A Modified Homotopy Analysis Method for Solving Boundary Layer Equations

Affiliation(s)

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China.

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China.

ABSTRACT

A new modification of the Homotopy Analysis Method (HAM) is presented for highly nonlinear ODEs on a semi-infinite domain. The main advantage of the modified HAM is that the number of terms in the series solution can be greatly reduced; meanwhile the accuracy of the solution can be well retained. In this way, much less CPU is needed. Two typical examples are used to illustrate the efficiency of the proposed approach.

Cite this paper

Y. Zhao, Z. Lin and S. Liao, "A Modified Homotopy Analysis Method for Solving Boundary Layer Equations,"*Applied Mathematics*, Vol. 4 No. 1, 2013, pp. 11-15. doi: 10.4236/am.2013.41003.

Y. Zhao, Z. Lin and S. Liao, "A Modified Homotopy Analysis Method for Solving Boundary Layer Equations,"

References

[1] S. Liao, “Proposed Homotopy Analysis Techniques for the Solution of Nonlinear Problem,” Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, 1992.

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

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

[4] V. Marinca and N. Herisanu, “Application of Optimal Homotopy Asymptotic Method for Solving Nonlinear Equations Arising in Heat Transfer,” International Communications in Heat and Mass Transfer, Vol. 35, No. 6, 2008, pp. 710-715. doi:10.1016/j.icheatmasstransfer.2008.02.010

[5] Z. Niu and C. Wang, “A One-Step Optimal Homotopy Analysis Method for Nonlinear Differential Equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 8, 2010, pp. 2026-2036. doi:10.1016/j.cnsns.2009.08.014

[6] S. Liao, “An Optimal Homotopy-Analysis Approach for Strongly Nonlinear Differential Equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 8, 2010, pp. 2003-2016. doi:10.1016/j.cnsns.2009.09.002

[7] Z. Lin, “Research and Application of Scaled Boundary FEM and Fast Multipole BEM,” Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, 2010.

[8] S. Liao, “Notes on the Homotopy Analysis Method: Some Definitions and Theorems,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 4, 2009, pp. 983-997. doi:10.1016/j.cnsns.2008.04.013

[9] S. J. Liao, “A Uniformly Valid Analytic Solution of Two-Dimensional Viscous Flow over a Semi-Infinite Flat Plate,” Journal of Fluid Mechanics, Vol. 385, 1999, pp. 101-128. doi:10.1017/S0022112099004292

[10] H. K. Kuiken, “A Backward Free-Convective Boundary Layer,” The Quarterly Journal of Mechanics and Applied Mathematics, Vol. 34, No. 3, 1981, pp. 397-413. doi:10.1093/qjmam/34.3.397

[1] S. Liao, “Proposed Homotopy Analysis Techniques for the Solution of Nonlinear Problem,” Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, 1992.

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

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

[4] V. Marinca and N. Herisanu, “Application of Optimal Homotopy Asymptotic Method for Solving Nonlinear Equations Arising in Heat Transfer,” International Communications in Heat and Mass Transfer, Vol. 35, No. 6, 2008, pp. 710-715. doi:10.1016/j.icheatmasstransfer.2008.02.010

[5] Z. Niu and C. Wang, “A One-Step Optimal Homotopy Analysis Method for Nonlinear Differential Equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 8, 2010, pp. 2026-2036. doi:10.1016/j.cnsns.2009.08.014

[6] S. Liao, “An Optimal Homotopy-Analysis Approach for Strongly Nonlinear Differential Equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 8, 2010, pp. 2003-2016. doi:10.1016/j.cnsns.2009.09.002

[7] Z. Lin, “Research and Application of Scaled Boundary FEM and Fast Multipole BEM,” Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, 2010.

[8] S. Liao, “Notes on the Homotopy Analysis Method: Some Definitions and Theorems,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 4, 2009, pp. 983-997. doi:10.1016/j.cnsns.2008.04.013

[9] S. J. Liao, “A Uniformly Valid Analytic Solution of Two-Dimensional Viscous Flow over a Semi-Infinite Flat Plate,” Journal of Fluid Mechanics, Vol. 385, 1999, pp. 101-128. doi:10.1017/S0022112099004292

[10] H. K. Kuiken, “A Backward Free-Convective Boundary Layer,” The Quarterly Journal of Mechanics and Applied Mathematics, Vol. 34, No. 3, 1981, pp. 397-413. doi:10.1093/qjmam/34.3.397