On the solution of the non-local parabolic partial differential equations via radial basis functions

Applied Mathematical Modelling - Tập 33 - Trang 1729-1738 - 2009
Mehdi Tatari1, Mehdi Dehghan2
1Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran
2Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No.424, Hafez Avenue, Tehran 15914, Iran

Tài liệu tham khảo

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 Sarra, 2005, Adaptive radial basis function method for time dependent partial differential equations, Appl. Numer. Math., 54, 79, 10.1016/j.apnum.2004.07.004 Shu, 2004, Solution of partial differential equations by a global radial basis function-based differential quadrature method, Eng. Anal. Bound. Elem., 28, 1217, 10.1016/j.enganabound.2003.02.001 Sarler, 2006, Meshfree explicit local radial basis function collocation method for diffusion problems, Comput. Math. Appl., 51, 1269, 10.1016/j.camwa.2006.04.013 Vertnik, 2006, Meshless local radial basis function collocation method for convective–diffusive solid–liquid phase change problems, Int. J. Numer. Methods Heat Fluid Flow, 16, 617, 10.1108/09615530610669148 Cannon, 1963, The solution of the heat equation subject to the specification of energy, Quart. Appl. Math., 21, 155, 10.1090/qam/160437 Day, 1992, Parabolic equations and thermodynamics, Quart. Appl. Math., 50, 523, 10.1090/qam/1178432 Bouziani, 1996, Mixed problem with boundary integral conditions for a certain parabolic equation, J. Appl. Math. Stoch. Anal., 9, 323, 10.1155/S1048953396000305 Bouziani, 1999, On a class of parabolic equations with a nonlocal boundary condition, Acad. Roy. Belg. Bull. Cl. Sci., 10, 61 Ionkin, 1980, Stability of a problem in heat transfer theory with a non-classical boundary condition, Differen. Equat., 15, 911 Sun, 1996, A second-order accurate finite difference scheme for a class of nonlocal parabolic equations with natural boundary conditions, J. Comput. Appl. Math., 76, 137, 10.1016/S0377-0427(96)00097-0 Ang, 2002, A method of solution for the one-dimensional heat equation subject to a nonlocal condition, SEA Bull. Math., 26, 197 Cahlon, 1995, Stepwise stability for the heat equation with a nonlocal constraint, SIAM J. Numer. Anal., 32, 571, 10.1137/0732025 Cannon, 1984, The one dimensional heat equation, vol. 23 Cannon, 1993, A numerical procedure for diffusion subject to the specification of mass, Int. J. Eng. Sci., 31, 347, 10.1016/0020-7225(93)90010-R Cannon, 1987, A Galerkin procedure for the diffusion equation subject to the specification of mass, SIAM J. Numer. Anal., 24, 499, 10.1137/0724036 Cannon, 1982, Implicit finite difference scheme for the diffusion of mass in porous media, 527 Cannon, 1986, Diffusion subject to specification of mass, J. Math. Anal. Appl., 115, 517, 10.1016/0022-247X(86)90012-0 Cannon, 1990, An implicit finite difference scheme for the diffusion equation subject to the specification of mass, Int. J. Eng. Sci., 28, 573, 10.1016/0020-7225(90)90086-X Cannon, 1990, A Galerkin procedure for diffusion equations with boundary integral conditions, Int. J. Eng. Sci., 28, 579, 10.1016/0020-7225(90)90087-Y Cannon, 1989, On a class of non-classical parabolic problems, Differen. Equat., 79, 266, 10.1016/0022-0396(89)90103-4 Deckert, 1963, Solutions for diffusion equations with integral type boundary conditions, Proc. Iowa Acad. Sci., 70, 345 Ionkin, 1991, Uniform stability of difference schemes for a nonlocal nonself-adjoint boundary value problem with variable coefficients, Differen. Equat., 27, 820 Kacur, 1995, On the numerical solution of semilinear parabolic problems in multicomponent structures with Volterra operators in the transmission conditions and in the boundary conditions, Z. Angew. Math. Mech., 75, 91, 10.1002/zamm.19950750202 Liu, 1999, Numerical solution of the heat equation with nonlocal boundary conditions, J. Comput. Appl. Math., 110, 115, 10.1016/S0377-0427(99)00200-9 Sapagovas, 1987, On some boundary value problems with a nonlocal condition, Differen. Equat., 23, 858 Dehghan, 2006, Implicit collocation technique for heat equation with non-classic initial condition, Int. J. Non-Linear Sci. Numer. Simul., 7, 447, 10.1515/IJNSNS.2006.7.4.461 Dehghan, 2006, Finite difference procedures for solving a problem arising in modeling and design of certain optoecletronic devices, Math. Comput. Simul., 71, 16, 10.1016/j.matcom.2005.10.001 Dehghan, 2006, A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications, Numer. Methods Partial Differen. Equat., 22, 220, 10.1002/num.20071 Fairweather, 1991, The reformulation and numerical solution of certain nonclassical initial-boundary value problems, SIAM J. Sci. Stat. Comput., 21, 127, 10.1137/0912007 Dehghan, 2005, On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, Numer. Meth. Part. D.E., 21, 24, 10.1002/num.20019 Makarov, 1985, Solution of a boundary value problem for a quasi-linear parabolic equation with nonclassical boundary conditions, Differen. Equat., 21, 296 Yurchuk, 1986, Mixed problem with an integral condition for certain parabolic equations, Differen. Equat., 22, 1457 Dehghan, 2005, Efficient techniques for the second-order parabolic equation subject to nonlocal specifications, Appl. Numer. Math., 52, 39, 10.1016/j.apnum.2004.02.002 Wang, 1990, A numerical method for the diffusion equation with nonlocal boundary specifications, Int. J. Eng. Sci., 28, 543, 10.1016/0020-7225(90)90056-O A. Bouziani, N. Merazga, S. Benamira, Galerkin method applied to parabolic evolution problem with nonlocal boundary conditions, Nonlinear Anal.: Theory Methods Appl., in press. Buhmann, 2003 Franke, 1982, Scattered data interpolation: tests of some methods, Math. Comput., 38, 181 Dehghan, 2006, Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions, Math. Comput. Modell., 44, 1160, 10.1016/j.mcm.2006.04.003 Brown, 2005, On approximate cardinal preconditioning methods for solving PDEs with radial basis functions, Eng. Anal. Bound. Elem., 29, 343, 10.1016/j.enganabound.2004.05.006 Dehghan, 2007, The one-dimensional heat equation subject to a boundary integral specification, Chaos Solitons Fract., 32, 661, 10.1016/j.chaos.2005.11.010 Dehghan, 2005, Parameter determination in a partial differential equation from the overspecified data, Math. Comput. Model., 41, 196, 10.1016/j.mcm.2004.07.010 Dehghan, 2007, Time-splitting procedures for the solution of the two-dimensional transport equation, Kybernetes, 36, 791, 10.1108/03684920710749857 Dehghan, 2001, An inverse problem of finding a source parameter in a semilinear parabolic equation, Appl. Math. Model., 25, 743, 10.1016/S0307-904X(01)00010-5 Dehghan, 2005, Identification of a time-dependent coefficient in a partial differential equation subject to an extra measurement, Numer. Meth. Part. D.E., 21, 611, 10.1002/num.20055