1992.

[3] J.-H. He, “A Coupling Method for a Homotopy Technique and a Perturbation Technique for Nonlinear Problems,” International Journal of Non-Linear Mechanics, Vol. 35, 2000, pp. 37-43.

[4] U. Ascher, M. Robert, M. M. Mattheij and R. D. Russell, “Numerical Solution of Boundary Value Problems for Ordinary Differential Equations,” SIAM, Philadelphia, 1995.

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

[6] D. D. Ganji, H. Tari and M. Bakhshi, “Variational Iteration Method and Homotopy Perturbation Method for Nonlinear Evalution Equations,” Computers & Mathematics with Applications, Vol. 54, No. 7-8, 2007, pp. 1018-1024.

[7] J. H. He, “Perturbation Method: A New Nonlinear Analytical Technique,” Applied Mathematics and Computation, Vol. 135, No. 1, 2003, pp. 73-79. doi:10.1016/S0096-3003(01)00312-5

[8] M. Inokuti, H. Sekine and T. Mura, “General Use of the Lagrange Multiplier in Non-Linear Mathematical Physics,” In: S. Nemat-Nassed, Ed., Variational Methods in the Mechanics of Solids, Pergamon Press, New York, 1978, pp. 156-162.

[9] S. J. Liao, “An Approximate Solution Technique Not Depending on Small Parameters: A Special Example,” International Journal of Non-Linear Mechanics, Vol. 30, No. 3, 1995, pp. 371-380. doi:10.1016/0020-7462(94)00054-E

[10] S. J. Liao, “Boundary Element Method for General Nonlinear Differential Operators,” Engineering Analysis with Boundary Element, Vol. 20, No. 2, 1997, pp. 91-99. doi:10.1016/S0955-7997(97)00043-X

[11] J. H. He, “Asymptotology by Homotopy Perturbation Method,” Applied Mathematics and Computation, Vol. 156, No. 3, 2004, pp. 591-596. doi:10.1016/j.amc.2003.08.011

[12] J. H. He, “Homotopy Perturbation Technique,” Applied Mechanics and Engineering, Vol. 178, 1997, pp. 257262.

[13] R. Memarbashi, “Variational Problems with Moving Boundaries Using Decomposition Method,” Mathematical Problems in Engineering, Vol. 2007, 2007, Article ID 10120. doi:10.1155/2007/10120

[14] J. H. He, “Homotopy Perturbation Method for Solving Boundary Value Problems,” Physics Letters A, Vol. 350, No. 1-2, 2006, pp. 87-88.

[15] M. L. krasnov, G. I. Makarenko and A. L. Kiselev, “Problems and Exercises in the Calculus of Variations,” George Yankovsky, Moscow, 1975.

[16] Y.-G. Wang and H.-F. Song and D. Li, “Solving TwoPoint Boundary Value Problems Using Combined Homotopy Perturbation Method and Green’s Function Method,” Applied Mathematics and Computation, Vol. 212, No. 2, 2009, pp. 366-376.