A conforming mixed finite-element method for the coupling of fluid flow with porous media flow

Salim Meddahi ^{We consider a porous medium entirely enclosed within a fluidregion and present a well-posed conforming mixed finite-elementmethod for the corresponding coupled problem. The interfaceconditions refer to mass conservation, balance of normal forcesand the Beavers–Joseph–Saffman law, which yieldsthe introduction of the trace of the porous medium pressureas a suitable Lagrange multiplier. The finite-element subspacesdefining the discrete formulation employ Bernardi–Raugeland Raviart–Thomas elements for the velocities, piecewiseconstants for the pressures and continuous piecewise-linearelements for the Lagrange multiplier. We show stability, convergenceand a priori error estimates for the associated Galerkin scheme.Finally, we provide several numerical results illustrating thegood performance of the method and confirming the theoreticalrates of convergence.}     

A Decomposition Theorem for the Solutions to the Interface Problems of Quasi-Linear Elliptic Equations

Interface problems for second order quasi-linear elliptic partial differential equations in a two-dimensional space are studied. We prove that each weak solution can be decomposed into two parts near singular points, one of which is a finite sum of functions of the form cr^a log^m rφ(θ), where the coefficients c depend on the H^1-norm of the solution, the C^(0,δ) -norm of the solution, and the equation only; and the other one of which is a regular one, the norm of which is also estimated.     

Optimal Control of Non-well-posed Heat Equations

This work is concerned with Pontryagin＇s maximum principle of optimal control problems governed by some non-well-posed semilinear heat equations. A type of approach to the non-well-posed optimal control problem is given.     

Homotopy analysis method for generalized Benjamin–Bona–lMahony equation

An analytic technique, the homotopy analysis method (HAM), is applied to solve the generalized Benjamin–Bona–Mahony (BBM) equation. An explicit series solution is given, different from traditional analytic techniques, our approach is independent of knowing some parameters. This analytic method provides us with a new way to obtain series solutions of such problems. The homotopy analysis method contains the auxiliary parameter ħ, which provides us with a simple way to adjust and control the convergence region of series solution.     

Stability Problem for Jensen-type Functional Equations of Cubic Mappings

In this paper, we establish the general solution and the generalized Hyers-Ulam-Rassias stability problem for a cubic Jensen-type functional equation,4f（（3x＋y）/4）＋4f（（x＋3y）/4）=6f（（x＋y）/2）＋f（x）＋f（y）,9f（（2x＋y/3）＋9f（（x＋2y）/3）=16f（（x＋y）/2＋f（x）＋f（y）in the spirit of D. H. Hyers, S. M. Ulam, Th. M. Rassias and P. Gaevruta.     

Characteristic Lie algebra and classification of semidiscrete models

We study characteristic Lie algebras of semi-discrete chains and attempt to use this notion to classify Darboux-integrable chains. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 1, pp. 413–423, July, 2007.     

Nonlinear Degenerate Parabolic Equation with Nonlinear Boundary Condition

In this paper, we obtain the necessary and sufficient conditions on the global existence of all positive （weak） solutions to a nonlinear degenerate parabolic equation with nonlinear boundary condition.     

Probabilistic approach for granular media equations in the non-uniformly convex case

We use here a particle system to prove both a convergence result (with convergence rate) and a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex. The proof of convergence is simpler than the one in Carrillo–McCann–Villani (Rev. Mat. Iberoamericana 19:971–1018, 2003; Arch. Rat. Mech. Anal. 179:217–263, 2006). All the results complete former results of Malrieu (Ann. Appl. Probab. 13:540–560, 2003) in the uniformly convex case. The main tool is an uniform propagation of chaos property and a direct control in Wasserstein distance of solutions starting with different initial measures. The deviation inequality is obtained via a T ₁ transportation cost inequality replacing the logarithmic Sobolev inequality which is no more clearly dimension free.      

The number of rooted Eulerian planar maps

In this paper we provide a solution of the functional equation unsolved in the paper, by the second author, "On functional equations arising from map enumerations" that appeared in Discrete Math, 123: 93-109 (1993). It is also the number of combinatorial distinct rooted general eulerian planar maps with the valency of root-vertex, the number of non-root vertices and non-root faces of the maps as three parameters. In particular, a result in the paper, by the same author, "On the number of eulerian planar map...     

Numerical solution to the optimal feedback control of continuous casting process

Using a semi-discrete model that describes the heat transfer of a continuous casting process of steel, this paper is addressed to an optimal control problem of the continuous casting process in the secondary cooling zone with water spray control. The approach is based on the Hamilton–Jacobi–Bellman equation satisfied by the value function. It is shown that the value function is the viscosity solution of the Hamilton–Jacobi–Bellman equation. The optimal feedback control is found numerically by solving the associated Hamilton–Jacobi–Bellman equation through a designed finite difference scheme. The validity of the optimality of the obtained control is experimented numerically through comparisons with different admissible controls. Detailed study of a low-carbon billet caster is presented.     

Scattering of vortices in the Abelian Higgs model

We study the scattering of vortices in the Abelian (2+1)-dimensional Higgs model. We show that in the case of the symmetric head-on collision of N vortices, their trajectories are rotated by the angle π/N after the collision. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 1, pp. 77–91, July, 2008.     

Asymptotics asN→∞ forN classical fermions or bosons

We study the asymptotics of the spectrum for certain averaged operators related to the Liouville operator. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 849–866, December, 1999.     

On Runge-Kutta-type methods for two-dimensional stochastic differential equations

In the multidimensional case, second-order weak Runge-Kutta methods for stochastic differential equation (SDE) need simulation of correlated random variables, unless the diffusion matrix of SDE satisfies the commutativity condition. In this paper, we show that this can be avoided for some types of diffusion matrices and test functions important for applications. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 403–412, July–September, 2006.     

On solving the second basic three-element boundary-value Carleman-type problem in classes of bianalytic functions in a disk

In this paper, we investigate one of the basic Carleman-type boundary-value problems for bianalytic functions. We obtain a constructive method for solving the problem in the case of circular domain. We establish that solving the problem can be reduced to a sequential solving two generalized Carleman-type problems for analytic functions in a disk. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 413–426, July–September, 2006.     

A geometric method for solving a series of integral Poincaré-Steklov equations

Eigenvalues and eigenfunctions are explicitly found for a family of singular integral equations. It is shown how their discrete spectrum becomes continuous as the equation degenerates. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 343–353, March, 1998.     

Integral transformation of solutions for a Fuchsian-class equation corresponding to the Okamoto transformation of the Painlevé VI equation

We show that under the Euler integral transformation with the kernel (x−z)^−α, some solutions of the Fuchs equations (the original pair for the Painlevé VI equation) pass into solutions of a system of the same form with the parameters changed according to the Okamoto transformation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 355–364, March, 2006.     

The Cauchy Problem for the Generalized Korteweg-de Vries-Benjamin-Ono Equation with Low Regularity Data

The Cauchy problem of the generalized Korteweg-de Vries-Benjamin-Ono equation is considered, and low regularity and limit behavior of the solutions are obtained. For k = 1, local well- posedness is obtained for data in H^s（R）（s 〉 -3/4）. For k = 2, local result for data in H^S（R）（s 〉1/4） is obtained. For k = 3, local result for data in H^S（R）（s 〉 -1/6） is obtained. Moreover, the solutions of generalized Korteweg-de Vries-Benjamin-Ono equation converge to the solutions of KdV equation if the term of Benjamin-Ono equation tends to zero.     

A nonconvex dissipative system and its applications (I)

In order to study the uniformly translating solution of some non-linear evolution equations such as the complex Ginzburg–Landau equation, this paper presents a qualitative analysis to a Duffing–van der Pol non-linear oscillator. Monotonic property of the bounded exact solution is established based on the construction of a convex domain. Under certain parametric choices, one first integral to the Duffing–van der Pol non-linear system is obtained by using the Lie symmetry analysis, which constitutes one of the bases for further work of obtaining uniformly translating solutions of the complex Ginzburg–Landau equation. Dedicated to Professor G. Strang on the occasion of his 70th birthday     

The structure of solutions of a hypergeometric equation system

We consider a system of ordinary differential equations which is a multidimensional analogue of a hypergeometric equation. We study the structure and asymptotics of solutions at the singular points and construct a fundamental system of solutions in a neighborhood of each singular point. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 52–64, January–March, 1999. Translated by V. Mackevičius     

The Compound Poisson Surplus Model with Interest and Liquid Reserves: Analysis of the Gerber–Shiu Discounted Penalty Function

We modify the compound Poisson surplus model for an insurer by including liquid reserves and interest on the surplus. When the surplus of an insurer is below a fixed level, the surplus is kept as liquid reserves, which do not earn interest. When the surplus attains the level, the excess of the surplus over the level will receive interest at a constant rate. If the level goes to infinity, the modified model is reduced to the classical compound Poisson risk model. If the level is set to zero, the modified model becomes the compound Poisson risk model with interest. We study ruin probability and other quantities related to ruin in the modified compound Poisson surplus model by the Gerber–Shiu function and discuss the impact of interest and liquid reserves on the ruin probability, the deficit at ruin, and other ruin quantities. First, we derive a system of integro-differential equations for the Gerber–Shiu function. By solving the system of equations, we obtain the general solution for the Gerber–Shiu function. Then, we give the exact solutions for the Gerber–Shiu function when the initial surplus is equal to the liquid reserve level or equal to zero. These solutions are the key to the exact solution for the Gerber–Shiu function in general cases. As applications, we derive the exact solution for the zero discounted Gerber–Shiu function when claim sizes are exponentially distributed and the exact solution for the ruin probability when claim sizes have Erlang(2) distributions. Finally, we use numerical examples to illustrate the impact of interest and liquid reserves on the ruin probability.