On a non-linear moving boundary problem for a diffusion–convection equation

We study a one-dimensional free boundary problem for a non-linear diffusion–convection equation whose diffusivity is heterogeneous in space as well as being non-linear. Under the Bäcklund transformation the problem is reduced to an associated free boundary problem. We prove the existence and uniqueness, local in time, of the solution by using the Friedman Rubinstein integral representation method and the Banach contraction theorem.     

Exact and asymptotic properties of multistate random walks

A method is presented which allows one to obtain explicit analytical expressions (both exact and asymptotic) for many of the physically interesting quantities related to a multistate random walk (MRW). The exact results include the Laplace-Fourier-transformed probability distribution (continuous time) and generating function (discrete time), and closed evolution equations for the propagators related to each internal state of the walker. Analytical expressions for the scattering dynamical structure function and the frequency-dependent diffusion coefficient are given as illustrations. Asymptotic approximations to the single-state propagators are derived, allowing a detailed analysis of the longtime behavior and the calculation of asymptotic properties by single-state random walk standard methods. As an example, analytical expressions for the drift and diffusion coefficients are given.One of the authors (M.O.C.) wants to thank the dean and research staff of the Facultad de Matemática, Astronomia y Física for their warm hospitality during his stay in Cordoba.     

A Stefan problem for a non-classical heat equation with a convective condition

We prove the existence and uniqueness, local in time, of the solution of a one-phase Stefan problem for a non-classical heat equation for a semi-infinite material with a convective boundary condition at the fixed face x = 0. Here the heat source depends on the temperature at the fixed face x = 0 that provides a heating or cooling effect depending on the properties of the source term. We use the Friedman-Rubinstein integral representation method and the Banach contraction theorem in order to solve an equivalent system of two Volterra integral equations. We also obtain a comparison result of the solution (the temperature and the free boundary) with respect to the one corresponding with null source term.     

Nonlinear Stefan problem with convective boundary condition in Storm’s materials

We consider a nonlinear one-dimensional Stefan problem for a semi-infinite material x > 0, with phase change temperature T_f. We assume that the heat capacity and the thermal conductivity satisfy a Storm’s condition, and we assume a convective boundary condition at the fixed face x = 0. A unique explicit solution of similarity type is obtained. Moreover, asymptotic behavior of the solution when \({h\rightarrow + \infty}\) is studied.     

Explicit solutions for a two-phase unidimensional Lamé-Clapeyron-Stefan problem with source terms in both phases

A two-phase Stefan problem with heat source terms of a general similarity type in both liquid and solid phases for a semi-infinite phase-change material is studied. We assume the initial temperature is a negative constant and we consider two different boundary conditions at the fixed face x=0, a constant temperature or a heat flux of the form (q₀>0). The internal heat source functions are given by (j=1 solid phase; j=2 liquid phase) where βj=βj(η) are functions with appropriate regularity properties, ρ is the mass density, l is the fusion latent heat by unit of mass, is the diffusion coefficient, x is the spatial variable and t is the temporal variable. We obtain for both problems explicit solutions with a restriction for data only for the second boundary conditions on x=0. Moreover, the equivalence of the two free boundary problems is also proved. We generalize the solution obtained in [J.L. Menaldi, D.A. Tarzia, Generalized Lamé-Clapeyron solution for a one-phase source Stefan problem, Comput. Appl. Math. 12 (2) (1993) 123-142] for the one-phase Stefan problem. Finally, a particular case where βj (j=1,2) are of exponential type given by βj(x)=exp(−²(x+dj)) with x and dj∈R is also studied in details for both boundary temperature conditions at x=0. This type of heat source terms is important through the use of microwave energy following [E.P. Scott, An analytical solution and sensitivity study of sublimation-dehydration within a porous medium with volumetric heating, J. Heat Transfer 116 (1994) 686-693]. We obtain a unique solution of the similarity type for any data when a temperature boundary condition at the fixed face x=0 is considered; a similar result is obtained for a heat flux condition imposed on x=0 if an inequality for parameter q₀ is satisfied.     

On a two-phase Stefan problem with convective boundary condition including a density jump at the free boundary

We consider a two-phase Stefan problem for a semi-infinite body with a convective boundary condition including a density jump at the free boundary with a time-dependent heat transfer coefficient of the type $h/\sqrt{t}$, whose solution was given in D. A. Tarzia, PAMM. Proc. Appl. Math. Mech. 7, 1040307–1040308 (2007). We demonstrate that the solution to this problem converges to the solution to the analogous one with a temperature boundary condition when the heat transfer coefficient $h\to +\infty$. Moreover, we analyze the dependence of the free boundary respecting to the jump density.     

An explicit solution for an instantaneous two-phase Stefan problem with nonlinear thermal coefficients

We consider a nonlinear heat conduction problem for a semi-infinitematerial x > 0, with phase-change temperature T₁, an initialtemperature T₂ (> T₁) and a heat flux of the type q (t) =q₀/t imposed on the fixed face x = 0. We assume that the volumetricheat capacity and the thermal conductivity are particular nonlinearfunctions of the temperature in both solid and liquid phases. We determine necessary and/or sufficient conditions on the parametersof the problem in order to obtain the existence of an explicitsolution for an instantaneous nonlinear twophase Stefan problem(solidification process).     

Analytical expressions for pulse profile of neutron stars in plasma environments

The European Physical Journal C - We construct the five loop anomalous dimensions of the basic fields in Quantum Chromodynamics in a linear covariant gauge in the modified Regularization Invariant...