Inna K. Shingareva

Capillary condensation as a morphological transition

The process of capillary condensation/evaporation in cylindrical pores is considered within the idea of symmetry breaking. Capillary condensation/evaporation is treated as a morphological transition between the wetting film configurations of different symmetry. We considered two models: (i) the classical Laplace theory of capillarity and (ii) the Derjaguin model which takes into account the surface forces expressed in terms of the disjoining pressure. Following the idea of Everett and Haynes, the problem of condensation/evaporation is considered as a transition from bumps/undulations to lenses. Using the method of phase portraits, we discuss the mathematical mechanisms of this transition hidden in the Laplace and Derjaguin equations. Analyzing the energetic barriers of the bump and lens formation, it is shown that the bump formation is a prerogative of capillary condensation: for the vapor-liquid transition in a pore, the bump plays the same role as the spherical nucleus in a bulk fluid. We show also that the Derjaguin model admits a variety of interfacial configurations responsible for film patterning at specific conditions.

Non-monotonic blow-up problems: Test problems with solutions in elementary functions, numerical integration based on non-local transformations

Abstract We consider blow-up problems having non-monotonic singular solutions that tend to infinity at a previously unknown point. For second-, third-, and fourth-order nonlinear ordinary differential equations, the corresponding multi-parameter test problems allowing exact solutions in elementary functions are proposed for the first time. A method of non-local transformations, that allows to numerically integrate non-monotonic blow-up problems, is described. A comparison of exact and numerical solutions showed the high efficiency of this method. It is important to note that the method of non-local transformations can be useful for numerical integration of other problems with large solution gradients (for example, in problems with solutions of boundary-layer type).

Analytical-Numerical Approach

One of the main points (related to computer algebra systems) is based on the implementation of a whole solution process, e.g., starting from an analytical derivation of exact governing equations, constructing discretizations and analytical formulas of a numerical method, performing numerical procedure, obtaining various visualizations, and analyzing and comparing the numerical solution obtained with other types of solutions.

Nonlinear problems with blow-up solutions: Numerical integration based on differential and nonlocal transformations, and differential constraints

Several new methods of numerical integration of Cauchy problems with blow-up solutions for nonlinear ordinary differential equations of the first- and second-order are described. Solutions of such problems have singularities whose positions are unknown a priori (for this reason, the standard numerical methods for solving problems with blow-up solutions can lead to significant errors). The first proposed method is based on the transition to an equivalent system of equations by introducing a new independent variable chosen as the first derivative, t=yx′, where x and y are independent and dependent variables in the original equation. The second method is based on introducing a new auxiliary nonlocal variable of the form ξ=∫x0xg(x,y,yx′)dx with the subsequent transformation to the Cauchy problem for the corresponding system of ODEs. The third method is based on adding to the original equation of a differential constraint, which is an auxiliary ODE connecting the given variables and a new variable. The proposed methods lead to problems whose solutions are represented in parametric form and do not have blowing-up singular points; therefore the transformed problems admit the application of standard fixed-step numerical methods. The efficiency of these methods is illustrated by solving a number of test problems that admit an exact analytical solution. It is shown that: (i) the methods based on nonlocal transformations of a special kind are more efficient than several other methods, namely, the method based on the hodograph transformation, the method of the arc-length transformation, and the method based on the differential transformation, and (ii) among the proposed methods, the most general method is the method based on the differential constraints. Some examples of nonclassical blow-up problems are considered, in which the right-hand side of equations has fixed singular points or zeros. Simple theoretical estimates are derived for the critical value of an independent variable bounding the domain of existence of the solution. It is shown by numerical integration that the first and the second Painleve equations with suitable initial conditions have non-monotonic blow-up solutions. It is demonstrated that the method based on a nonlocal transformation of the general form as well as the method based on the differential constraints admit generalizations to the nth-order ODEs and systems of coupled ODEs.

Hypersingular nonlinear boundary-value problems with a small parameter

For the first time, some hypersingular nonlinear boundary-value problems with a small parameter~

Approximate Analytical Approach

\varepsilon

General Analytical Approach. Integrability

at the highest derivative are described. These problems essentially (qualitatively and quantitatively) differ from the usual linear and quasilinear singularly perturbed boundary-value problems and have the following unusual properties: (i) in hypersingular boundary-value problems, super thin boundary layers arise, and the derivative at the boundary layer can have very large values of the order of

Geometric-Qualitative Approach

e^{1/\varepsilon}

Special Functions and Orthogonal Polynomials

and more (in standard problems with boundary layers, the derivative at the boundary has the order of

The excitation of tsunami by low-frequency non-stationary point and distributed centres of expansion

\varepsilon^{-1}