Invariant imbedding and hyperbolic heat waves

Journal of Mathematical Physics

Volumn 38, Issue 3, 1997, Pages 1723-1749

  Wall, David J N ^a   Olsson, Peter ^b  

^a UNIVERSITY OF CANTERBURY   (New Zealand)

^b CHALMERS UNIVERSITY OF TECHNOLOGY   (Sweden)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031502257     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.531825     Document Type: Article

References (27)

1. C. Cattaneo, "Sulla conduzione de calore," Atti Semin. Mat. Fis. Univ. Modena, 3, 3 (1948).
2. J. O. Powell, "Two coefficient reconstruction via a trace formulae in a one-dimensional inverse scattering problem," Inv. Prob. 11, 275 (1995).
3. C. R. Vogel, "Wave splitting for some nonhyperbolic time-dependent PDEs," in Ref. 13, Chap. 10, p. 129-138.
4. D. D. Joseph and L. Preziosi, "Heat waves," Rev. Mod. Phys. 61, 41-73 (1989).
5. D. D, Joseph and L. Preziosi, "Addendum to the paper: Heat waves," Rev. Mod. Phys. 62, 375-391 (1990).
6. -9 m for metallic conductors at room temperature.
7. K. Mitra, S. Kumar, A. Vedavarz, and M. K. Moallemi, "Experimental evidence of hyperbolic heat conduction in processed meat," J. Heat Transfer 117, 568-573 (1995).
8. l is the heat flux in the lateral direction, say y, the problem is no longer one-dimensional. Of course no heat problem can be truly one-dimensional, however if X is very small the problem can be approximately treated as if it were.
9. I. Åberg, G. Kristensson, and D. J. N. Wall, "Propagation of transient electromagnetic waves in time-varying media -Direct and inverse scattering problems," Inv. Prob. 11, 29-49 (1995).
10. For convenience it is expedient for us to define diffusivity K as the square of the usual terminology.
11. S. He, "Factorization of a dissipative wave equation and the Green function technique for axially symmetric fields in a stratified slab," J. Math. Phys. 33, 953-966 (1992).
12. V. H. Weston, "Factorisation of the dissipative wave equation and inverse scattering," J. Math. Phys. 29, 2205-2218 (1988).
13. Invariant Imbedding and Inverse Problems, edited by J. P. Corones, G. Kristensson, P. Nelson, and D. L. Seth (SIAM, Philadelphia, 1992).
14. A. P. Prudnikov, Yu. A. Brychkov, and O. I. L. Marichev, Integrals and Series (Gordon and Breach, New York, 1986).
15. M. E. Taylor, Pseudodifferential Operators, Princeton Mathematical Series No. 34 (Princeton University Press, Princeton, NJ, 1981).
16. K. R. Oldham and J. Spanier, The Fractional Calculus (Academic, New York, 1974).
17. I. Åberg, G. Kristensson, and D. J. N. Wall, "Transient waves in non-stationary media," J. Math. Phys. 37, 2229-2252 (1996).
18. R. J. Krueger and R. L. Ochs, "A Green's function approach to the determination of internal fields," Wave Motion 11, 525-543 (1989).
19. G. Kristensson and R. J. Krueger, "Direct and inverse scattering in the time domain for a dissipative wave equation. Part 3: Scattering operators in the presence of a phase velocity mismatch," J. Math. Phys. 28, 360-370 (1987).
20. Because of previous assumptions only γ has a jump at x = l.
21. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1.
22. J. Lundstedt and S. He, "Time-domain direct and inverse problems for a nonuniform LCRG line with internal sources," IEEE Trans. Electromagn. Compatibility (in press).
23. There should not be any confusion with the symbol for the characteristic curves and that for the relaxation time, τ(x), as the former has a superscript and three arguments.
24. The end boundary conditions for the clamped spline are found by using cubic interpolants at each of the endpoints x=0 and x=l, and then using the first derivative of these interpolants to estimate the end derivatives of ζ̃.
25. 2), t>0. Note the lack of dependence on τ, this has the effect of making the boundary condition more slowly varying with respect to time when the relaxation time is smaller.
26. G. B. Nagy, O. E. Oritz, and O. A. Reula, "The behaviour of hyperbolic heat equations' solutions near their parabolic limits," J. Math. Phys, 8, 4334-4356 (1994).
27. D. J. N. Wall, "On the numerical solution of a functional differential equation pertaining to a wave equation," in Ref. 13, Chap. 13, pp. 169-186.