Numerical and asymptotic analysis of a localized heat source undergoing periodic motion. (English) Zbl 1239.45003
Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 12, e-Suppl., e2168-e2172 (2009).
Summary: A highly localized heat source moves with simple periodic motion along a one-dimensional medium with reactive-diffusive properties. It is known that the system will experience a blow-up in finite time. Numerical results suggest that this blow-up, though unavoidable, can be delayed by increasing either the amplitude or the frequency of the motion. These numerical results are compared to known analytical results. Numerical and analytical results are also compared for the related case in which the heat source moves at a constant speed in one direction. The asymptotic behavior of the temperature of the material at the location of the heat source near the time of blow-up is also analyzed. It can be of interest to consider the asymptotic behavior of the solution in the context of the numerical solution in order to gain confidence in the numerical results.
MSC:
| 45D05 | Volterra integral equations |
| 65R20 | Numerical methods for integral equations |
| 80M35 | Asymptotic analysis for problems in thermodynamics and heat transfer |
| 80A20 | Heat and mass transfer, heat flow (MSC2010) |
Keywords:
nonlinear Volterra integral equations; blow-up; moving source; numerical analysis; asymptotic analysisReferences:
| [1] | C.M. Kirk, A localized heat source undergoing periodic motion: analysis of blow-up and a numerical solution, Cubo, Math. J. (in press); C.M. Kirk, A localized heat source undergoing periodic motion: analysis of blow-up and a numerical solution, Cubo, Math. J. (in press) · Zbl 1179.65163 |
| [2] | Kirk, C. M.; Olmstead, W. E., Blow-up in a reactive-diffusive medium with a moving heat source, Z. Angew. Math. Phys., 53, 147-159 (2002) · Zbl 0994.35068 |
| [3] | Roberts, C. A., Recent results on blow-up and quenching for nonlinear Volterra equations, J. Comput. Appl. Math., 205, 736-743 (2007) · Zbl 1114.35096 |
| [4] | Roberts, C. A., Analysis of explosion for nonlinear Volterra integral equations, J. Comp. Appl. Math., 97, 153-166 (1998) · Zbl 0932.45007 |
| [5] | Baker, C. T.H., A perspective on the numerical treatment of Volterra Equations, J. Comput. Appl. Math., 125, 217-249 (2000) · Zbl 0976.65121 |
| [6] | Bandle, C.; Brunner, H., Numerical analysis of semilinear parabolic problems with blow-up solutions, Rev. Real Acad. Cienc. Exact. Fís. Natur. Madrid, 88, 203-222 (1994) · Zbl 0862.65053 |
| [7] | Bandle, C.; Brunner, H., Blow-up in diffusion equations: A survey, J. Comp. Appl. Math., 97, 3-22 (1998) · Zbl 0932.65098 |
| [8] | Brunner, H., Collocation methods for Volterra integral and related functional differential equations (2004), Cambridge University Press · Zbl 1059.65122 |
| [9] | Chan, C. Y.; Tian, H. Y., Single-point blow-up for a degenerate parabolic problem due to a concentrated nonlinear source, Quart. Appl. Math., 61, 363-385 (2003) · Zbl 1032.35105 |
| [10] | Chan, C. Y.; Tian, H. Y., Single-point blow-up for a degenerate parabolic problem with a nonlinear source of local and nonlocal features, Appl. Math. Comput., 145, 371-390 (2003) · Zbl 1038.35033 |
| [11] | Glenn Lasseigne, D.; Olmstead, W. E., Ignition of a combustible solid with reactant consumption, SIAM J. Appl. Math., 47, 332-342 (1987) · Zbl 0622.45005 |
| [12] | Ritter, L. R.; Olmstead, W. E.; Volpert, V. A., Initiation of free radical polymerization waves, SIAM J. Appl. Math., 63, 1831-1848 (2003) · Zbl 1037.35090 |
| [13] | Roberts, C. A.; Olmstead, W. E., Growth rates for blow-up solutions of nonlinear Volterra equations, Quart. Appl. Math., 153-159 (1996) · Zbl 0916.45007 |
| [14] | Roberts, C. A., Characterizing the blow-up solutions for nonlinear Volterra integral equations, Proc. 2nd World Congress of Nonlinear Analysts. Proc. 2nd World Congress of Nonlinear Analysts, Nonlinear analysis, TMA, 30, 923-933 (1997) · Zbl 0891.45003 |
| [15] | Roberts, C. A.; Lasseigne, D. G.; Olmstead, W. E., Volterra equations which model explosion in a diffusive medium, J. Integral Equations Appl., 5, 531-546 (1993) · Zbl 0804.45002 |
| [16] | Kirk, C. M.; Olmstead, W. E., Blow-up solutions of the two-dimensional heat equation due to a localized moving source, Anal. Appl., 3, 1-16 (2005) · Zbl 1086.35006 |
| [17] | Olmstead, W. E., Critical speed for the avoidance of blow-up in a reactive-diffusive medium, Z. Angew. Math. Phys., 48, 701-710 (1997) · Zbl 0884.35067 |
| [18] | N. Bleistein, R.A. Handelsman, Asymptotic expansion of integrals, Holt, Rinehardt, and Winston, New York, 1975; N. Bleistein, R.A. Handelsman, Asymptotic expansion of integrals, Holt, Rinehardt, and Winston, New York, 1975 · Zbl 0327.41027 |
| [19] | Brunner, H., The numerical analysis of functional integral and integro-differential equations of Volterra type, Acta Numer., 13, 55-145 (2004) |
| [20] | Diogo, T.; Lima, P., Collocation solutions of a weakly singular Volterra integral equation, TEMA Tend. Mat. Apl. Comput., 8, 229-238 (2007) · Zbl 1208.65184 |
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