Abstract
The singular boundary method (SBM) is employed to solve the two-dimensional telegraph equation on arbitrary domains. The Houbolt finite difference method is used to discretize the time derivatives. The original equations are then split into a system of partial differential equations, which is solved using the method of particular solution, in combination with the singular boundary method to obtain the homogeneous solution. Finally, three numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method.
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References
Brebbia CA, Telles JCF, Wrobel L (2012) Boundary element techniques: theory and applications in engineering. Springer Science and Business Media, Berlin
Guiggiani M, Gigante A (1990) A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method. J Appl Mech 57(4):906–915
Banerjee PK (1994) The boundary element methods in engineering. McGRAW-HILL Book Company Europe, New York
Kansa E (1990) 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):127–145
Dehghan M, Shokri A (2009) Numerical solution of the nonlinear klein-gordon equation using radial basis functions. J Comput Appl Math 230(2):400–410
Aslefallah M, Shivanian E (2015) Nonlinear fractional integro-differential reaction–diffusion equation via radial basis functions. Eur Phys J Plus 130(47):1–9
Shivanian E (2015) A meshless method based on radial basis and spline interpolation for 2-D and 3-D inhomogeneous biharmonic BVPs. Z Naturforschung A 70(8):673–682
Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37(2):229–256
Abbasbandy S, Shirzadi A (2010) A meshless method for two-dimensional diffusion equation with an integral condition. Eng Anal Bound Elem 34(12):1031–1037
Shivanian E (2015) Meshless local Petrov–Galerkin (MLPG) method for three-dimensional nonlinear wave equations via moving least squares approximation. Eng Anal Bound Elem 50:249–257
Shivanian E, Aslefallah M (2017) Stability and convergence of spectral radial point interpolation method locally applied on two-dimensional pseudoparabolic equation. Numer Methods Partial Differ Equ 33(3):724–741
Aslefallah M, Shivanian E (2018) An efficient meshless method based on RBFs for the time fractional diffusion-wave equation. Afr Mat 29(7–8):1203–1214
Shivanian E, Aslefallah M (2019) Numerical solution of two-dimensional hyperbolic equations with nonlocal integral conditions using radial basis functions. Int J Ind Math 11(1):25–34
Fairweather G, Karageorghis A (1998) The method of fundamental solutions for elliptic boundary value problems. Adv Comput Math 9(1–2):69
Marin L (2010) Regularized method of fundamental solutions for boundary identification in two-dimensional isotropic linear elasticity. Int J Solids Struct 47(24):3326–3340
Karageorghis A, Lesnic D, Marin L (2011) A survey of applications of the MFS to inverse problems. Inverse Probl Sci Eng 19(3):309–336
Marin L, Lesnic D (2005) The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations. Comput Struct 83:267–78
Lin J, Reutskiy SY (2018) An accurate meshless formulation for the simulation of linear and fully nonlinear advection diffusion reaction problems. Adv Eng Softw 126:127–146
Lin J, Reutskiy SY, Lu J (2018) A novel meshless method for fully nonlinear advection–diffusion–reaction problems to model transfer in anisotropic media. Appl Math Comput 339:459–476
Lin J, Chen CS, Liu C-S, Lu J (2016) Fast simulation of multi-dimensional wave problems by the sparse scheme of the method of fundamental solutions. Comput Math Appl 72(3):555–567
Chen W (2009) Singular boundary method: a novel, simple, mesh-free, boundary collocation numerical method. Chin J Solid Mech 30(6):592–599
Li JP, Chen W, Fu ZJ, Sun LL (2016) Explicit empirical formula evaluating original intensity factors of singular boundary method for potential and Helmholtz problems. Eng Anal Bound Elem 73:161–169
Li JP, Chen W, Fu ZJ (2016) Numerical investigation on convergence rate of singular boundary method. Math Probl Eng 2016:1–13
Lin J, Chen W, Chen CS (2014) Numerical treatment of acoustic problems with boundary singularities by the singular boundary method. J Sound Vib 333(14):3177–3188
Chen W, Fu Z, Wei X (2009) Potential problems by singular boundary method satisfying moment condition. CMES 54(1):65–85
Tang Z, Fu Z, Zheng D, Huang J (2018) Singular boundary method to simulate scattering of SH wave by the canyon topography. Adv Appl Math Mech 10:912–924
Qu WZ, Chen W, Gu Y (2015) Fast multipole accelerated singular boundary method for the 3D Helmholtz equation in low frequency regime. Comput Math Appl 70:679–690
Lin Ji, Zhang C, Sun L, Lu J (2018) Simulation of seismic wave scattering by embedded cavities in an elastic half-plane using the novel singular boundary method. Adv Appl Math Mech 10(2):322–342
Wang F, Chen W, Zhang C, Lin J (2017) Analytical evaluation of the origin intensity factor of time-dependent diffusion fundamental solution for a matrix-free singular boundary method formulation. Appl Math Modell 49:647–662
Chen W, Tanaka M (2002) A meshless, integration-free, and boundary-only RBF technique. Comput Math Appl 43:379–91
Young DL, Chen KH, Lee CW (2005) Novel meshless method for solving the potential problems with arbitrary domain. J Comput Phys 209(1):290–321
Chen W, Zhang JY, Fu ZJ (2014) Singular boundary method for modified Helmholtz equations. Eng Anal Bound Elem 44:112–119
Sarler B (2009) Solution of potential flow problems by the modified method of fundamental solutions: formulations with the single layer and the double layer fundamental solutions. Eng Anal Bound Elem 33(12):1374–1382
Liu YJ (2010) A new boundary meshfree method with distributed sources. Eng Anal Bound Elem 34(11):914–919
Gu Y, Chen W, Zhang CZ (2011) Singular boundary method for solving plane strain elastostatic problems. Int J Solids Struct 48(18):2549–2556
Gu Y, Chen W, He XQ (2012) Singular boundary method for steady-state heat conduction in three dimensional general anisotropic media. Int J Heat Mass Transfer 55(17–18):4837–4848
Gu Y, Chen W (2013) Infinite domain potential problems by a new formulation of singular boundary method. Appl Math Model 37:1638–51
Gu Y, Chen W, Zhang J (2012) Investigation on near-boundary solutions by singular boundary method. Eng Anal Bound Elem 36:1173–82
Shivanian E, Abbasbandy S, Alhuthali MS, Alsulami HH (2015) Local integration of 2-D fractional telegraph equation via moving least squares approximation. Eng Anal Bound Elem 56:98–105
Lin CY, Gu MH, Young DL (2010) The time-marching method of fundamental solutions for multi-dimensional telegraph equations, CMC: computers. Mater Contin 18(1):43–68
Abbasbandy S, Ghehsareh HR, Hashim I, Alsaedi A (2014) A comparison study of meshfree techniques for solving the two-dimensional linear hyperbolic telegraph equation. Eng Anal Bound Elem 47:10–20
Ramachandran PA, Balakrishnan K (2000) Radial basis functions as approximate particular solutions: review of recent progress. Eng Anal Bound Elem 24:575–582
Muleshkov AS, Golberg MA, Chen CS (1999) Particular solutions of Helmholtz-type operators using higher order polyharmonic splines. Comput Mech 24(5–6):411–419
Chen CS, Fan CM, Wen PH (2011) The method of approximate particular solutions for solving elliptic problems with variable coefficients. Int J Comput Methods 8(3):545–559
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Aslefallah, M., Rostamy, D. Application of the singular boundary method to the two-dimensional telegraph equation on arbitrary domains. J Eng Math 118, 1–14 (2019). https://doi.org/10.1007/s10665-019-10008-8
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DOI: https://doi.org/10.1007/s10665-019-10008-8