Explicit Solution for a non classical heat conduction problem for a semi-infinite strip with non-uniform heat source

D. TARZIA, A. CERETANI; LUIS T. VILLA

BOUNDARY VALUE PROBLEMS

HINDAWI PUBLISHING CORPORATION

Lugar: New York; Año: 2015

1687-2762

A non-classical initial and boundary value problem for a non-homogeneous one- dimensionalheat equation for a semi-infinite material x > 0 with a zero temperature boundarycondition at the face x = 0 is studied with the aim of finding explicit solutions. It is not astandard heat conduction problem because a heat source −Φ(x)F(V (t); t) is considered,where Φ and F are real functions and V represents the heat flux at the face x = 0.Explicit solutions independents of the space or temporal variables are given. Solutionswith separated variables when the data functions are defined from the solution X = X(x)of a linear initial value problem of second order and the solution T = T(t) of a non-linear(in general) initial value problem of first order which involves the function F, are alsogiven and explicit solutions corresponding to different definitions of the function F areobtained. A solution by an integral representation depending on the heat flux at theboundary x = 0 for the case in which F = F(V; t) = V , for some > 0, is obtainedand explicit expressions for the heat flux at the boundary x = 0 and for its correspondingsolution are calculated when h = h(x) is a potential function and Φ = Φ(x) is given byΦ(x) = x, Φ(x) = − sinh (x) or Φ(x) = − sin (x), for some > 0 and > 0.The limit when the temporal variable t tends to +∞ of each explicit solution obtainedin this paper is studied and the ?controlling? effects of the source term −ΦF are analysedby comparing the asymptotic behaviour of each solution with the asymptotic behaviourof the solution to the same problem but in absence of source term.Finally, a relationship between this problem with another non-classical initial andboundary value problem for the heat equation is established and explicit solutions for thissecond problem are also obtained.As a consequence of our study, several problems which can be used as benchmarkproblems for testing new numerical methods for solving partial differential equations areobtained.