×

A boundary element approach for parabolic differential equations using a class of particular solutions. (English) Zbl 0766.65075

This interesting paper introduces two alternative direct boundary element approaches to solve a class of parabolic differential equations. Both techniques avoid domain integration terms using a dual-reciprocity like procedure: radial basis functions are employed to generate particular solutions which eliminate the need for time consuming interior integrals. The techniques use finite differences in time and differ in the way in which the generalized “forcing function” is defined. Two examples \((2D)\) are discussed and the authors point out the advantages and limitations of the two approaches.

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Rizzo, F. J.; Shippy, D. J., A method of solution for certain problems of transient heat conduction, AIAA J., 8, 2004-2009 (1970) · Zbl 0237.65074
[2] Liggett, J. A.; Liu, P. L.F., Unsteady flow in confined aquifers: A comparison of two boundary integral methods, Water Resour. Res., 15, 861-866 (1979)
[3] Wrobel, L. C.; Brebbia, C. A., The boundary element method for steady-state and transient heat conduction, (Lewis, R. W.; Morgan, K., Numerical Methods in Thermal Problems, Vol. 1 (1979), Pineridge Press: Pineridge Press Swansea, UK), 58-73 · Zbl 0457.65074
[4] Taigbenu, A. E.; Liggett, J. A., Boundary element calculations for the diffusion equation, J. Eng. Mech., 111, 3, 311-328 (1985)
[5] Curran, D.; Cross, M.; Lewis, B. A., Solution of parabolic differential equations using discretization in time, Appl. Math. Modelling, 4, 398-400 (1980) · Zbl 0457.65076
[6] Ingber, M. S., An efficient boundary element method for a class of parabolic differential equations using discretization in time, Numer. Methods PDE, 3, 187-197 (1987) · Zbl 0652.65087
[7] Nardini, D.; Brebbia, C. A., A new approach to free vibration analysis using boundary elements, (Brebbia, C. A., Boundary Elements IV (1982), Springer-Verlag: Springer-Verlag Berlin), 312-348 · Zbl 0541.73104
[8] Wrobel, L. C.; Brebbia, C. A., The dual-reciprocity boundary element formulation for non-linear diffusion problems, Comput. Methods Appl. Mech. Eng., 65, 147-164 (1987) · Zbl 0612.76094
[9] Wrobel, L. C.; Brebbia, C. A., Boundary elements for non-linear head conduction problems, Commun. Appl. Numer. Methods, 4, 617-622 (1988) · Zbl 0652.65085
[10] Ahmad, S.; Banerjee, P. K., Free vibration analysis by BEM using particular integrals, J. Eng. Mech., 112, 682-695 (1986)
[11] Herry, D. P.; Banerjee, P. K., A new boundary element formulation for two- and three-dimensional thermoelasticity using particular integrals, Int. J. Numer. Methods Eng., 26, 2061-2078 (1988) · Zbl 0662.73009
[12] Herry, D. P.; Banerjee, P. K., A new boundary element formulation for two- and three-dimensional elastoplasticity using particular integrals, Int. J. Numer. Methods Eng., 26, 2079-2098 (1988)
[13] Brebbia, C. A., On the treatment of domain integrals in boundary elements, (Brebbia, C. A.; Zamani, N. G., Boundary Element Techniques: Applications in Engineering (1989), Computational Mechanics Publications: Computational Mechanics Publications Southampton, UK), 431-439
[14] Zheng, R.; Coleman, C. J.; Phan-Thien, N., A boundary element approach for non-homogeneous potential problems, Comput. Mech., 7, 4, 279-288 (1991) · Zbl 0735.73086
[15] Brebbia, C. A.; Telles, J. C.F.; Wrobel, L. C., Boundary Element Techniques (1984), Springer-Verlag: Springer-Verlag Berlin · Zbl 0556.73086
[16] Powell, M. J.D., Radial basis functions for multivariate interpolation, (Mason, J. C.; Cox., M. G., Algorithms for Approximation (1987), Clarendon Press: Clarendon Press Oxford, UK) · Zbl 0638.41001
[17] Mitra, A. K.; Ingber, M. S., Resolving the difficulties in the BIEM caused by geometric corners and discontinuous boundary conditions, (Brebbia, C. A., Boundary Elements IX (1987), Springer-Verlag: Springer-Verlag Berlin), 519-534
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.