# zbMATH — the first resource for mathematics

A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions. (English) Zbl 1155.65379
Summary: The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical scheme to solve the two-dimensional damped/undamped sine-Gordon equation. The proposed scheme is based on using collocation points and approximating the solution employing the thin plate splines (TPS) radial basis function (RBF). The new scheme works in a similar fashion as finite difference methods. Numerical results are obtained for various cases involving line and ring solitons.

##### MSC:
 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 35Q51 Soliton equations 35Q53 KdV equations (Korteweg-de Vries equations)
Full Text:
##### References:
 [1] Ahmed, S.G., A collocation method using new combined radial basis functions of thin plate and multiquadraic types, Eng. anal. bound. elem., 30, 697-701, (2006) · Zbl 1195.80035 [2] Argyris, J.; Haase, M., An engineer’s guide to soliton phenomena, application of the finite element method, Comput. methods appl. mech. eng., 61, 71-122, (1987) · Zbl 0624.76020 [3] Argyris, J.; Haase, M.; Heinrich, J.C., Finite element approximation to two-dimensional sine-Gordon solitons, Comput. methods appl. mech. eng., 86, 1-26, (1991) · Zbl 0762.65073 [4] Bratsos, A.G., A third order numerical scheme for the two-dimensional sine-Gordon equation, Math. comput. simulation, 76, 271-282, (2007) · Zbl 1135.65358 [5] Bratsos, A.G., The solution of the two-dimensional sine-Gordon equation using the method of lines, J. comput. appl. math., 206, 251-277, (2007) · Zbl 1117.65126 [6] Bratsos, A.G., An explicit numerical scheme for the sine-Gordon equation in $$2 + 1$$ dimensions, Appl. numer. anal. comput. math., 2, 2, 189-211, (2005) · Zbl 1075.65111 [7] Bratsos, A.G., A modified predictor-corrector scheme for the two-dimensional sine-Gordon equation, Numer. algor., 43, 295-308, (2006) · Zbl 1112.65077 [8] Chen, J.T.; Chen, I.L.; Chen, K.H.; Lee, Y.T.; Yeh, Y.T., A meshless method for free vibration analysis of circular and rectangular clamped plates using radial basis function, Eng. anal. bound. elem., 28, 5, 535-545, (2004) · Zbl 1130.74488 [9] Christiansen, P.L.; Lomdahl, P.S., Numerical solution of $$2 + 1$$ dimensional sine-Gordon solitons, Physica, 2D, 482-494, (1981) · Zbl 1194.65122 [10] Christiansen, P.L.; Olsen, O.H., On dynamical two-dimensional solutions to the sine-Gordon equation, Z. angew. math. mech., 59, 30, 10, (1979) · Zbl 0409.35069 [11] Dehghan, M., Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math. comput. simulation, 71, 16-30, (2006) · Zbl 1089.65085 [12] Dehghan, M.; Shokri, A., A numerical method for KdV equation using collocation and radial basis functions, Nonlinear dyn., 50, 111-120, (2007) · Zbl 1185.76832 [13] Dehghan, M.; Mirzaei, D., The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation, Comput. methods appl. mech. eng., 197, 476-486, (2008) · Zbl 1169.76401 [14] Djidjeli, K.; Price, W.G.; Twizell, E.H., Numerical solutions of a damped sine-Gordon equation in two space variables, J. eng. math., 29, 347-369, (1995) · Zbl 0841.65083 [15] Franke, C.; Schaback, R., Convergence orders of meshless collocation methods using radial basis functions, Adv. comput. math., 8, 4, 381-399, (1997) · Zbl 0909.65088 [16] Franke, C.; Schaback, R., Solving partial differential equations by collocation using radial basis functions, Appl. math. comput., 93, 1, 73-82, (1998) · Zbl 0943.65133 [17] Gorria, C.; Gaididei, Yu.B.; Soerensen, M.P.; Christiansen, P.L.; Caputo, J.G., Kink propagation and trapping in a two-dimensional curved Josephson junction, Phys. rev. B, 69, 1-10, (2004) [18] Guo, B.-Y.; Pascual, P.J.; Rodriguez, M.J.; Vzquez, L., Numerical solution of the sine-Gordon equation, Appl. math. comput., 18, 1-14, (1986) · Zbl 0622.65131 [19] Hardy, R.L., Multiquadric equations of topography and other irregular surfaces, Geophys. res., 176, 1905-1915, (1971) [20] Helal, M.A., Soliton solution of some nonlinear partial differential equations and its application in fluid mechanics, Chaos solitons fractals, 13, 1917-1929, (2002) · Zbl 0997.35063 [21] Hirota, R., Exact three-soliton solution of the two-dimensional sine-Gordon equation, J. phys. soc. jpn., 35, 15-66, (1973) [22] Hon, Y.C.; Wu, Z., Additive Schwarz domain decomposition with radial basis approximation, Int. J. appl. math., 4, 1, 599-606, (2000) · Zbl 1051.65121 [23] Josephson, J.D., Supercurrents through barriers, Adv. phys., 14, 419-451, (1965) [24] Kaliappan, P.; Lakshmanan, M., Kadomtsev-Petviashvili and two-dimensional sine-Gordon equations: reduction to painlev transcendents, J. phys. A: math. gen., 249, 23, (1979) · Zbl 0425.35080 [25] Kansa, E.J., Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics. I, Comput. math. appl., 19, 127-145, (1990) · Zbl 0692.76003 [26] Kansa, E.J., Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics. II, Comput. math. appl., 19, 147-161, (1990) · Zbl 0850.76048 [27] Kaup, D.J.; Newell, A.C., Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory, Proc. roy. soc. London ser. A, 361, 413-446, (1978) [28] Liew, K.L., Mesh-free radial basis function method for buckling analysis of non-uniformly loaded arbitrarily shaped shear deformable plates, Comput. methods appl. mech. eng., 193, 3, 205-224, (2004) · Zbl 1075.74700 [29] Madych, W.R.; Nelson, S.A., Multivariate interpolation and conditionally positive definite functions II, Math. comput., 54, 189, 211-230, (1990) · Zbl 0859.41004 [30] Madych, W.R.; Nelson, S.A., Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation, J. approx. theory, 70, 94-114, (1992) · Zbl 0764.41003 [31] Nakajima, K.; Onodera, Y.; Nakamura, T.; Sato, R., Numerical analysis of vortex motion on Josephson structures, J. appl. phys., 45, 9, 4095-4099, (1974) [32] Nardini, D.; Brebbia, C.A., A new approach to free vibration analysis using boundary elements, boundary element methods in engineering, (1982), Computational Mechanics Publications Southampton · Zbl 0541.73104 [33] Sheng, Q.; Khaliq, A.Q.M.; Voss, D.A., Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme, Math. comput. simulation, 68, 355-373, (2005) · Zbl 1073.65095 [34] Vitor, M., RBF-based meshless methods for 2D elastostatic problems, Eng. anal. bound. elem., 28, 10, 1271-1281, (2004) · Zbl 1077.74056 [35] Wendland, H., Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. comput. math., 4, 389-396, (1995) · Zbl 0838.41014 [36] Xin, J.X., Modeling light bullets with the two-dimensional sine-Gordon equation, Physica D, 135, 345-368, (2000) · Zbl 0936.78006 [37] Zagrodzinsky, J., Particular solutions of the sine-Gordon equation in $$2 + 1$$ dimensions, Phys. lett., 72A, 284-286, (1979) [38] Zerroukat, M.; Power, H.; Chen, C.S., A numerical method for heat transfer problem using collocation and radial basis functions, Int. J. numer. methods eng., 42, 1263-1278, (1992) · Zbl 0907.65095 [39] 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. · Zbl 1059.65072 [40] M. Dehghan, Parameter determination in a partial differential equation from the overspecified data, Math. Comput. Model. 41 (2005) 196-213. · Zbl 1080.35174 [41] M. Dehghan, The one-dimensional heat equation subject to a boundary integral specification, Chaos, Solitons Fractals 32 (2007) 661-675. · Zbl 1139.35352 [42] M. Dehghan, Implicit collocation technique for heat equation with non-classic initial condition, Int. J. Nonlin. Sci. Numer. Simul. 7 (2006) 447-450. [43] 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. · Zbl 1084.65099 [44] F. Shakeri, M. Dehghan, Numerical solution of the Klein-Gordon equation via He’s variational iteration method, Nonlin. Dynam. 51 (2008) 89-97. · Zbl 1179.81064 [45] 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. · Zbl 1135.65381
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.