A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions

Mathematics and Computers in Simulation - Tập 79 Số 3 - Trang 700-715 - 2008
Mehdi Dehghan1, Ali Shokri1
1Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Avenue, Tehran, Iran

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Ahmed, 2006, A collocation method using new combined radial basis functions of thin plate and multiquadraic types, Eng. Anal. Bound. Elem., 30, 697, 10.1016/j.enganabound.2006.03.001

Argyris, 1987, An engineer’s guide to soliton phenomena, Application of the finite element method, Comput. Methods Appl. Mech. Eng., 61, 71, 10.1016/0045-7825(87)90117-4

Argyris, 1991, Finite element approximation to two-dimensional sine-Gordon solitons, Comput. Methods Appl. Mech. Eng., 86, 1, 10.1016/0045-7825(91)90136-T

Bratsos, 2007, A third order numerical scheme for the two-dimensional sine-Gordon equation, Math. Comput. Simulation, 76, 271, 10.1016/j.matcom.2006.11.004

Bratsos, 2007, The solution of the two-dimensional sine-Gordon equation using the method of lines, J. Comput. Appl. Math., 206, 251, 10.1016/j.cam.2006.07.002

Bratsos, 2005, An explicit numerical scheme for the sine-Gordon equation in 2+1 dimensions, Appl. Numer. Anal. Comput. Math., 2, 189, 10.1002/anac.200410035

Bratsos, 2006, A modified predictor-corrector scheme for the two-dimensional sine-Gordon equation, Numer. Algor., 43, 295, 10.1007/s11075-006-9061-3

Chen, 2004, A meshless method for free vibration analysis of circular and rectangular clamped plates using radial basis function, Eng. Anal. Bound. Elem., 28, 535, 10.1016/S0955-7997(03)00106-1

Christiansen, 1981, Numerical solution of 2+1 dimensional sine-Gordon solitons, Physica, 2D, 482

Christiansen, 1979, On dynamical two-dimensional solutions to the sine-Gordon equation, Z. Angew. Math. Mech., 59, 10

Dehghan, 2006, Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math. Comput. Simulation, 71, 16, 10.1016/j.matcom.2005.10.001

Dehghan, 2007, A numerical method for KdV equation using collocation and radial basis functions, Nonlinear Dyn., 50, 111, 10.1007/s11071-006-9146-5

Dehghan, 2008, The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation, Comput. Methods Appl. Mech. Eng., 197, 476, 10.1016/j.cma.2007.08.016

Djidjeli, 1995, Numerical solutions of a damped sine-Gordon equation in two space variables, J. Eng. Math., 29, 347, 10.1007/BF00042761

Franke, 1997, Convergence orders of meshless collocation methods using radial basis functions, Adv. Comput. Math., 8, 381, 10.1023/A:1018916902176

Franke, 1998, Solving partial differential equations by collocation using radial basis functions, Appl. Math. Comput., 93, 73, 10.1016/S0096-3003(97)10104-7

Gorria, 2004, Kink propagation and trapping in a two-dimensional curved Josephson junction, Phys. Rev. B, 69, 1, 10.1103/PhysRevB.69.134506

Guo, 1986, Numerical solution of the sine-Gordon equation, Appl. Math. Comput., 18, 1, 10.1016/0096-3003(86)90025-1

Hardy, 1971, Multiquadric equations of topography and other irregular surfaces, Geophys. Res., 176, 1905, 10.1029/JB076i008p01905

Helal, 2002, Soliton solution of some nonlinear partial differential equations and its application in fluid mechanics, Chaos Solitons Fractals, 13, 1917, 10.1016/S0960-0779(01)00189-8

Hirota, 1973, Exact three-soliton solution of the two-dimensional sine-Gordon equation, J. Phys. Soc. Jpn., 35, 15, 10.1143/JPSJ.35.1566

Hon, 2000, Additive Schwarz domain decomposition with radial basis approximation, Int. J. Appl. Math., 4, 599

Josephson, 1965, Supercurrents through barriers, Adv. Phys., 14, 419, 10.1080/00018736500101091

Kaliappan, 1979, Kadomtsev-Petviashvili and two-dimensional sine-Gordon equations: reduction to Painlev transcendents, J. Phys. A: Math. Gen., 249, 23

Kansa, 1990, Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics. I, Comput. Math. Appl., 19, 127, 10.1016/0898-1221(90)90270-T

Kansa, 1990, Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics. II, Comput. Math. Appl., 19, 147, 10.1016/0898-1221(90)90271-K

Kaup, 1978, Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory, Proc. Roy. Soc. London Ser. A, 361, 413, 10.1098/rspa.1978.0110

Liew, 2004, Mesh-free radial basis function method for buckling analysis of non-uniformly loaded arbitrarily shaped shear deformable plates, Comput. Methods Appl. Mech. Eng., 193, 205, 10.1016/j.cma.2003.10.002

Madych, 1990, Multivariate interpolation and conditionally positive definite functions II, Math. Comput., 54, 211, 10.1090/S0025-5718-1990-0993931-7

Madych, 1992, Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation, J. Approx. Theory, 70, 94, 10.1016/0021-9045(92)90058-V

Nakajima, 1974, Numerical analysis of vortex motion on Josephson structures, J. Appl. Phys., 45, 4095, 10.1063/1.1663917

Nardini, 1982

Sheng, 2005, Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme, Math. Comput. Simulation, 68, 355, 10.1016/j.matcom.2005.02.017

Vitor, 2004, RBF-based meshless methods for 2D elastostatic problems, Eng. Anal. Bound. Elem., 28, 1271, 10.1016/j.enganabound.2003.06.003

Wendland, 1995, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math., 4, 389, 10.1007/BF02123482

Xin, 2000, Modeling light bullets with the two-dimensional sine-Gordon equation, Physica D, 135, 345, 10.1016/S0167-2789(99)00128-1

Zagrodzinsky, 1979, Particular solutions of the sine-Gordon equation in 2+1 dimensions, Phys. Lett., 72A, 284, 10.1016/0375-9601(79)90469-9

Zerroukat, 1992, A numerical method for heat transfer problem using collocation and radial basis functions, Int. J. Numer. Methods Eng., 42, 1263, 10.1002/(SICI)1097-0207(19980815)42:7<1263::AID-NME431>3.0.CO;2-I

M. Dehghan, On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, Numer. Methods Partial Differ. Equations 21 (2005) 24–40.

M. Dehghan, Parameter determination in a partial differential equation from the overspecified data, Math. Comput. Model. 41 (2005) 196–213.

M. Dehghan, The one-dimensional heat equation subject to a boundary integral specification, Chaos, Solitons Fractals 32 (2007) 661–675.

M. Dehghan, Implicit collocation technique for heat equation with non-classic initial condition, Int. J. Nonlin. Sci. Numer. Simul. 7 (2006) 447–450.

M. Dehghan, A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications, Numer. Methods Partial Differ. Equations 22 (2006) 220–257.

F. Shakeri, M. Dehghan, Numerical solution of the Klein-Gordon equation via He’s variational iteration method, Nonlin. Dynam. 51 (2008) 89–97.

M. Dehghan, F. Shakeri, Application of He’s variational iteration method for solving the Cauchy reaction–diffusion problem J. Comput. Appl. Math. 214 (2008) 435–446.