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Analytic solutions of initial-boundary-value problems of transient conduction using symmetries. (English) Zbl 1187.35005

Summary: Lie symmetry method is applied to find analytic solutions of initial-boundary-value problems of transient conduction in semi-infinite solid with constant surface temperature or constant heat flux condition. The solutions are obtained in a manner highlighting the systematic procedure of extending the symmetry method for a PDE to investigate BVPs of the PDE. A comparative analysis of numerical and closed form solutions is carried out for a physical problem of heat conduction in a semi-infinite solid bar made of AISI 304 stainless steel.

MSC:

35B06 Symmetries, invariants, etc. in context of PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
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[1] Ames, W. F., Nonlinear Partial Differential Equations in Engineering, vols. I and II (1965-1972), Academic Press: Academic Press New York · Zbl 0255.35001
[2] Baumann, G., Symmetry Analysis of Differential Equations with Mathematica (2000), Springer-Verlag: Springer-Verlag New York · Zbl 0898.34003
[3] Bluman, G. W., Applications of the general similarity solution of the heat equation to boundary-value problems, Quart. Appl. Math., 31, 403 (1974) · Zbl 0276.35053
[4] Bluman, G. W.; Anco, S. C., Symmetry and Integration Methods for Differential Equations (2002), Springer-Verlag: Springer-Verlag New York · Zbl 1013.34004
[5] Bluman, G. W.; Cole, J. D., Similarity Methods for Differential Equations (1974), Springer-Verlag: Springer-Verlag New York · Zbl 0292.35001
[6] Bluman, G. W.; Kumei, S., Symmetries and Differential Equations (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0718.35004
[7] Cantwell, Brian. J., Introduction to Symmetry Analysis (2002), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1082.34001
[9] Euler, N.; Steeb, W. H., Continuous Symmetries, Lie Algebras and Differential Equations (1992), Bibliographisches Institut: Bibliographisches Institut Mannheim · Zbl 0755.35112
[10] Hansen, A. G., Similarity Analyses of Boundary Value Problems in Engineering (1964), Prentice Hall: Prentice Hall Englewood Cliffs · Zbl 0137.22603
[11] Hydon, P. E., Symmetry Methods for Differential Equations (2000), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0991.39005
[12] Ibragimov, N. H., Group analysis of ordinary differential equations and the invariance principle in mathematical physics (to the 150th anniversary of Sophus Lie), Usp. Matem. Nauk, 47, 83 (1992)
[13] (Ibragimov, N. H., CRC handbook of Lie group analysis of differential equations. CRC handbook of Lie group analysis of differential equations, Symmetries, Exact Solutions and Conservation Laws, vol. 1 (1994), CRC Press: CRC Press Boca Raton) · Zbl 0864.35001
[14] (Ibragimov, N. H., CRC handbook of Lie group analysis of differential equations. CRC handbook of Lie group analysis of differential equations, Applications in Engineering and Physical Sciences, vol. 2 (1995), CRC Press: CRC Press Boca Raton) · Zbl 0864.35002
[15] (Ibragimov, N. H., CRC handbook of Lie group analysis of differential equations. CRC handbook of Lie group analysis of differential equations, New Trends in Theoretical Developments and Computational Methods, vol. 3 (1996), CRC Press: CRC Press Boca Raton) · Zbl 0864.35003
[16] Ibragimov, N. H., Elementary Lie Group Analysis and Ordinary Differential Equations (1999), John Wiley & Sons: John Wiley & Sons Chichester · Zbl 1047.34001
[18] Miller, W., Symmetry and Separation of Variables (1977), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0368.35002
[19] Olver, P. J., Applications of Lie Groups to Differential Equations (1986), Springer-Verlag: Springer-Verlag New York · Zbl 0656.58039
[20] Ovsiannikov, L. V., Group Analysis of Differential Equations (1982), Academic Press: Academic Press New York · Zbl 0485.58002
[21] Polyanin, A. D.; Zaitsev, V. F., Handbook of Nonlinear Partial Differential Equations (2004), CRC Press: CRC Press Boca Raton - London · Zbl 1024.35001
[22] Stephani, H., Differential Equations. Their Solution using Symmetries (1989), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0704.34001
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