An immersed interface method for the Navier-Stokes equations on irregular domains

An augmented method based on a Cartesian grid is proposed for the incompressible Navier Stokes equations on an irregular domain. The irregular domain is embedded into a rectangular one so that a fast Poisson solver can be used in the projection method. Unlike several methods suggested in the literature that set the force strengths as unknowns, which often results an ill-conditioned linear system, we set the jump in the normal derivative of the velocity as the augmented variable. The new approach improve the condition number of the system for the augmented variable significantly. Using the immersed interface method, we achieve second order accuracy for the velocity. Numerical results and comparisons are given to validate the new method. Some interesting new numerical experiments results are also presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The attractor problem for nonlinear wave equations in plane domains

We consider a model example of a quasilinear wave equation in the unit square with zero boundary conditions and use the method of quasinormal forms to prove that there are quite a few dichotomic cycles and tori bifurcating from zero equilibrium. A conjecture concerning the attractor structure is presented. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 217–229, August, 2000.     

Integral equations in a diffraction problem on a locally inhomogeneous medium interface

We study the diffraction of an E-polarized field on a locally inhomogeneous interface of transparent media. We prove the unique solvability of the boundary value diffraction problem and obtain integral representations of the solution. We derive a system of integral equations equivalent to the original boundary value problem and prove a solvability theorem for this system.     

On the solvability of a boundary value problem for nonlinear wave equations in angular domains

For a one-dimensional wave equation with a weak nonlinearity, we study the Darboux boundary value problem in angular domains, for which we analyze the existence and uniqueness of a global solution and the existence of local solutions as well as the absence of global solutions.     

Integral equations in the linear theory of elasticity in semiinfinite domains

We investigate linear integral equations on a semiaxis that appear in the course of construction of solutions of boundary-value problems in the theory of elasticity in such domains as a semiinfinite strip or a cylinder. By using the Mellin transformation and the theory of perturbations of linear operators, we establish general results concerning the solvability and asymptotic properties of solutions of the equations considered. We give examples of application of the general statements obtained to specific integral equations in the theory of elasticity. Translated from Ukrainskii Matematicheskii Zhumal, Vol. 50, No. 5, pp. 613–622, May, 1998.     

Finite-element method for the wave propagation problem in an irregular waveguide (E-polarization)

Translated from Chislennye Metody Resheniya Obratnykh Zadach Matematicheskoi Fiziki, pp. 111–119.     

Integral equations with Cauchy kernel in the contact problem

The contact problem of two elastic bodies of arbitrary shape with a general kernel form, investigated from Hertz problem, is reduced to an integral equation of the second kind with Cauchy kernel. A numerical method is adapted to determine the unknown potential function between the two surfaces under certain conditions. Many cases are derived and discussed from the work.     

Integral equations and the interior Dirichlet potential problem

The two standard approaches for reformulating the interior Dirichlet potential problem as a boundary integral equation of the second kind are discussed. The integral equation derived from the representation of the solution as a double layer is shown to be more general than the one derived from Green's theorem. The boundary integral equation of the latter method, however, has definite analytical and numerical value. From it a new integral equation is derived whose solution can be represented as a convergent Neumann series and it is shown that the Green's function of the first kind can be obtained from it. An example is supplied to illustrate the method.     

Non-homogeneous non-linear damped wave equations in unbounded domains

We present a global existence theorem for solutions of u^tt ? ?_ia_ik (x)?_ku + u_t = ?(t, x, u, u_t, ?u, ?u_t, ?²u), u(t = 0) = u⁰, u(=0)=u¹, u(t, x), t ? 0, x?Ω.Ω equals ?³ or Ω is an exterior domain in ?³ with smoothly bounded star-shaped complement. In the latter case the boundary condition u|_?Ω = 0 will be studied. The main theorem is obtained for small data (u⁰, u¹) under certain conditions on the coefficients a_ik. The L^p - L^q decay rates of solutions of the linearized problem, based on a previously introduced generalized eigenfunction expansion ansatz, are used to derive the necessary a priori estimates.     

A one-sweep method for linear elliptic equations over irregular domains

The determination of the configuration of equilibrium in a number of problems in mechanics and structures such as torsion, deflection of elastic membranes,etc., involve the solution of variational problems defined over irregular regions. This problem, in turn, may be reduced to the solution of elliptic differential equations subject to boundary conditions. In this paper, we study a method for the solution of such a problem when the region is of irregular shape. The method consists in solving the problem over a larger, imbedding, rectangular domain subject to appropriate constraints such as to satisfy the conditions of the original problem at the boundary. In this paper, we introduce the constraints by considering appropriate factors on the Green's function of the auxiliary problem. A conveniently discretized version of the problem is then treated by invariant imbedding, yielding some earlier results plus some new ones, namely, a direct one-sweep procedure that minimizes storage requirements. In addition, the present solution appears to be very convenient when the solution is required at a limited number of points. The derivations are specialized to Laplace's equation, but the method can be applied readily to general systems of second-order elliptic equations with no essential modifications. Finally, the existence of the necessary matrices in the imbedding equations is established.     

Global existence of null-form wave equations in exterior domains

We provide a proof of global existence of solutions to quasilinear wave equations satisfying the null condition in certain exterior domains. In particular, our proof does not require estimation of the fundamental solution for the free wave equation. We instead rely upon a class of Keel–Smith–Sogge estimates for the perturbed wave equation. Using this, a notable simplification is made as compared to previous works concerning wave equations in exterior domains: one no longer needs to distinguish the scaling vector field from the other admissible vector fields.     

Asymptotics of optimal control in the problem of harmonic wave scattering by an obstacle

Optimal impedance control for the Helmholtz equation in an unbounded domain is studied. Asymptotics of the optimal control with respect to a regularization parameter are constructed.     

Solution of an inverse scattering problem for the acoustic wave equation in three-dimensional media

A three-dimensional inverse scattering problem for the acoustic wave equation is studied. The task is to determine the density and acoustic impedance of a medium. A necessary and sufficient condition for the unique solvability of this problem is established in the form of an energy conservation law. The interpretation of the solution to the inverse problem and the construction of medium images are discussed.     

Integral equations related to the study of an inverse coefficient problem for a system of partial differential equations

We consider an inverse coefficient problem for a linear system of partial differential equations. The values of one solution component on a given curve are used as additional information for determining the unknown coefficient. The proof of the uniqueness of the solution of the inverse problem is based on the analysis of the unique solvability of a homogeneous integral equation of the first kind. The existence of a solution of the inverse problem is proved by reduction to a system of nonlinear integral equations.