Inverse problem for a pseudoparabolic equation with integral overdetermination conditions

We consider inverse problems of finding an unknown coefficient in the leading term of a linear pseudoparabolic equation of filtration type on the basis of integral data over the entire boundary or its part under the assumption that the unknown coefficient depends on time. We derive conditions for the time-global solvability and uniqueness of the solution of the inverse problem.     

Finite abelian group methods in iterative algorithms for solving boundary-value problems with a boundary surface of complex structure

We consider external boundary-value problems for the Laplace equation on surfaces of complex structure. Various iterative computational schemes are constructed for numerical solution of the corresponding integral equations using set-theoretical group methods. Simulation results of electron-optical images are reported. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996, pp. 16–27.     

The Contact Problems of the Mathematical Theory of Elasticity for Plates with an Elastic Inclusion

The contacts problem of the theory of elasticity and bending theory of plates for finite or infinite plates with an elastic inclusion of variable rigidity are considered. The problems are reduced to integral differential equation or to the system of integral differential equations with variable coefficient of singular operator. If such coefficient varies with power law we can manage to investigate the obtained equations, to get exact or approximate solutions and to establish behavior of unknown contact stresses at the ends of elastic inclusion.      

Correspondence principle and the method of approximations for certain nonlinear hereditary media

Problems for two classes of nonlinear hereditary media are reduced by integral transforms to problems of the nonlinear theory of elasticity. An approximation of the elastic solution that makes it possible to solve the formulated problems for nonlinear hereditary materials is demonstrated in the case when the viscoelastic problems reduce to problems of the theory of small elastoplastic strains with active loading.Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 1, pp. 66–73, January–February, 1971.     

Analysis of glass-reinforced plastic bottoms of cylindrical pressure vessels

Solutions are proposed for boundary value problems associated with spherical, ellipsoidal and optimal-design glass-reinforced plastic bottoms filament-wound as an integral part of cylindrical shells.Moscow Region. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 918–926, September–October, 1968.     

Exact solution to the behavior of the electron plasma in a metal layer in an alternating electric field

Exact expressions are obtained for the first time for the response of the conduction electrons in a metal layer to an external alternating electric field. Case's well-known method of separation of the variables is modified, and the methods of singular integral equations and boundary-value problems of complex analysis are used. The solutions can be used to determined the screened field and also the absorption in a metal layer in a wide range of frequencies.N. K. Krupskaya Moscow Region Pedagogical Institute; Moscow Institute of Chemical Mechanical Engineering. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 2, pp. 179–189, February, 1992.     

Numerical method for solving an inverse problem for a population model

The inverse problem of determining the growth rate coefficient of biological objects from additional information on their time-dependent density is considered. Two nonlinear integral equations are derived for the unknown coefficient, which is determined on part of its domain from one equation and on the remaining part from the other equation. The nonlinear integral equations are solved by iterative methods. The convergence conditions for the iterative methods are formulated, and results of numerical experiments are presented.     

A method for calculating the current distribution in an unsymmetric oscillator with several excitation points

A method is proposed for calculating the current distribution in an unsymmetric oscillator with several excitation points joined by a common distribution line. The radiator and the distribution line are represented by multipoles, which are described by scattering matrices. The elements of the radiator scattering matrix are calculated by the integral equation method. The calculation results are compared with experimental findings.Translated from Matematicheskie Modeli Estestvoznaniya, Published by Moscow University, Moscow, 1995, pp. 71–76.     

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.     

Analysis of united boundary‐domain integro‐differential and integral equations for a mixed BVP with variable coefficient

The mixed (Dirichlet–Neumann) boundary‐value problem for the ‘Laplace’ linear differential equation with variable coefficient is reduced to boundary‐domain integro‐differential or integral equations (BDIDEs or BDIEs) based on a specially constructed parametrix. The BDIDEs/BDIEs contain integral operators defined on the domain under consideration as well as potential‐type operators defined on open sub‐manifolds of the boundary and acting on the trace and/or co‐normal derivative of the unknown solution or on an auxiliary function. Some of the considered BDIDEs are to be supplemented by the original boundary conditions, thus constituting boundary‐domain integro‐differential problems (BDIDPs). Solvability, solution uniqueness, and equivalence of the BDIEs/BDIDEs/BDIDPs to the original BVP, as well as invertibility of the associated operators are investigated in appropriate Sobolev spaces. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical methods for some inverse problems of heart electrophysiology

We consider numerical methods for solving inverse problems that arise in heart electrophysiology. The first inverse problem is the Cauchy problem for the Laplace equation. Its solution algorithm is based on the Tikhonov regularization method and the method of boundary integral equations. The second inverse problem is the problem of finding the discontinuity surface of the coefficient of conductivity of a medium on the basis of the potential and its normal derivative given on the exterior surface. For its numerical solution, we suggest a method based on the method of boundary integral equations and the assumption on a special representation of the unknown surface.     

Estimation of the error of approximation of the analytic solution of problems for nonlinear viscoelastic materials

Estimates are given for the error of approximation of the solution of problems of nonlinear viscoelastic media, for which the nonlinear elastic solution depends analytically on the material characteristics, the body and surface forces, and on the boundary displacements. From the integral estimates particular estimates are derived for viscoelastic materials in cases when the nonlinear elastic solution is majorized by a geometric progression or the sum of products of geometric progressions.Moscow Institute of Electronic Engineering. Translated from Mekhanika Polimerov, No. 5, pp. 827–834, September–October, 1971.     

Galerkin method for the linear-radiator integral equation

The article examines the general Galerkin method scheme for the singular integral equation in the problem of radiation from a finite-thickness linear radiator. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 150–158.     

Regularization for Higher Order Singular Integral Equations

By means of the definition of Hadamard principal value permutation formula and composition formula for higher order singular integrals developing in [5], the regularization problems for higher order singular integral equations with variable coefficients are discussed, and the higher order singular integral equations with constant coefficient are solved. It is the first time to treat the high order singular integral equations of arbitrary degree in the theory of singular integral equations.      

Symmetric boundary integral formulations for Helmholtz transmission problems

In this work we propose and analyze numerical methods for the approximation of the solution of Helmholtz transmission problems in two or three dimensions. This kind of problems arises in many applications related to scattering of acoustic, thermal and electromagnetic waves. Formulations based on boundary integral methods are powerful tools to deal with transmission problems in unbounded media. Different formulations using boundary integral equations can be found in the literature. We propose here new symmetric formulations based on a paper by Martin Costabel and Ernst P. Stephan (1985), that uses the Calderón projector for the interior and exterior problems to develop closed expressions for the interior and exterior Dirichlet-to-Neumann operators. These operators are then matched to obtain an integral system that is equivalent to the Helmholtz transmission problem and uses Cauchy data on the transmission boundary as unknowns. We show how to simplify the aspect and analysis of the method by employing an additional mortar unknown with respect to the ones used in the original paper, writing it in an appropriate way to devise Krylov type iterations based on the separate Dirichlet-to-Neumann operators.     

Numerical Solution of Inverse Problems for a Hyperbolic Equation with a Small Parameter Multiplying the Highest Derivative

We consider two inverse problems for a hyperbolic equation with a small parameter multiplying the highest derivative. The inverse problems are reduced to systems of linear Volterra integral equations of the second kind for the unknown functions. These systems are used to prove the existence and uniqueness of the solution of the inverse problems and numerically solve them. The applicability of the methods developed here to the approximate solution of the problem on an unknown source in the heat equation is studied numerically.     

3‐D Contact problems for elastic wedges with Coulomb friction

3‐D quasi‐static contact problems for elastic wedges with Coulomb friction are reduced to integral equations and integral inequalities with unknown contact normal pressures. To obtain these equations and inequalities, Green's functions for the wedges, where one face of the wedges is either stress‐free or fixed, are needed. Using Fourier and Kontorovich–Lebedev integral transformations, all the stresses and displacements in the wedges can be constructed in terms of solutions of Fredholm integral equations of the second kind on the semiaxis. The Green's functions can be calculated as uniformly convergent power series in (1‐2ν), where νis Poisson's ratio. An exponential decay of the kernels and right‐hand sides of the Fredholm integral equations provides the applicability of the collocation method for simple and fast calculation of the Green's functions. For a half‐space, which is a special case of an elastic wedge, the kernels degenerate and the functions reduce to the well‐known Boussinesq and Cerruti solutions. Analysing the contact problems reveals that the Green's functions govern the kernels of the above mentioned integral equations and inequalities. Under the assumption that the punch has a smooth shape, the contact pressure is zero on the boundary of the unknown contact zone. Solving the contact problems with the help of the Galanov–Newton method, the normal contact pressure, the contact zone and the normal displacement around the contact zone can be determined simultaneously. In view of the numerical results, the influence of the friction forces on the punch force and the punch settlement is discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

一般形式的奇异对偶积分方程组正交多项式求解法

基于Jacobi正交多项式法,直接求解一般形式的对偶积分方程组,将对偶积分方程组中的未知函数,表示成n次Jacobi正交多项式级数,用正交多项式将奇异对偶积分方程组,化成线性代数方程组,通过求解级数中的各项系数,由此给出奇异对偶积分方程组的一般性解,并严格证明了奇异对偶积分方程组和由它化成的线性代数方程组的等价性,解的存在性和解的表示形式不唯一性．本文给出的理论解和解法,可供求解复杂的数学、物理、软科学中的混合边值问题应用．     

Inverse problem of simultaneously determining the right-hand side and the coefficient of a lower order derivative for a parabolic equation on the plane

We obtain existence and uniqueness theorems for the solution of the inverse problem of simultaneously determining the right-hand side and the coefficient of a lower-order derivative in a parabolic equation under an integral observation condition. We give explicit estimates for the maximum absolute value of the unknown right-hand side and the unknown coefficient of the equation with constants expressed via the input data of the problem. We present a nontrivial example of an inverse problem to which our theorems apply.     

The three-dimensional contact problem with friction for an elastic wedge

Solutions of three-dimensional boundary-value problems of the theory of elasticity are given for a wedge, on one face of which a concentrated shearing force is applied, parallel to its edge, while the other face is stress-free or is in a state of rigid or sliding clamping. The solutions are obtained using the method of integral transformations and the technique of reducing the boundary-value problem of the theory of elasticity to a Hilbert problem, as generalized by Vekua (functional equations with a shift of the argument when there are integral terms). Using these and previously obtained equations, quasi-static contact problems of the motion of a punch with friction at an arbitrary angle to the edge of the wedge are considered. In a similar way the contact area can move to the edge of a tooth in Novikov toothed gears. The method of non-linear boundary integral equations is used to investigate contact problems with an unknown contact area.