An analysis of the solution set to a homotopy equation between polynomials with real coefficients

We investigate the structure of the solution setS to a homotopy equationH(Z,t)=0 between two polynomialsF andG with real coefficients in one complex variableZ. The mapH is represented asH(x+iy, t)=h ₁(x, y, t)+ih ₂(x, y, t), whereh ₁ andh ₂ are polynomials from ℝ² × [0,1] into ℝ and i is the imaginary unit. Since all the coefficients ofF andG are real, there is a polynomialh ₃ such thath ₂(x, y, t)=yh₃(x, y, t). Then the solution setS is divided into two sets {(x, t)∶h ₁(x, 0, t)=0} and {(x+iy, t)∶h ₁(x, y, t)=0,h ₃(x, y, t)=0}. Using this division, we make the structure ofS clear. Finally we briefly explain the structure of the solution set to a homotopy equation between polynomial systems with real coefficients in several variables.     

Concentration and cavitation phenomena of Riemann solutions for the isentropic Euler system with the logarithmic equation of state

The analytical solutions of the Riemann problem for the isentropic Euler system with the logarithmic equation of state are derived explicitly for all the five different cases. The concentration and cavitation phenomena are observed and analyzed during the process of vanishing pressure in the Riemann solutions. It is shown that the solution consisting of two shock waves converges to a delta shock wave solution as well as the solution consisting of two rarefaction waves converges to a solution consisting of four contact discontinuities together with vacuum states with three different virtual velocities in the limiting situation.     

Steady state analysis of a nonlinear renewal equation

In the analysis of a nonlinear renewal equation it is natural to anticipate the existence of nonzero steady states and deal with the question of their stability. Sufficient conditions for existence and uniqueness for these steady states are given. The study of the linearized version of the renewal equation around the steady state helps to a great extent to have insight into some complicated dynamics of the full problem. At this stage the first eigenvalue of the steady state plays a vital role. The characteristic equation, a functional equation whose roots are the eigenvalues, is derived. We give various structures showing that the steady state may be stable or unstable (though the fertility rate is decreasing with competition). A similar study is carried out on a nonlinear model motivated by neuroscience in which the total population is conserved.     

Complex variable approach to the analysis of a fractional differential equation in the real line

The first aim of this work is to establish a Peano-type existence theorem for an initial value problem involving a complex fractional derivative, and then, as a consequence of this theorem, to give a partial answer for the local existence of the continuous solution to the initial value problem:

$\{\begin{array}{cc}\hfill & D_x^{q}u(x)=f(x,u(x)),\hfill \\ \hfill & u(0)=b,(b\ne 0).\hfill \end{array}$

Moreover, for some special cases of the problem, we investigate the corresponding geometric properties of the solutions.     

On the possibility of simplifying the thermoelastic equation

By some very simple algebraic manipulations, it is shown as the linear thermoelastic equation can be solved by looking for the solutions of much simpler vectorial equations. As an example the displacements of the reflecting surface of a mirror illuminated by a high power laser beam have been evaluated.

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Sommario Utilizzando alcuni semplici passaggi algebrici, viene mostrato come l'equazione degli spostamenti termoelastici possa essere risolta cercando le soluzioni di equazioni vettoriali molto più semplici. Come esempio vengono calcolate le deformazioni della superficie riflettente di uno specchio investito da un fascio laser di grande potenza.

     

On the justification of the Reynolds equation,describing isentropic compressible flows through a tiny pore

Abstract We consider the isentropic compressible flow through a tiny pore. Our approach is to adapt the recent results by N. Masmoudi on the homogenization of compressible flows through porous media to our situation. The major difference is in the a priori estimates for the pressure field. We derive the appropriate ones and then Masmoudi’s results allow to conclude the convergence. In this way the compressible Reynolds equation in the lubrication theory is rigorously justified. Keywords: Compressible Navier-Stokes equations, Lubrication, Pressure estimates Mathematics Subject Classification (2000): 35B27, 76M50, 35D05     

An optimal control problem for a nonlinear elliptic equation with a phase constraint and state variation

We study an optimal control problem for a system described by a nonlinear elliptic equation with a state constraint in the form of an inclusion. We prove the solvability of the problem under consideration and by varying the state of the system obtain necessary optimality conditions.     

Singular integral equation with Cauchy kernel on the real axis

We find closed-form formulas for the solution of the simplest singular integral equation with Cauchy kernel on the real axis and use them to reduce the full singular integral equation considered in the paper to a Fredholm equation. We construct numerical schemes for the above-mentioned equations and estimate the accuracy order of the approximate solution.     

Methods for solving a singular integral equation with Cauchy kernel on the real line

We study exact and approximate methods for solving a singular integral equation with Cauchy kernel on the real line. On the basis of the theory of positive operators, we prove an existence and uniqueness theorem for this equation in the space of Lebesgue square integrable functions. This theorem is then used to give a theoretical justification of general projection and projection-iteration methods as well as an iteration method for solving this equation.     

Accessory coefficients of the second-order Fuchs equation with real singular points

The author investigates the equations of Schwarz and Fuchs related with the conformal mapping of a semiplane onto a circular polygon. One investigates the question of the dependence of the accessory coefficients as functions of the singularities of the equations. As an intermediate result one constructs formulas for the Fourier coefficients of the Siegel-Selberg series and of the absolute Klein invariant. At the conclusion of the paper one considers the example of a rectangle for which the problem of the determination of the accessory coefficients is simplified.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 129, pp. 17–29, 1983.     

Axisymmetric solutions of the two-dimensional Euler equations with a two-constant equation of state

A family of self-similar and global bounded weak solutions are constructed rigorously for all positive time to the two-dimensional isentropic Euler equations with the equation of state $p=A_1{\rho }^{{\gamma }_1}+A_2{\rho }^{{\gamma }_2}$ for initial pure radial velocity. The main difficulty is that the equations cannot be directly reduced to an autonomous system of ordinary differential equations. To deal with it, we first establish the global existence and the detailed structures of solutions to a new system under the axisymmetric and self-similar assumptions.     

Existence analysis of an optimal shape design problem with non coercive state equation

In this paper, a shape optimization problem, modelling a welding process, governed by a second order non coercive PDE is considered. The well posedness of the shape optimal design problem is showed using the degree of Leray–Schauder. Then the existence of an optimal solution is proved.     

An integral equation with a difference kernel

In this note a new method of solving a class of integral equations with difference kernels is given. It is based on establishing a connection between the solution of the given equation and that of the corresponding equation on the half-axis. This method allows us to reduce the given equation to a new integral equation with the kernel of a simple structure.Translated from Matematicheskie Zametki, Vol. 19, No. 6, pp. 927–932, June, 1976.     

Cauchy-type singular integral equation with constant coefficients on the real line

In this paper, exact solutions of the characteristic and dominant equation with Cauchy kernel on the real line are presented. Next, trigonometric polynomials are used to derive approximate solutions of these equations. Moreover, estimations of errors of the approximated solutions are presented and proved.