×

zbMATH — the first resource for mathematics

On exact solution of Laplace equation with Dirichlet and Neumann boundary conditions by the homotopy analysis method. (English) Zbl 1203.65275
Summary: We present the homotopy analysis method (shortly HAM) for obtaining the numerical solutions of Laplace equation with Dirichlet and Neumann boundary conditions. The series solution is developed and the recurrence relations are given explicitly. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. The comparison of the HAM results with the variational iteration method (shortly VIM) results is made. The HAM contains the auxiliary parameter \(\hbar\), therefore we control with a simple way the convergence region of solution series.

MSC:
65N99 Numerical methods for partial differential equations, boundary value problems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University, 1992
[2] Liao, S.J., Int. J. non-linear mech., 34, 759, (1999)
[3] Liao, S.J., Beyond perturbation: introduction to the homotopy analysis method, (2003), Champan & Hall/CRC Press Boca Raton
[4] Liao, S.J., J. fluid. mech., 488, 189, (2003)
[5] Liao, S.J., Appl. math. comput., 147, 499, (2004)
[6] Liao, S.J., Int. J. heat mass transfer, 48, 2529, (2005)
[7] Ayub, M.; Rasheed, A.; Hayat, T., Int. J. eng. sci., 41, 2091, (2003)
[8] Hayat, T.; Khan, M., Nonlinear dynam., 42, 395, (2005)
[9] Hayat, T.; Khan, M.; Ayub, M., Z. angew. math. phys., 56, 1012, (2005)
[10] Wu, W.; Liao, S.J., Chaos solitons fractals, 26, 177, (2005)
[11] Hayat, T.; Sajid, M., Phys. lett. A, 361, 316, (2007) · Zbl 1170.76307
[12] Abbasbandy, S., Phys. lett. A, 360, 109, (2006)
[13] Abbasbandy, S., Phys. lett. A, 361, 478, (2007) · Zbl 1273.65156
[14] Wazwaz, A.M., Phys. lett. A, 363, 260, (2007)
[15] Liao, S.J., Int. J. non-linear mech., 37, 1-18, (2002)
[16] Liao, S.J., Appl. math. comput., 144, 495, (2003)
[17] Wang, C.; Wu, Y.; Wu, W., Wave motion, 41, 329, (2004)
[18] Wu, Y.; Wang, C.; Liao, S.J., Chaos solitons fractals, 23, 1733, (2005)
[19] Sajid, M.; Hayat, T.; Asghar, S., Phys. lett. A, 355, 18, (2006)
[20] Abbas, Z.; Sajid, M.; Hayat, T., Theor. comput. fluid dyn., 20, 229, (2006) · Zbl 1109.76065
[21] M. Sajid, T. Hayat, S. Asghar, unpublished
[22] He, J.H., Int. J. mod. phys. B, 20, 1141, (2006)
[23] He, J.H., Commun. nonlinear sci. numer. simul., 3, 92, (1998)
[24] He, J.H., Comput. math. appl. mech. eng., 167, 57, (1998)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.