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**A meshless method for numerical solution of a linear hyperbolic equation with variable coefficients in two space dimensions.**
*(English)*
Zbl 1159.65084

Summary: A meshless method is proposed for the numerical solution of the two space dimensional linear hyperbolic equation subject to appropriate initial and Dirichlet boundary conditions. The new developed scheme uses collocation points and approximates the solution employing thin plate splines radial basis functions. Numerical results are obtained for various cases involving variable, singular and constant coefficients, and are compared with analytical solutions to confirm the good accuracy of the presented scheme.

### MSC:

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

35L15 | Initial value problems for second-order hyperbolic equations |

### Keywords:

collocation; radial basis functions; thin plate splines; two-dimensional linear hyperbolic equation; two-dimensional telegraph equation; meshless method; numerical results
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\textit{M. Dehghan} and \textit{A. Shokri}, Numer. Methods Partial Differ. Equations 25, No. 2, 494--506 (2009; Zbl 1159.65084)

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### References:

[1] | Gao, Unconditionally stable difference schemes for a one-space-dimensional linear hyperbolic equation, Appl Math Comput 187 pp 1272– (2007) · Zbl 1114.65347 |

[2] | Mohanty, An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation, Numer Methods Partial Diff Eq 17 pp 684– (2001) |

[3] | Mohanty, An unconditionally stable ADI method for the linear hyperbolic equation in three space dimensions, Int J Comput Math 79 pp 133– (2002) · Zbl 0995.65093 |

[4] | Mohanty, An operator splitting method for an unconditionally stable difference scheme for a linear hyperbolic equation with variable coefficients in two space dimensions, Appl Math Comput 152 pp 799– (2004) · Zbl 1077.65093 |

[5] | Dehghan, High order implicit collocation method for the solution of two dimensional linear hyperbolic equation, Numer Methods Partial Diff Eq |

[6] | Dehghan, A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications, Numer Methods Partial Diff Eq 22 pp 220– (2006) |

[7] | Dehghan, Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math Comput Simul 71 pp 16– (2006) · Zbl 1089.65085 |

[8] | Ahmed, A collocation method using new combined radial basis functions of thin plate and multiquadraic types, Eng Anal Bound Elem 30 pp 697– (2006) · Zbl 1195.80035 |

[9] | Kansa, Multiquadrics-A scattered data approximation scheme with applications to computational fluid dynamics - I, Comput Math Appl 19 pp 127– (1990) · Zbl 0692.76003 |

[10] | Kansa, Multiquadrics-A scattered data approximation scheme with applications to computational fluid dynamics - II, Comput Math Appl 19 pp 147– (1990) · Zbl 0692.76003 |

[11] | Hon, An efficient numerical scheme for Burgers equation, Appl Math Comput 95 pp 37– (1998) · Zbl 0943.65101 |

[12] | Hon, Multiquadric solution for shallow water equations, ASCE J Hydraulic Eng 125 pp 524– (1999) |

[13] | Zerroukat, A numerical method for heat transfer problem using collocation and radial basis functions, Int J Numer Meth Eng 42 pp 1263– (1992) |

[14] | Dehghan, A numerical method for KdV equation using collocation and radial basis functions, Nonlinear Dynam 50 pp 111– (2007) · Zbl 1185.76832 |

[15] | Dehghan, A numerical method for two-dimensional Schrödinger equation using collocation and radial basis functions, Comput Math Appl 54 pp 136– (2007) · Zbl 1126.65092 |

[16] | Hon, A radial basis function method for solving options pricing model, Financial Eng 8 pp 31– (1999) |

[17] | Marcozzi, On the use of boundary conditions for variational formulations arising in financial mathematics, Appl Math Comput 124 pp 197– (2001) · Zbl 1047.91033 |

[18] | Fasshauer, Solving partial differential equations by collocation with radial basis functions (1997) · Zbl 0938.65140 |

[19] | Dehghan, Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions, Math Comput Model 44 pp 1160– (2006) · Zbl 1137.65408 |

[20] | Lapidus, Numerical solution of partial differential equations in science and engineering (1982) · Zbl 0584.65056 |

[21] | Dehghan, Implicit collocation technique for heat equation with non-classic initial condition, Int J Nonlinear Sci Numer Simulation 7 pp 447– (2006) · Zbl 06942230 |

[22] | Dehghan, On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, Numer Methods Partial Diff Eq 21 pp 24– (2005) |

[23] | Dehghan, The one-dimensional heat equation subject to a boundary integral specification, Chaos, Solitons Fractals 32 pp 661– (2007) · Zbl 1139.35352 |

[24] | Dehghan, Parameter determination in a partial differential equation from the overspecified data, Math Comput Modelling 41 pp 196– (2005) · Zbl 1080.35174 |

[25] | Dehghan, Efficient techniques for the second-order parabolic equation subject to nonlocal specifications, Appl Numer Math 52 pp 39– (2005) · Zbl 1063.65079 |

[26] | Dehghan, The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation, Computer Methods in Applied Mechanics and Engineering 197 pp 476– (2008) · Zbl 1169.76401 |

[27] | Shakeri, Numerical solution of the Klein-Gordon equation via He’s variational iteration method, Nonlinear Dynamics 51 pp 89– (2008) · Zbl 1179.81064 |

[28] | Dehghan, Identification of a time-dependent coefficient in a partial differential equation subject to an extra measurement, Numer Methods Partial Diff Eq 21 pp 611– (2005) · Zbl 1069.65104 |

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