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Analytic-numerical solutions of diffusion mathematical models with delays. (English) Zbl 1155.65387

Summary: This work deals with the construction of analytic-numerical solutions of mixed problems for diffusion and reaction-diffusion equations with delay. Using the method of separation of variables, exact theoretical infinite series solutions are derived. Bounds on the truncation errors, when these series are approximated by finite sums, are given, thus providing constructive continuous numerical solutions with prescribed accuracy in bounded domains. In order to improve the computational efficiency of these numerical solutions, polynomial approximations to the initial functions are also considered.

MSC:

65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
35K57 Reaction-diffusion equations
35C10 Series solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
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[1] Kolmanovskii, V.; Myshkis, A., Applied theory of functional differential equations, (1992), Kluwer Academic Publishers Dordrecht · Zbl 0917.34001
[2] Kolmanovskii, V.; Myshkis, A., Introduction to the theory and applications of functional differential equations, (1999), Kluwer Academic Publishers Dordrecht · Zbl 0917.34001
[3] Wu, J., Theory and applications of partial functional differential equations, (1996), Springer-Verlag New York
[4] Cattaneo, C., Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée, C. R. acad. sci., 247, 431-433, (1958) · Zbl 1339.35135
[5] Vernotte, P., LES paradoxes de la théorie continue de l’équation de la chaleur, C. R. acad. sci., 246, 3154-3155, (1958) · Zbl 1341.35086
[6] Vernotte, P.; Vernotte, P., Some possible complications in the phenomena of thermal conduction, C. R. acad. sci., 252, 2190-2191, (1961)
[7] Joseph, D.D.; Preziosi, L., Heat waves, Rev. modern phys., 61, 41-73, (1989) · Zbl 1129.80300
[8] Tzou, D.Y., Experimental support for the lagging behavior in heat propagation, AIAA J. termophys. heat transfer, 9, 686-693, (1995)
[9] Qiu, T.Q.; Tien, C.L., Short-pulse laser heating on metals, Int. J. heat mass transfer, 35, 719-726, (1992)
[10] Qiu, T.Q.; Tien, C.L., Heat transfer mechanisms during short-pulse laser heating of metals, ASME J. heat transfer, 115, 835-841, (1993)
[11] Bertman, B.; Sandiford, D.J., Second sound in solid helium, Sci. amer., 222, 92, (1970)
[12] Tzou, D.Y., The generalized lagging response in small-scale and high-rate heating, Int. J. heat mass transfer, 38, 3231-3240, (1995)
[13] Tzou, D.Y., Macro- to microscale heat transfer: the lagging behavior, (1996), Taylor & Francis Washington
[14] Kulish, V.V.; Novozhilov, V.B., An integral equation for the dual-lag model of heat transfer, ASME J. heat transfer, 126, 805-808, (2004)
[15] Xu, M.; Wang, L., Dual-phase-lagging heat conduction based on Boltzmann transport equation, Int. J. heat mass transfer, 48, 5616-5624, (2005) · Zbl 1188.76242
[16] Kuang, Y., Delay differential equations. with applications in population dynamics, (1993), Academic Press San Diego · Zbl 0777.34002
[17] Kot, M., Elements of mathematical ecology, (2001), Cambridge University Press Cambridge
[18] Turyn, L., A partial functional differential equation, J. math. anal appl., 263, 1-13, (2001) · Zbl 1064.35203
[19] Murray, J.D., Mathematical biology. I: an introduction, (2002), Springer-Verlag New York
[20] Farlow, S.J., Partial differential equations for scientists and engineers, (1993), Dover New York · Zbl 0851.35001
[21] Folland, G.B., Fourier analysis and its applications, (1992), Wadsworth & Brooks Pacific Groove · Zbl 0371.35008
[22] Scott, E.J., On a class of linear partial differential equations with retarded argument in time, Bul. inst. Pol. din. iasi., 15, 99-103, (1969) · Zbl 0189.10501
[23] Wiener, J.; Debnath, L., Boundary value problems for the diffusion equation with piecewise continuous time delay, Internat. J. math. math. sci., 20, 187-195, (1997) · Zbl 0879.35065
[24] Martín, J.A.; Rodríguez, F.; Company, R., Analytic solution of mixed problems for the generalized diffusion equation with delay, Math. comput. modelling, 40, 361-369, (2004) · Zbl 1062.35157
[25] Bellman, R.; Cooke, K.L., Differential-difference equations, (1963), Academic Press New York · Zbl 0115.30102
[26] El’sgol’ts, L.E.; Norkin, S.B., Introduction to the theory and application of differential equations with deviating arguments, (1973), Academic Press New York · Zbl 0287.34073
[27] ()
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