## On the solution of the non-local parabolic partial differential equations via radial basis functions.(English)Zbl 1168.65403

Summary: The problem of solving the one-dimensional parabolic partial differential equation subject to given initial and non-local boundary conditions is considered. The approximate solution is found using the radial basis functions collocation method. There are some difficulties in computing the solution of the time dependent partial differential equations using radial basis functions. If time and space are discretized using radial basis functions, the resulted coefficient matrix will be very ill-conditioned and so the corresponding linear system cannot be solved easily. As an alternative method for solution, we can use finite-difference methods for discretization of time and radial basis functions for discretization of space. Although this method is easy to use but an accurate solution cannot be provided. In this work an efficient collocation method is proposed for solving non-local parabolic partial differential equations using radial basis functions. Numerical results are presented and are compared with some existing methods.

### MSC:

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

 [1] 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 [2] Sarra, S.A., Adaptive radial basis function method for time dependent partial differential equations, Appl. numer. math., 54, 79-94, (2005) · Zbl 1069.65109 [3] Shu, C.; Ding, H.; Yeo, K.S., Solution of partial differential equations by a global radial basis function-based differential quadrature method, Eng. anal. bound. elem., 28, 1217-1226, (2004) · Zbl 1081.65546 [4] Sarler, B.; Vertnik, R., Meshfree explicit local radial basis function collocation method for diffusion problems, Comput. math. appl., 51, 1269-1282, (2006) · Zbl 1168.41003 [5] Vertnik, R.; Sarler, B., Meshless local radial basis function collocation method for convective – diffusive solid – liquid phase change problems, Int. J. numer. methods heat fluid flow, 16, 617-640, (2006) · Zbl 1121.80014 [6] Cannon, J.R., The solution of the heat equation subject to the specification of energy, Quart. appl. math., 21, 155-160, (1963) · Zbl 0173.38404 [7] Day, W.A., Parabolic equations and thermodynamics, Quart. appl. math., 50, 523-533, (1992) · Zbl 0794.35069 [8] Bouziani, A., Mixed problem with boundary integral conditions for a certain parabolic equation, J. appl. math. stoch. anal., 9, 323-330, (1996) · Zbl 0864.35049 [9] Bouziani, A., On a class of parabolic equations with a nonlocal boundary condition, Acad. roy. belg. bull. cl. sci., 10, 61-77, (1999) · Zbl 1194.35200 [10] Ionkin, N.I., Stability of a problem in heat transfer theory with a non-classical boundary condition, Differen. equat., 15, 911-914, (1980) · Zbl 0431.35045 [11] Sun, Z.Z., A second-order accurate finite difference scheme for a class of nonlocal parabolic equations with natural boundary conditions, J. comput. appl. math., 76, 137-146, (1996) · Zbl 0873.65129 [12] Ang, W.T., A method of solution for the one-dimensional heat equation subject to a nonlocal condition, SEA bull. math., 26, 197-203, (2002) · Zbl 1032.35073 [13] Cahlon, B.; Kulkrani, D.M.; Shi, P., Stepwise stability for the heat equation with a nonlocal constraint, SIAM J. numer. anal., 32, 571-593, (1995) · Zbl 0831.65094 [14] Cannon, J.R., The one dimensional heat equation, Encyclopedia of mathematics and its applications, vol. 23, (1984), Addison-Wesley Men 10 Park, CA [15] Cannon, J.R.; Matheson, A.L., A numerical procedure for diffusion subject to the specification of mass, Int. J. eng. sci., 31, 347-355, (1993) · Zbl 0773.65069 [16] Cannon, J.R.; Prez-Esteva, S.; van der Hoek, J.A., A Galerkin procedure for the diffusion equation subject to the specification of mass, SIAM J. numer. anal., 24, 499-515, (1987) · Zbl 0677.65108 [17] Cannon, J.R.; van der Hoek, J., Implicit finite difference scheme for the diffusion of mass in porous media, (), 527-539 [18] Cannon, J.R.; van der Hoek, J., Diffusion subject to specification of mass, J. math. anal. appl., 115, 517-529, (1986) · Zbl 0602.35048 [19] Cannon, J.R.; Lin, Y.; Wang, S., An implicit finite difference scheme for the diffusion equation subject to the specification of mass, Int. J. eng. sci., 28, 573-578, (1990) · Zbl 0721.65046 [20] Cannon, J.R.; Lin, Y., A Galerkin procedure for diffusion equations with boundary integral conditions, Int. J. eng. sci., 28, 579-587, (1990) · Zbl 0721.65054 [21] Cannon, J.R.; Yin, H.M., On a class of non-classical parabolic problems, Differen. equat., 79, 266-288, (1989) · Zbl 0702.35120 [22] Deckert, K.L.; Maple, C.G., Solutions for diffusion equations with integral type boundary conditions, Proc. iowa acad. sci., 70, 345-361, (1963) · Zbl 0173.12803 [23] Ionkin, N.T.; Furleov, D.G., Uniform stability of difference schemes for a nonlocal nonself-adjoint boundary value problem with variable coefficients, Differen. equat., 27, 820-826, (1991) · Zbl 0818.65081 [24] Kacur, J.; van Keer, R., 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, 91-103, (1995) · Zbl 0823.65088 [25] Liu, Y., Numerical solution of the heat equation with nonlocal boundary conditions, J. comput. appl. math., 110, 115-127, (1999) · Zbl 0936.65096 [26] Sapagovas, M.; Chegis, R.Yu., On some boundary value problems with a nonlocal condition, Differen. equat., 23, 858-863, (1987) · Zbl 0641.34014 [27] Dehghan, M., Implicit collocation technique for heat equation with non-classic initial condition, Int. J. non-linear sci. numer. simul., 7, 447-450, (2006) [28] Dehghan, M., Finite difference procedures for solving a problem arising in modeling and design of certain optoecletronic devices, Math. comput. simul., 71, 16-30, (2006) · Zbl 1089.65085 [29] Dehghan, M., A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications, Numer. methods partial differen. equat., 22, 220-257, (2006) · Zbl 1084.65099 [30] Fairweather, G.; Saylor, RD., The reformulation and numerical solution of certain nonclassical initial-boundary value problems, SIAM J. sci. stat. comput., 21, 127-144, (1991) · Zbl 0722.65062 [31] Dehghan, M., 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, 24-40, (2005) · Zbl 1059.65072 [32] Makarov, V.L.; Kulyev, D.T., Solution of a boundary value problem for a quasi-linear parabolic equation with nonclassical boundary conditions, Differen. equat., 21, 296-305, (1985) · Zbl 0573.35048 [33] Yurchuk, N.I., Mixed problem with an integral condition for certain parabolic equations, Differen. equat., 22, 1457-1463, (1986) · Zbl 0654.35041 [34] Dehghan, M., Efficient techniques for the second-order parabolic equation subject to nonlocal specifications, Appl. numer. math., 52, 39-62, (2005) · Zbl 1063.65079 [35] Wang, S.; Lin, Y., A numerical method for the diffusion equation with nonlocal boundary specifications, Int. J. eng. sci., 28, 543-546, (1990) · Zbl 0718.76096 [36] A. Bouziani, N. Merazga, S. Benamira, Galerkin method applied to parabolic evolution problem with nonlocal boundary conditions, Nonlinear Anal.: Theory Methods Appl., in press. · Zbl 1155.35053 [37] Buhmann, M.D., Radial basis functions, (2003), Cambridge University Press Cambridge · Zbl 1004.65015 [38] Franke, R., Scattered data interpolation: tests of some methods, Math. comput., 38, 181-200, (1982) · Zbl 0476.65005 [39] Dehghan, M.; Tatari, M., Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions, Math. comput. modell., 44, 1160-1168, (2006) · Zbl 1137.65408 [40] Brown, D.; Ling, L.; Kansa, E.; Levesley, J., On approximate cardinal preconditioning methods for solving PDEs with radial basis functions, Eng. anal. bound. elem., 29, 343-353, (2005) · Zbl 1182.65174 [41] Dehghan, M., The one-dimensional heat equation subject to a boundary integral specification, Chaos solitons fract., 32, 661-675, (2007) · Zbl 1139.35352 [42] Dehghan, M., Parameter determination in a partial differential equation from the overspecified data, Math. comput. model., 41, 196-213, (2005) · Zbl 1080.35174 [43] Dehghan, M., Time-splitting procedures for the solution of the two-dimensional transport equation, Kybernetes, 36, 791-805, (2007) · Zbl 1193.93013 [44] Dehghan, M., An inverse problem of finding a source parameter in a semilinear parabolic equation, Appl. math. model., 25, 743-754, (2001) · Zbl 0995.65098 [45] Dehghan, M., Identification of a time-dependent coefficient in a partial differential equation subject to an extra measurement, Numer. meth. part. D.E., 21, 611-622, (2005) · Zbl 1069.65104
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