We consider the fourth-order thin film equation (TFE)with the unstable second-order diffusion term. We show that, for the first critical exponentwhere N ≥ 1 is the space dimension, the free-boundary problem the with zero contact angle and zero-flux conditions admits continuous sets (branches) of self-similar similarity solutions of the formFor the Cauchy problem, we describe families of self-similar patterns, which admit a regular limit as n → 0+ and converge to the similarity solutions of the semilinear unstable limit Cahn-Hilliard equationstudied earlier in [12]. Using both analytic and numerical evidence, we show that such solutions of the TFE are oscillatory and of changing sign near interfaces for all n ∈ (0,nh), where the valuecharacterizes a heteroclinic bifurcation of periodic solutions in a certain rescaled ODE. We also discuss the cases p ⧧ = p0, the interface equation, and regular analytic approximations for such TFEs as an approach to the Cauchy problem.

In this article the linear theory of thermoviscoelastic mixtures is considered. The fundamental solution of the system of linear-coupled partial differential equations of steady oscillations (steady vibrations) of the theory of thermoviscoelastic mixtures is constructed in terms of elementary functions and basic properties are established.

We analyse the two-dimensional, gravitationally-driven spreading of fluid through a porous medium overlying a horizontal impermeable boundary from which fluid can drain freely at one end. Under the assumption that none of the intruding fluid is retained within the pores in the trail of the current, the motion of the current is described by the dipole self-similar solution of the first kind derived by Barenblatt and Zel'dovich (1957). We show that small perturbations of arbitrary shape imposed on this solution decay in time, indicating that the self-similar solution is linearly stable. We use the connection between the perturbation eigenfunctions and symmetry transformations of the self-similar solution to demonstrate that variables can always be specified in terms of which the rate of decay of the perturbations is maximised. Unsaturated flow can be modelled by assuming that a constant fraction of the fluid is retained within the pores by capillary action in the trail of the current. It has been shown (Barenblatt and Zel'dovich, 1998; Ingerman and Shvets, 1999) that in this case, the motion of the current is described by a self-similar solution of the second kind characterised by an anomalous exponent. We derive leading-order analytic expressions for the anomalous exponent and the self-similar quantities valid for small values of the fraction of fluid retained using direct asymptotic analysis and by using a novel application of the method of multiple scales. The latter offers a number of advantages and permits the evolution of the current to be clearly connected with its initial conditions in a way not possible with conventional approaches. We demonstrate that the theoretical predictions provided by these expressions are in excellent agreement with results from the numerical integration of the governing equations.

This article provides a borrower's optimal strategies to terminate a mortgage with a fixed interest rate by paying the outstanding balance all at once. The problem is modelled as a free boundary problem for the appropriate analogue of the Black-Scholes pricing equation under the assumption of the Vasicek model for the short-term rate of investment. Here the free boundary provides the optimal time at which the mortgage contract is to be terminated. A number of integral identities are derived and then used to design efficient numerical codes for computing the free boundary. For numerical simulation, parameters for the Vasicek model are estimated via the method of maximum likelihood estimation using 40 years of data from US government bonds. The asymptotic behaviour of the free boundary for the infinite horizon is fully analysed. Interpolating this infinite horizon behaviour and a known near-expiry behaviour, two simple analytical approximation formulas for the optimal exercise boundary are proposed. Numerical evidence shows that the enhanced version of the approximation formula is amazingly accurate; in general, its relative error is less than 1%, for all time before expiry.

Fluids in unsaturated porous media are described by the relationship between pressure (p) and saturation (u). Darcy's law and conservation of mass provides an evolution equation for u, and the capillary pressure provides a relation between p and u of the form p∈ pc(u,∂tu). The multi-valued function pc leads to hysteresis effects. We construct weak and strong solutions to the hysteresis system and homogenize the system for oscillatory stochastic coefficients. The effective equations contain a new dependent variable that encodes the history of the wetting process and provide a better description of the physical system.