Meshless local Petrov-Galerkin (MLPG) approximation to the two dimensional sine-Gordon equation

Journal of Computational and Applied Mathematics

Volumn 233, Issue 10, 2010, Pages 2737-2754

  Mirzaei, Davoud ^a   Dehghan, Mehdi ^a  

^a AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

Author keywords

Meshless local Petrov Galerkin (MLPG) method; Moving least squares (MLS) approximation; Sine Gordon (SG) equation

Indexed keywords

CONSERVATION OF ENERGY; LARGE DOMAIN; MESH-LESS METHODS; MESHLESS; MESHLESS LOCAL PETROV-GALERKIN; MESHLESS LOCAL PETROV-GALERKIN METHOD; MLPG METHOD; MOVING LEAST SQUARES APPROXIMATION; NON-LINEAR; NON-LINEARITY; NONLINEAR PROBLEMS; NUMERICAL INTEGRATIONS; NUMERICAL RESULTS; NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS; PREDICTOR-CORRECTOR SCHEMES; RING SOLITONS; SCIENTIFIC COMMUNITY; SINE-GORDON EQUATION; SUB-DOMAINS; THREE DIMENSIONS; TIME DERIVATIVE; TIME STEPPING METHOD; TWO-DIMENSION;

ALGORITHMS; DENSITY (SPECIFIC GRAVITY); GALERKIN METHODS; INTEGRATION; NUMERICAL METHODS; SOLITONS; TWO DIMENSIONAL;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 73449115631     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2009.11.022     Document Type: Article

References (54)

1. Belytschko T., Lu Y.Y., and Gu L. Element-free Galerkin methods. Internat. J. Numer. Methods Engrg. 37 (1994) 229-256
2. Duarte C.A., and Oden J.T. H-p clouds-an hp meshless method. Numer. Methods Partial Differential Equations 12 (1996) 673-705
3. Liu W., Jun S., and Zhang Y. Reproducing kernel particle method. Internat. J. Numer. Methods Fluids 20 (1995) 1081-1106
4. Lucy L.B. A numerical approach to the testing of the fission hypothesis. Astron. J. 82 12 (1977) 1013-1024
5. Nayroles B., Touzot G., and Villon P. Generalizing the finite element method: Diffuse approximation and diffuse elements. Comput. Mech. 10 (1992) 307-318
6. Melenk J.M., and Babuska I. The partition of unity finite element method: Basic theory and applications. Comput. Methods Appl. Mech. Engrg. 139 (1996) 289-314
7. Sukumar N., Moran B., and Belytschko T. The natural element method in solid mechanics. Internat. J. Numer. Methods Engrg. 43 (1998) 839-887
8. Lancaster P., and Salkauskas K. Surfaces generated by moving least squares methods. Math. Comp. 37 (1981) 141-158
9. Atluri S.N., and Zhu T. A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput. Mech. 22 2 (1998) 117-127
10. Atluri S.N., and Shen S.P. The meshless local Petrov-Galerkin (MLPG) method: A simple and less-costly alternative to the finite element methods. Comput. Model. Eng. Sci. 3 1 (2002) 11-51
11. Atluri S.N. The Meshless Method (MLPG) for Domain and BIE Discretizations (2004), Tech Science Press
12. Batra R.C., Porfiri M., and Spinello D. Treatment of material discontinuity in two meshless local Petrov-Galerkin (MLPG) formulations of axisymmetric transient heat conduction. Internat. J. Numer. Methods Engrg. 61 (2004) 2461-2479
13. Dehghan M., and Mirzaei D. The meshless local Petrov-Galerkin (MLPG) method for the generalized two-dimensional non-linear Schrödinger equation. Eng. Anal. Bound. Elem. 32 (2008) 747-756
14. Dehghan M., and Mirzaei D. Meshless Local Petrov-Galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity. Appl. Numer. Math. 59 (2009) 1043-1058
15. Sladek J., Sladek V., Krivacek J., Wen P.H., and Zhang Ch. Meshless local Petrov-Galerkin (MLPG) method for Reissner-Mindlin plates under dynamic load. Comput. Methods Appl. Mech. Engrg. 196 (2007) 2681-2691
16. Sladek J., Sladek V., and Hon Y.C. Inverse heat conduction problems by meshless local Petrov-Galerkin method. Eng. Anal. Bound. Elem. 30 (2006) 650-661
17. Sladek J., Sladek V., Zhang Ch., and Schanz M. Meshless local Petrov-Galerkin method for continuously nonhomogeneous linear viscoelastic solids. Comput. Mech. 37 (2006) 279-289
18. Sladek J., Sladek V., Zhang Ch., Krivacek J., and Wen Ph. Analysis of orthotropic thick plates by meshless local Petrov-Galerkin (MLPG) method. Internat. J. Numer. Methods Engrg. 67 (2006) 1830-1850
19. Xiao J.R., and McCarthy M.A. Meshless analysis of the obstacle problem for beams by the MLPG method and subdomain variational formulations. Eur. J. Mech. A Solids 22 (2003) 385-399
20. Dehghan M. Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math. Comput. Simulation 71 (2006) 16-30
21. Pedersen N.F., and Madsen S. Search for the in-phase flux flow mode in stacked Josephson junctions. Physica C 437 (2006) 262-266
22. Darminto H.S., and van Gils S.A. Static sand dynamic properties of fluxons in a zig-zag 0 - π Josephson junction. Phys. Lett. A 361 (2007) 270-276
23. Argyris J., Haase M., and Heinrich J.C. Finite element approximation to two-dimensional sine-Gordon solitons. Comput. Methods Appl. Mech. Engrg. 86 (1991) 1-26
24. Bratsos A.G. The solution of the two-dimensional sine-Gordon equation using the method of lines. J. Comput. Appl. Math. 206 (2007) 251-277
25. Djidjeli K., Price W.G., and Twizell E.H. Numerical solutions of a damped sine-Gordon equation in two space variables. J. Engrg. Math. 29 (1995) 347-369
26. Christiansen P.L., and Lomdahl P.S. Numerical solutions of 2+1 dimensional sine-Gordon solitons. Physica D 2 3 (1981) 482-494
27. Hirota R. Exact three-soliton solution of the two-dimensional sine-Gordon equation. J. Phys. Soc. Japan 35 (1973) 1566
28. Christiansen P.L., and Olsen O.H. Return effect for rotationally symmetric solitary wave solutions to the sine-Gordon equation. Phys. Lett. A 68 2 (1978) 185-188
29. Zagrodzinsky J. Particular solutions of the sine-Gordon equation in 2+1 dimensions. Phys. Lett. A 72 (1979) 284-286
30. Kaliappan P., and Lakshmanan M. Kadomtsev-Petviashvili and two-dimensional sine-Gordon equations: Reduction to Painlevè transcendents. J. Phys. A: Math. Gen. 12 (1979) L249-L252
31. Leibbrandt G. New exact solutions of the classical sine-Gordon equation in 2+1 and 3+1 dimensions. Phys. Rev. Lett. 41 (1978) 435-438
32. Guo B.Y., Pascual P.J., Rodriguez M.J., and Vázquez L. Numerical solution of the sine-Gordon equation. Appl. Math. Comput. 18 (1986) 1-14
33. Sheng Q., Khaliq A.Q.M., and Voss D.A. Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme. Math. Comput. Simulation 68 (2005) 355-373
34. Bratsos A.G. An explicit numerical scheme for the sine-Gordon equation in 2+1 dimensions. Appl. Num. Anal. Comp. Math. 2 2 (2005) 189-211
35. Bratsos A.G. A modified predictor-corrector scheme for the two-dimensional sine-Gordon equation. Numer. Algorithms 43 (2006) 295-308
36. Bratsos A.G. A third order numerical scheme for the two-dimensional sine-Gordon equation. Math. Comput. Simulation 76 (2007) 271-282
37. Bratsos A.G. An improved numerical scheme for the sine-Gordon equation in 2+1 dimensions. Internat. J. Numer. Methods Engrg. 75 (2008) 787-799
38. Dehghan M., and Mirzaei D. The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation. Comput. Methods Appl. Mech. Engrg. 197 (2008) 476-486
39. Mirzaei D., and Dehghan M. Boundary element solution of the two-dimensional sine-Gordon equation using continuous linear elements. Eng. Anal. Bound. Elem. 33 (2009) 12-24
40. Dehghan M., and Shokri A. A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions. Math. Comput. Simulation 79 (2008) 700-715
41. Dehghan M., and Mirzaei D. The boundary integral equation approach for numerical solution of the one-dimensional sine-Gordon equation. Numer. Methods Partial Differential Equations 24 (2008) 1405-1415
42. Wazwaz A.M. The tanh method and a variable separated ODE method for solving double sine-Gordon equation. Phys. Lett. A 350 (2006) 367-370
43. Wazwaz A.M. Exact solutions for the generalized sine-Gordon and the generalized sinh-Gordon equations. Chaos Solitons Fractals 28 (2006) 127-135
44. Helal M.A. Soliton solution of some nonlinear partial differential equations and its application in fluid mechanics. Chaos Solitons Fractals 13 (2002) 1917-1929
45. Hu D.A., Long S.Y., Liu K.Y., and Li G.Y. A modified meshless local Petrov-Galerkin method to elasticity problems in computer modelling and simulation. Eng. Anal. Bound. Elem. 30 (2006) 399-404
46. Liu K.Y., Long S.Y., and Li G.Y. A simple and less-costly meshless local Petrov-Galerkin (MLPG) method for the dynamic fracture problem. Eng. Anal. Bound. Elem. 30 (2006) 72-76
47. Belytschko T., Krongauz Y., Organ D., Fleming M., and Krysl P. Meshless methods: An overview and recent developments. Comput. Methods Appl. Mech. Engrg. 139 (1996) 3-47
48. Bogolyubskiǐ I.L., and Makhankov V.G. Lifetime of pulsating solitons in certain classical models. JETP Lett. 24 1 (1976) 12-14
49. Bogolyubskiǐ I.L. Oscillating particle-like solutions of the nonlinear Klein-Gordon equation. JETP Lett. 24 10 (1976) 535-538
50. Christiansen P.L., Grønbech-Jensen N., Lomdahl P.S., and Malomed B.A. Oscillations of eccentric Pulsons. Phys. Scr. 55 (1997) 131-134
51. Malomed B.A. Decay of shrinking solitons in multidimensional sine-Gordon equation. Physica D 24 (1987) 155-171
52. Malomed B.A. Dynamic of quasi-one-dimensional kinks in the two-dimensional sine-Gordon model. Physica D 52 (1991) 157-170
53. Maslov E.M. Dynamics of rotationally symmetric solitons in near-SG field model with applications to large-area Josephson junctions and ferromagnets. Physica D 15 (1985) 433-443
54. Maslov E.M. Rotationally symmetric SG oscillator with tunable frequency. Phys. Lett. A 131 6 (1988) 364-367