Conjugation Boundary Research Articles

Reduction of Vector Boundary Value Problems on Riemann Surfaces to One-Dimensional Problems

This article lays foundations for the theory of vector conjugation boundary value problems on a compact Riemann surface of arbitrary positive genus. The main constructions of the classical theory of vector boundary value problems on the plane are carried over to Riemann surfaces: reduction of the problem to a system of integral equations on a contour, the concepts of companion and adjoint problems, as well as their connection with the original problem, the construction of a meromorphic matrix solution. We show that each vector conjugation boundary value problem reduces to a problem with a triangular coefficient matrix, which in fact reduces the problem to a succession of one-dimensional problems. This reduction to the well-understood one-dimensional problems opens up a path towards a complete construction of the general solution of vector boundary value problems on Riemann surfaces.

Connection between holomorphic vector bundles and cohomology on a Riemann surface and conjugation boundary value problems

This paper studies interconnections between holomorphic vector bundles on compact Riemann surfaces and the solution of the homogeneous conjugation boundary value problem for analytic functions on the one hand, and cohomology and the solution of the inhomogeneous problem on the other. We establish that constructing the general solution to the homogeneous problem with arbitrary coefficients in the boundary conditions is equivalent to classifying holomorphic vector bundles. Solving the inhomogeneous problem is equivalent to checking the solvability of 1-cocycles with coefficients in the sheaf of sections of a bundle; in particular, the solvability conditions in the inhomogeneous problem determine obstructions to the solvability of 1-cocycles, i.e. the first cohomology group. Using this connection, we can apply the methods of boundary value problems to vector bundles. The results enable us to elucidate the role of boundary value problems in the general theory of Riemann surfaces.

Influence of the twin microstructure on the mechanical properties in magnetic shape memory alloys

The microstructure evolution, i.e. reorientation of martensite variants, is an important deformation mechanism in shape-memory alloys. This microstructure evolution occurs by the motion of twin boundaries and the nucleation and annihilation of twins in the hierarchical microstructure. An appropriate discrete disclination model for the description of the internal elastic fields and microstructure evolution is introduced for representative volume elements. The model is applied to an experimentally characterized microstructure, i.e. conjugation boundary, and the predicted mechanical response is verified by comparison to experimental measurements. The influence of the twin microstructure on the homogenized stress-strain curve is studied. It is found that regular twinned microstructures have a low strain energy and a high resistance against deformation. These simulations also reason the origin of the microstructural stability of conjugation boundaries.

An ℝ-linear conjugation problem for two concentric annuli

We consider an infinite planar four-phase heterogeneous medium with three concentric circles as a boundary between isotropic medium’s components of distinct resistivities/conductivities. It is supposed that the velocity field in this structure is generated by a finite set of arbitrary multipoles. We distinguish two cases when multipoles are inside of medium’s components or at the interface. An exact analytical solution of the corresponding ℝ-linear conjugation boundary value problem is derived for both cases. Examples of flow nets (isobars and streamlines) are presented.

Simulation of conjugate heat transfer under conditions of aero-gas-dynamic heating of anisotropic blunt forebodies of hypersonic aircraft

An integrated physical and mathematical model of conjugate heat and mass transfer in super- and hypersonic flows around the blunt anisotropic bodies with determination of their thermal state is proposed. Also proposed is a technique for the numerical solution of the problems for conjugate heat exchange between gas-dynamic near-wall flows and bodies, the thermal conductivity of which is described by the second rank tensor. It is shown that variation of the tensor components of the material external layer thermal conductivity influences significantly the conjugation boundary temperature, which, in its turn, changes the heat flows to the body at the expense of temperature gradient at the conjugation boundary, density and dynamic viscosity.

Perfect lensing with phase-conjugating surfaces: toward practical realization

It is theoretically known that a pair of phase-conjugating surfaces can function as a perfect lens, focusing propagating waves and enhancing evanescent waves. However, the known experimental approaches based on thin sheets of nonlinear materials cannot fully realize the required phase conjugation boundary condition. In this paper, we show that the ideal phase-conjugating surface is, in principle, physically realizable and investigate the necessary properties of nonlinear and nonreciprocal particles which can be used to build a perfect lens system. The physical principle of the lens operation is discussed in detail and directions of possible experimental realizations are outlined.

Darcian Seepage through a Parallelogrammic Ramp: Toth’s Model Revisited

AbstractFully saturated, 2-D steady seepage in a homogeneous, isotropic, parallelogrammic ramp is studied using the methods of complex analysis (a conformal mapping and integral representation of mixed boundary-value problem). A porous medium flow is induced by an overland free-surface flow with the latter determining the “shearing” pressure (hydraulic head) of the former through a boundary condition along one side of the polygon serving as a conjugation line. For a uniform overland flow, the stream function along the conjugation line has a single maximum corresponding to the Toth hinge line. The inflow rate into the ramp is divided into a part exfiltrating through the toe and through a toe-adjacent segment of the conjugation boundary. A wavy free surface is shown to generate a more complex seepage topology.

The Over-determined Boundary Value Problem Method in the Electromagnetic Waves Propagation and Diffraction Theory

The over-determined boundary value problems in the partial domains are proposed to be used as the auxiliary problems to investigate the wave processes in the complex structures. The necessary and su-cient conditions of solvability of the over-determined problem are the dependencies between the boundary functions. These dependencies can be obtained in terms of the Fourier transforms or Fourier coe-cients of the boundary functions. The difiraction problems for the electromagnetic waves on the conducting screens in the space and in the waveguides with metallic walls are considered as the examples. The method of partial domains is widely used in the electromagnetic wave propagation and difirac- tion theory to solve the conjugation problems and boundary value problems with mixed boundary conditions. In the case when the integral or summatorial representations of the fleld to be found are obtained in some parts of the waveguide structure it is possible to get the integral or summatorial equations equivalent to the initial problem. It is convenient to consider the over-determined boundary value problems in the partial domains as the auxiliary problems. By this we are to consider more boundary conditions on some pieces of the domains boundaries than it is necessary to choose the unique solution. The review of articles devoted to over-determined boundary value problem method for the partial difierential equations is given in the paper (1). The necessary and su-cient conditions of solvability of the over-determined problems have the form of the supplementary connections between the auxiliary boundary functions. These con- nections together with the initial boundary conditions and the conjunction conditions form the complete set of equations to determine the electromagnetic fleld. In many cases the conditions at the inflnity for the unbounded domains (the radiation conditions) can be formulated also as the auxiliary conditions for the boundary functions in the over-determined problem.

Scattering of the plane wave by a transparent wedge

In the paper it is proved that the problem of scattering of the plane wave by a transparent wedge has a unique solution, provided that the radiation condition should be meant in the following form: if one subtracts from the solution the incident wave and all reflected and refracted waves, then the remainder satisfies the radiation condition in integral form. The problem is scalar, the velocities of the wave inside and outside the wedge are not equal, the wave process is described by the classical Helmholtz equations, and the conjugation boundary condition is satisfied on the sides of the wedge. Bibliography: 8 titles.

Solvability of the conjugation boundary value problem for the Vekua nonlinear equation on a Riemann surface

Solvability of the conjugation boundary value problem for the Vekua nonlinear equation on a Riemann surface