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Time-dependent fundamental solutions for homogeneous diffusion problems. (English) Zbl 1098.76622
Summary: This paper describes applications of the method of fundamental solutions (MFS) to 1-, 2- and 3-D diffusion equations. The time-dependent fundamental solutions for diffusion equations are used directly to obtain the solution as a linear combination of fundamental solutions of diffusion operator. The proposed scheme is free from conventionally used Laplace transform or from finite difference schemes to deal with the time derivative of the governing equation. By properly placing the field points and source points at a given time level, the solution is advanced in time until steady state solutions are reached. Test results obtained for 1-, 2- and 3-D diffusion problems show good comparisons with analytical solutions and MFS-based modified Helmholtz fundamental solutions. Thus the presented MFS with space-time unification has been demonstrated as a promising mesh-free numerical tool to solve homogeneous diffusion problem.

##### MSC:
 76R50 Diffusion 76M25 Other numerical methods (fluid mechanics) (MSC2010)
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##### References:
 [1] Zhu, S.P., Solving transient diffusion problems: time-dependent fundamental solution approaches versus LTDRM approaches, Eng anal bound elem, 21, 87-90, (1998) · Zbl 0979.76525 [2] Zhu, S.P.; Liu, H.W.; Lu, X.P., A combination of LTDRM and ATPS in solving diffusion problems, Eng anal bound elem, 21, 285-289, (1998) · Zbl 0941.80008 [3] Bulgakov, V.; Sarler, B.; Kuhn, G., Iterative solution of systems of equations in the dual reciprocity boundary element method for the diffusion equation, Int J numer meth eng, 43, 713-732, (1998) · Zbl 0948.76050 [4] Zerroukat, M., A boundary element scheme for diffusion problems using compactly supported radial basis functions, Eng anal bound elem, 23, 201-209, (1999) · Zbl 0968.76568 [5] Sutradhar, A.; Paulino, G.H.; Gray, L.J., Transient heat conduction in homogeneous and non-homogeneous materials by the Laplace transform Galerkin boundary element method, Eng anal bound elem, 26, 119-132, (2002) · Zbl 0995.80010 [6] Bialecki, R.A.; Jurgas, P.; Kuhn, G., Dual reciprocity BEM without matrix inversion for transient heat conduction, Eng anal bound elem, 26, 227-236, (2002) · Zbl 1002.80019 [7] Nardini, D.; Brebbia, C.A., A new approach to free vibration analysis using boundary elements, () · Zbl 0541.73104 [8] Tsai CC, Meshless numerical methods and their engineering applications. PhD Thesis. Department of Civil Engineering, National Taiwan University, Taipei, Taiwan; 2002. [9] Golberg, M.A., The method of fundamental solutions for Poisson’s equation, Eng anal bound elem, 16, 205-213, (1995) [10] Golberg, M.A.; Chen, C.S., The method of fundamental solutions for potential, Helmholtz and diffusion problems, () · Zbl 0945.65130 [11] Fairweather, G.; Karageoghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv comput math, 9, 69-95, (1998) · Zbl 0922.65074 [12] Poullikkas, A.; Karageorghis, A.; Grorgiou, G., Methods of fundamental solutions for harmonic and biharmonic boundary value problems, Comput mech, 21, 416-423, (1998) · Zbl 0913.65104 [13] Poullikkas, A.; Karageorghis, A.; Grorgiou, G., The method of fundamental solutions for inhomogeneous elliptic problems, Comput mech, 22, 100-107, (1998) · Zbl 0913.65103 [14] Chen, C.S.; Golberg, M.A.; Hon, Y.C., The method of fundamental solutions and quasi-Monte-Carlo method for diffusion equations, Int J numer meth eng, 43, 1421-1435, (1998) · Zbl 0929.76098 [15] Balakrishnan, K.; Ramachandran, P.A., The method of fundamental solutions for linear diffusion-reaction equations, Math comput model, 31, 2/3, 221-237, (2000) · Zbl 1042.35569 [16] Walker, S.P., Diffusion problems using transient discrete source superposition, Int J numer meth eng, 35, 165-178, (1992) · Zbl 0764.76044 [17] Fairweather, G.; Karageorghis, A.; Martin, P.A., The method of fundamental solutions for scattering and radiation problems, Eng anal bound elem, 27, 759-769, (2003) · Zbl 1060.76649
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