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Numerical treatment of moving and free boundary value problems with the tau method. (English) Zbl 0999.65110

Summary: This paper reports numerical experiments on the implementation of the operational formulation of the tau method for moving and free boundary value problems. We consider problems defined by linear and nonlinear ordinary differential equations and by linear partial differential equations. We compare the accuracy attainable with the technique introduced in this paper with that of standard numerical techniques. We find that the tau method provides accurate results, even using approximations of a low degree.

MSC:

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
35R35 Free boundary problems for PDEs
80A22 Stefan problems, phase changes, etc.
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[1] Ortiz, E. L., The Tau Method, SIAM J. Numer. Anal., 480-492 (1969) · Zbl 0195.45701
[2] Ortiz, E. L.; Samara, H., An operational approach to the Tau Method for the numerical solution of nonlinear differential equations, Computing, 27, 15-25 (1981) · Zbl 0449.65053
[3] Ortiz, E. L.; Samara, H., Numerical solution of partial differential equations with variable coefficients, Computing, 10, 5-13 (1984) · Zbl 0575.65118
[4] Ortiz, E. L., On the numerical solution of nonlinear and functional differential equations with the Tau Method, (Ansorge, R.; Töring, W., Numerical Treatment of Differential Equations in Applications. Lecture Notes in Math. (1978), Springer-Verlag,: Springer-Verlag, Berlin), 127-139, No. 679 · Zbl 0387.65053
[5] Ortiz, E. L.; Pham, A., Linear recursive schemes associated with some nonlinear partial differential equations and their numerical solution with the Tau Method, SIAM Journal on Mathematical Analysis, 18, 452-464 (1987) · Zbl 0619.35017
[6] Onumanyi, P.; Ortiz, E. L., Numerical solution of stiff and singularly perturbed boundary value problems with a segmented-adaptive formulation of the Tau Method, Mathematics of Computation, 43, 189-203 (1984) · Zbl 0574.65091
[7] Freilich, J. H.; Ortiz, E. L., Numerical solution of systems of differential equations with the Tau Method: An error analysis, Mathematics of Computation, 39, 160, 467-479 (1982) · Zbl 0501.65042
[8] Marquina, A.; Martinez, V., Shooting methods for 1D steady-state free boundary problems, Computers Math. Applic., 25, 2, 39 (1992) · Zbl 0769.65057
[9] El Misiery, A. M.; Ortiz, E. L., Tau-lines: A new hybrid approach to the numerical treatment of crack problems based on the Tau Method, Comput. Meth. Appl. Mech. Engng., 56, 265-282 (1986) · Zbl 0576.73095
[10] Liu, K. M.; Ortiz, E. L., Numerical solution of eigenvalue problems for partial differential equations with the Tau-Lines Method, Computers Math. Applic., 12B, 5/6, 1153-1168 (1986) · Zbl 0626.65109
[11] Crank, J., Free and Moving Boundary Problems (1984), OUP,: OUP, Oxford · Zbl 0547.35001
[12] Hansen, E. B.; Hougaard, P., On a moving boundary problem from biomechanics, J. Inst. Math. Appl., 13, 385-398 (1974) · Zbl 0307.45016
[13] Miller, J. V.; Morton, K. W.; Baines, M. J., A finite element moving boundary computation with an adaptive mesh, J. Inst. Math. Appl., 22, 467-477 (1978) · Zbl 0394.65032
[14] Cottle, R. W., Numerical methods for complementarity problems in engineering and applied science, (Proc. Comput. Meth. Appl. Sci. Engng. Springer Lec. Notes, No. 704 (1979), Springer-Verlag,: Springer-Verlag, Berlin), 37-52
[15] Elliott, C.; Ockendon, J. R., Weak and variational methods for moving boundary problems, (Research Notes in Mathematics, No. 59 (1982), Pitman,: Pitman, London) · Zbl 0476.35080
[16] Crank, J.; Gupta, R. S., A moving boundary problem arising from the difussion of oxygen in absorbing tissue, J. Inst. Math. Appl., 10, 19-33 (1972) · Zbl 0247.65064
[17] Gupta, R. S., Ph.D. Thesis (1973), Brunel University
[18] Gupta, R. S., Moving grid methods without interpolations, Comput. Meth. Appl. Mech. Engng., 4, 143-152 (1974) · Zbl 0284.76072
[19] Murray, W. D.; Landis, F., Numerical and machine solutions of transient heat-condution problems involving melting or freezing. Part 1—Method of analysis and sample solutions, J. Heat Transfer, 81, 2, 106-112 (1959)
[20] Gupta, R. S.; Kumar, D., A modified variable time step method for the one-dimensional Stefan problem, Comput. Meth. Appl. Mech. Engng., 23, 101-109 (1980) · Zbl 0446.76070
[21] Gupta, R. S.; Kumar, D., Complete numerical solution of the oxygen diffusion problem involving a moving boundary, Comput. Meth. Appl. Mech. Engng., 29, 233-239 (1981) · Zbl 0469.65087
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