We present a bibliography on moving and free boundary problems for the heatdiffusion equation, particularly regarding the Stefan and related problems. It contains 5869 titles referring to 588 scientific Journals, 122 books, 88 symposia (having at least 3 contributions on the subject), 30 collections, 59 thesis and 247 technical reports. It tries to give a comprehensive account of the western existing mathematicalphysical-engineering literature on this research field. RESUMEN Se presenta una bibliografía sobre problemas de frontera móvil y libre para la ecuación del calor-difusión, en particular sobre el problema de Stefan y problemas relacionados. Contiene 5869 títulos distribuidos en 588 revistas científicas, 122 libros, 88 simposios (teniendo al menos 3 contribuciones en el tema), 30 colecciones, 59 tesis y 247 informes técnicos o prepublicaciones. Se da un informe amplio de la bibliografía matemática, física y de las ingenierías existente en occidente sobre este tema de investigación. Primary Mathematics Subject Classification Number (*): 35R35, 80A22 Secondary Mathematics Subject Classification Number (*): 35B40, 35C05, 35C15, 35Kxx, 35R30, 46N20, 49J20, 65Mxx, 65Nxx, 76R50, 76S05, 76T05, 93C20. (*) Following the 1991 Mathematics Subject Classification compiled by Mathematical Reviews and Zentralblatt fur Mathematik. Primary key words: Enthalpy formulation or method, Filtration, Free boundary problems, Freezing, Melting, Moving boundary problems, Mushy region, Phase-change problem, Solidification, Stefan problem. Secondary key words: Continuous mechanics, Diffusion process, Functional analysis, Heat conduction, Mathematical methods, Numerical methods, Partial differential equations, Variational inequalities, Weak solutions. Palabras claves primarias: Método o formulación en entalpía, Filtración, Problemas de frontera libre, Congelación, Derretimiento, Problemas de frontera móvil, Región pastosa, Problema de cambio de fase, Solidificación, Problema de Stefan. Palabras claves secundarias: Mecánica del continuo, Procesos difusivos, Análisis funcional, Conducción del calor, Métodos matemáticos, Métodos numéricos, Ecuaciones diferenciales a derivadas parciales, Inecuaciones variacionales, Soluciones débiles. El manuscrito fue recibido y aceptado en octubre de 1999. D.A. Tarzia, A bibliography on FBP. The Stefan problem, MAT Serie A, # 2 (2000). 3

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A bibliography on moving-free boundary problems for the heat-diffusion equation. The stefan and related problems

In certain polymer-penetrant ,systems, nonlinear viscoelastic effects dominate those of Fickian diffusion. This behavior is often embodied in a memory integral incorporating nonlocal time effects into the dynamics; this integral can be derived from an augmented chemical potential. The mathematical framework presented is a moving boundary-value problem. The boundary separates the polymer into two distinct states: glassy and rubbery, where different physical processes dominate. The moving boundary condition that results is not solvable by similarity solutions, but can be solved by perturbation and integral equation techniques. Asymptotic solutions are obtained where sharp fronts move with constant speed. The resultant profiles are quite similar to experimental results in a dissolving polymer. It is then demonstrated that such a model has a limit on the allowable front speed and a self-regulating mass uptake.

http://www.math.udel.edu/%7Eedwards/download/pubdir/pubj4.pdf

A mathematical model for a dissolving polymer

We consider a class of second‐order partial differential equations which arises in diffusion phenomena and, following a new approach, we look for a Lie invariance classification via equivalence transformations. A class of exact invariant solutions containing an arbitrary function is obtained.

A group analysis approach for a nonlinear differential system arising in diffusion phenomena

We study the structure of diffusive layers in solutions of unstable nonlinear diffusion equations. These equations are regularizations of the forward-backward heat equation and have diffusion coefficients that become negative. Such models include the Cahn-Hilliard equation and the pseudoparabolic viscous diffusion equation. Using singular perturbation methods we show that the balance between diffusion and higher-order regularization terms uniquely determines the interface structure in these equations. It is shown that the well-known “equal area” rule for the Cahn-Hilliard equation is a special case of a more general rule for shock construction in the viscous Cahn-Hilliard equation.

The structure of internal layers for unstable nonlinear diffusion equations

We examine a model for non-Fickian “sorption overshoot” behavior in diffusive polymer-penetrant systems. The equations of motion proposed by Cohen and White [SIAM J. Appl. Math., 51 (1991), pp. 472–483] are solved for two-dimensional problems using matched asymptotic expansions. The phenomenon of shock formation predicted by the model is examined and contrasted with similar behavior in classical reaction-diffusion systems. Mass uptake curves produced by the model are examined and shown to compare favorably with experimental observations.

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Shock formation in a multidimensional viscoelastic diffusive system

Diffusion of a solvent into a polymer slab is sometimes characterized by a sharp front that moves into the medium and lasts for a long time. The behavior of such a front cannot be explained by a standard diffusion equation. Model equations are proposed that include solvent flux due to stress gradients in the polymer in addition to the Fickian flux. The stress in turn obeys a concentration-dependent evolution equation. In the limit of small diffusivity, the solution to a penetration problem is shown to have a steep front that progresses into the medium. The equations that govern the front are derived. A method for approximating the front is presented that uses a piece of the long time similarity solution to represent the concentration profile behind the front. The addition of a convective term to the equations is shown to raise the possibility of a traveling wave solution.

Shocks in a model for stress-driven diffusion

A Stefan problem is formulated for a diffusion equation where the Fickian flux is supplemented by a stress gradient term. The stress in turn obeys a concentration-dependent evolution equation that includes impulsive and relaxation effects. The speed of the free boundary is taken to be proportional to the flux at the front. For short times, the front position is found to be proportional to $t^{1/2} $. Using singular perturbation and numerical methods, three types of “long-time” behavior are found, depending on relationships between the coefficient functions and the parameters in the problem: (i) the front position is proportional to $t^{1/2} $; (ii) the front position is proportional to t; or (iii) the front stops in a finite time and the solution becomes singular (infinite gradients) at that point.

Robert W. Cox

Papers

Shocks in a model for stress-driven diffusion

Stress-assisted diffusion: a free boundary problem