Homotopy method of fundamental solutions for solving certain nonlinear partial differential equations

Engineering Analysis with Boundary Elements

Volumn 36, Issue 8, 2012, Pages 1226-1234

  Tsai, Chia Cheng ^a  

^a NATIONAL KAOHSIUNG UNIVERSITY OF APPLIED SCIENCES   (Taiwan)

Author keywords

Augmented polyharmonic spline; Homotopy analysis method; Method of fundamental solutions; Nonlinear partial differential equation

Indexed keywords

APPLICATION AREA; FUNDAMENTAL SOLUTIONS; GOVERNING EQUATIONS; HIGH-ORDER; HOMOTOPY ANALYSIS METHODS; HOMOTOPY METHOD; LINEAR TERMS; MESHLESS NUMERICAL METHOD; METHOD OF FUNDAMENTAL SOLUTIONS; NON-LINEAR PDES; NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS; NUMERICAL EXPERIMENTS; PARTICULAR SOLUTION; POLYHARMONIC SPLINES; SOLUTION PROCEDURE; STRONGLY NONLINEAR;

NONLINEAR EQUATIONS; NUMERICAL METHODS;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 84858316758     PISSN: 09557997     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.enganabound.2012.02.012     Document Type: Article

References (60)

1. A.D. Polyanin, and V.F. Zaitsev Handbook of nonlinear partial differential equations 2004 Chapman & Hall/CRC
2. A.H. Nayfeh Perturbation methods (Wiley classics library) 2000 Wiley-VCH
3. S.J. Liao The proposed homotopy analysis technique for the solution of nonlinear problems 1992 Shanghai Jiao Tong University
4. S.J. Liao Beyond perturbation: introduction to the homotopy analysis method 2003 Boca Raton, Chapman & Hall/CRC Press
5. S.-J. Liao A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate J Fluid Mech 385 1999 101 128
6. S.-J. Liao, and A. CAMPO Analytic solutions of the temperature distribution in Blasius viscous flow problems J Fluid Mech 453 2002 411 425
7. S.-J. Liao, and K.F. Cheung Homotopy analysis of nonlinear progressive waves in deep water J Eng Math 45 2 2003 105 116 (Pubitemid 36352562)
8. L. Tao, H. Song, and S. Chakrabarti Nonlinear progressive waves in water of finite depth - an analytic approximation Coast Eng 54 11 2007 825 834 (Pubitemid 47610328)
9. S.-J. Liao On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet J Fluid Mech 488 2003 189 212 (Pubitemid 37054134)
10. S.-J. Liao High-order BEM formulations for strongly non-linear problems governed by quite general non-linear differential operators, Part 2: some 2D examples Int J Numer Methods Fluids 24 9 1997 863 873 (Pubitemid 127619259)
11. S.-J. Liao Boundary element method for general nonlinear differential operators Eng Anal Bound Elem 20 2 1997 91 99 (Pubitemid 127412967)
12. S.-J. Liao General boundary element method for non-linear heat transfer problems governed by hyperbolic heat conduction equation Comput Mech 20 5 1997 397 406
13. S.-J. Liao On the general boundary element method Eng Anal Bound Elem 21 1 1998 39 51 (Pubitemid 128394057)
14. S.-J. Liao The general boundary element method and its further generalizations Int J Numer Methods Fluids 31 3 1999 627 655 (Pubitemid 29492790)
15. S.-J. Liao General boundary-element method for unsteady nonlinear heat transfer problems Numer Heat Transfer, Part B: Fund: Int J Comput Methodol 35 2 1999 225 242
16. S.-J. Liao A direct boundary element approach for unsteady non-linear heat transfer problems Eng Anal Bound Elem 26 1 2002 55 59 (Pubitemid 33034374)
17. S.J. Liao, and A.T. Chwang General boundary element method for non-linear problems Int J Numer Methods Fluids 23 5 1996 467 483 (Pubitemid 126622730)
18. S.J. Liao, and Z.H.U. Ji-Mao A short note on high-order streamfunction-vorticity formulations of 2D steady state Navier-Stokes equations Int J Numer Methods Fluids 22 1 1996 1 9
19. Y.Y. Wu, S.J. Liao, and X.Y. Zhao Some notes on the general boundary element method for highly nonlinear problems Commun Nonlinear Sci Numer Simul 10 7 2005 725 735 (Pubitemid 40133616)
20. M.M. Grigoriev, and G.F. Dargush A poly-region boundary element method for incompressible viscous fluid flows Int J Numer Methods Eng 46 7 1999 1127 1158
21. E.J. Kansa Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics - I: surface approximations and partial derivative estimates Comput Math Appl 19 8-9 1990 127 145 (Pubitemid 20668364)
22. E.J. Kansa Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics - II: solutions to parabolic, hyperbolic and elliptic partial differential equations Comput Math Appl 19 8-9 1990 147 161 (Pubitemid 20668365)
23. V.D. Kupradze, and M.A. Aleksidze The method of functional equations for the approximate solution of certain boundary value problems USSR Comput Math Math Phys 4 4 1964 82 126
24. R. Mathon, and R.L. Johnston The approximate solution of elliptic boundary-value problems by fundamental solutions SIAM J Numer Anal 14 4 1977 638 650
25. A. Bogomolny Fundamental solutions method for elliptic boundary value problems SIAM J Numer Anal 22 4 1985 644 669 (Pubitemid 15528321)
26. R.L. Johnston, and G. Fairweather The method of fundamental solutions for problems in potential flow Appl Math Modelling 8 4 1984 265 270 (Pubitemid 16519700)
27. C.C. Tsai, D.L. Young, C.W. Chen, and C.M. Fan The method of fundamental solutions for eigenproblems in domains with and without interior holes Proc R Soc A - Math Phys Eng Sci 462 2069 2006 1443 1466 (Pubitemid 44922098)
28. C.C. Tsai, D.L. Young, C.L. Chiu, and C.M. Fan Numerical analysis of acoustic modes using the linear least squares method of fundamental solutions J Sound Vib 324 3-5 2009 1086 1110
29. A. Karageorghis The method of fundamental solutions for the calculation of the eigenvalues of the Helmholtz equation Appl Math Lett 14 7 2001 837 842 (Pubitemid 33675418)
30. G. Fairweather, A. Karageorghis, and P.A. Martin The method of fundamental solutions for scattering and radiation problems Eng Anal Bound Elem 27 7 2003 759 769
31. A. Karageorghis, and G. Fairweather The method of fundamental solutions for the numerical solution of the biharmonic equation J Comput Phys 69 2 1987 434 459
32. M.A. Golberg The method of fundamental solutions for Poissons equation Eng Anal Bound Elem 16 3 1995 205 213
33. C.C. Tsai, D.L. Young, D.C. Lo, and T.K. Wong Method of fundamental solutions for three-dimensional stokes flow in exterior field J Eng Mech - ASCE 132 3 2006 317 326 (Pubitemid 43257522)
34. C.J.S. Alves, and A.L. Silvestre Density results using Stokeslets and a method of fundamental solutions for the Stokes equations Eng Anal Bound Elem 28 10 2004 1245 1252
35. D.L. Young, S.J. Jane, C.M. Fan, K. Murugesan, and C.C. Tsai The method of fundamental solutions for 2D and 3D Stokes problems J Comput Phys 211 1 2006 1 8 (Pubitemid 41363971)
36. C.C. Tsai, D.L. Young, C.M. Fan, and C.W. Chen MFS with time-dependent fundamental solutions for unsteady Stokes equations Eng Anal Bound Elem 30 10 2006 897 908 (Pubitemid 44354117)
37. C.C. Tsai The method of fundamental solutions for three-dimensional elastostatic problems of transversely isotropic solids Eng Anal Bound Elem 31 7 2007 586 594 (Pubitemid 46843544)
38. C.C. Tsai The method of fundamental solutions with dual reciprocity for three-dimensional thermoelasticity under arbitrary body forces Eng Comput 26 3-4 2009 229 244
39. D.L. Young, C.C. Tsai, and C.M. Fan Direct approach to solve nonhomogeneous diffusion problems using fundamental solutions and dual reciprocity methods J Chin Inst Eng 27 4 2004 597 609
40. D.L. Young, C.C. Tsai, K. Murugesan, C.M. Fan, and C.W. Chen Time-dependent fundamental solutions for homogeneous diffusion problems Eng Anal Bound Elem 28 12 2004 1463 1473
41. G. Fairweather, and A. Karageorghis The method of fundamental solutions for elliptic boundary value problems Adv Comput Math 9 1 1998 69 95 (Pubitemid 128403959)
42. M.A. Golberg, and C.S. Chen The method of fundamental solutions for potential, Helmholtz and diffusion problems M.A. Golberg, Boundary integral methods: numerical and mathematical aspects 1999 Computational Mechanics Publications Southampton 103 176
43. D. Nardini, and C.A. Brebbia A new approach to free vibration analysis using boundary elements Appl Math Modelling 7 3 1983 157 162 (Pubitemid 13592360)
44. M.A. Golberg, and C.S. Chen The theory of radial basis functions applied to the BEM for inhomogeneous partial differential equations Bound Elem Commun 5 1994 57 61
45. J. Duchon Splines minimizing rotation-invariant semi-norms in Sobolev spaces W. Schempp, K. Zeller, Constructive theory of functions of several variables 1977 Springer Berlin/Heidelberg 85 100
46. R.L. Hardy Multiquadric equations of topography and other irregular surfaces J Geophys Res 76 8 1971 1905 1915
47. M.A. Golberg Numerical evaluation of particular solutions in the BEM - a review Bound Elem Commun 6 3 1995 99 106
48. C.S. Chen The method of fundamental solutions for non-linear thermal explosions Commun Numer Methods Eng 11 8 1995 675 681
49. S.R. Karur, and P.A. Ramachandran Radial basis function approximation in the dual reciprocity method Math Comput Modelling 20 7 1994 59 70
50. M.A. Golberg, C.S. Chen, and S.R. Karur Improved multiquadric approximation for partial differential equations Eng Anal Bound Elem 18 1996 9 17 (Pubitemid 126358815)
51. C.-C. Tsai Automatic particular solutions of arbitrary high-order splines associated with polyharmonic and poly-Helmholtz equations Eng Anal Bound Elem 35 7 2011 925 934
52. C.C. Tsai, A.H.D. Cheng, and C.S. Chen Particular solutions of splines and monomials for polyharmonic and products of Helmholtz operators Eng Anal Bound Elem 33 4 2009 514 521
53. C.C. Tsai Particular solutions of Chebyshev polynomials for polyharmonic and poly-Helmholtz equations CMES - Comput Modeling Eng Sci 27 3 2008 151 162
54. C.C. Tsai, Y.C. Lin, D.L. Young, and S.N. Aturi Investigations on the accuracy and condition number for the method of fundamental solutions CMES - Comput Modeling Eng Sci 16 2 2006 103 114 (Pubitemid 46142893)
55. A.H.D. Cheng, H. Antes, and N. Ortner Fundamental solutions of products of Helmholtz and polyharmonic operators Eng Anal Bound Elem 14 2 1994 187 191
56. C.Y. Ku, W.C. Yeih, C.S. Liu, and C.C. Chi Applications of the fictitious time integration method using a new time-like function CMES - Comput Modeling Eng Sci 43 2 2009 173 190
57. C.C. Tsai, C.S. Liu, and W.C. Yeih Fictitious time integration method of fundamental solutions with Chebyshev polynomials for solving Poisson-type nonlinear PDEs CMES - Comput Modeling Eng Sci 56 2 2010 131 151
58. IEEE Standard for Floating-Point Arithmetic. IEEE Std: 754-2008, 2008; pp. 1-58.
59. Hanrot G, Lefevre V, Pelissier P, Zimmermann P. The GNU MPFR library; 2005.
60. L. Fousse, G. Hanrot, V. Lefevre, P. Pelissier, and P. Zimmermann MPFR: a multiple-precision binary floating-point library with correct rounding ACM Trans Math Softw 33 2 2007 13