Inverse Boundary Value Problem Research Articles

An inverse boundary value problem for certain anisotropic quasilinear elliptic equations

In this paper we prove uniqueness in the inverse boundary value problem for quasilinear elliptic equations whose linear part is the Laplacian and nonlinear part is the divergence of a function analytic in the gradient of the solution. The main novelty in terms of the result is that the coefficients of the nonlinearity are allowed to be “anisotropic”. As in previous works, the proof reduces to an integral identity involving the tensor product of the gradients of 3 or more harmonic functions. Employing a construction method using Gaussian quasi-modes, we obtain a convenient family of harmonic functions to plug into the integral identity and establish our result.

Inverse Boundary Value Problem for a Fractional Differential Equations of Mixed Type with Integral Redefinition Conditions

In this paper, we consider an inverse boundary value problem for a mixed type partial differential equation with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. The differential equation depends from another positive parameter in mixed derivatives. With respect to first variable this equation is a fractional-order nonhomogeneous differential equation in the positive part of the considering segment, and with respect to second variable is a second-order differential equation with spectral parameter in the negative part of this segment. Using the Fourier series method, the solutions of direct and inverse boundary value problems are constructed in the form of a Fourier series. Theorems on the existence and uniqueness of the problem are proved for regular values of the spectral parameter. It is proved the stability of the solution with respect to redefinition functions, and with respect to parameter given in mixed derivatives. For irregular values of the spectral parameter, an infinite number of solutions in the form of a Fourier series are constructed.

Boundary Value Problems for a Fourth Order Partial Differential Equation with an Unknown Right-hand Part

In this article in a rectangular domain inverse boundary value problems with local and non-local boundary conditions for a fourth order partial differential equation were formulated and investigated. The solutions of the inverse problems were obtained in the form of Fourier series. It was proved the theorem on uniqueness and existence of the solution of the inverse problems under consideration.

The interior inverse boundary value problem for the impulsive Sturm-Liouville operator with the spectral boundary conditions

In this study, we discuss the inverse problem for the Sturm-Liouville operator with the impulse and with the spectral boundary conditions on the finite interval (0, pi). By taking the Mochizuki-Trooshin's method, we have shown that some information of eigenfunctions at some interior point and parts of two spectra can uniquely determine the potential function q(x) and the boundary conditions.

Finding the Shape of the Profile of Aero-Hydrodynamical Lattice by Given Chord Diagram of Velocity Distrubution

We consider the inverse boundary value problem of aero-hydrodynamical lattices. It is necessary to find the shape of the profile of a lattice streamlined by a potential flow of an incompressible non-viscous liquid, when the distribution of the velocity value on the profile is given as a function of the abscissa of the profile point (chord diagram); the lattice period and the ratio of complex liquid flow rates before and after the lattice are known.

Application of transfer function for quick estimation of gas flow parameters—A useful model‐based approach to enhancing measurements

AbstractAt the core of chemical engineering, process design is fundamental to reach a goal of productivity. Axial dispersion is one of the key phenomena that can affect productivity. Therefore, its characterization is necessary for adequate modeling and designing devices. Applications of the transfer function (TF) to the derivation of a high precision model of tracer flow and, next, determination of an axial dispersion coefficient in a commercial measurement system are presented. A TF concept makes easier customization of model ideas to different systems and consequently allows for obtaining a model that matches in the best way a physical system. The method takes advantage of the efficient numerical algorithm by den Iseger, that is; inverse Laplace transform to solve the model both at the stage of model development (boundary value problem) and model application (inverse boundary value problem). As a result, a very precise model of commercial measurement instrument was developed and, next, it was employed to the determination of axial dispersion coefficients for an empty tube and packed bed. The method presented is precise in a wide range of operating conditions. The paper shows that mathematical modeling can be exploited to enhance measurements for a commercial measurement instrument, that is, unlock the system's full potential with no equipment design changes. The method is also a good alternative to computational fluid dynamics for high precision calculations—it fits experimental results better than a CFD model.

Optimality of Increasing Stability for an Inverse Boundary Value Problem

In this work we study the optimality of increasing stability of the inverse boundary value problem (IBVP) for Schr\"{o}dinger equation. The rigorous justification of increasing stability for the IBVP for Schr\"{o}dinger equation were established by Isakov \cite{Isa11} and by Isakov, Nagayasu, Uhlmann, Wang of the paper \cite{INUW14}. In \cite{Isa11}, \cite{INUW14}, the authors showed that the stability of this IBVP increases as the frequency increases in the sense that the stability estimate changes from a logarithmic type to a H\"{o}lder type. In this work, we prove that the instability changes from an exponential type to a H\"older type when the frequency increases. This result verifies that results in \cite{Isa11}, \cite{INUW14} are optimal.

Interior decay of solutions to elliptic equations with respect to frequencies at the boundary

We prove decay estimates in the interior for solutions to elliptic equations in divergence form with Lipschitz continuous coefficients. The estimates explicitly depend on the distance from the boundary and on suitable notions of frequency of the Dirichlet boundary datum. We show that, as the frequency at the boundary grows, the square of a suitable norm of the solution in a compact subset of the domain decays in an inversely proportional manner with respect to the corresponding frequency. Under Lipschitz regularity assumptions, these estimates are essentially optimal and they have important consequences for the choice of optimal measurements for corresponding inverse boundary value problem.

An inverse heat conduction boundary problem for a two-part rod with different thermal conductivity

The paper studies the problem of determining the boundary condition in the heat conduction equation for a rod consisting of homogeneous parts with different thermophys- ical properties. We consider the Dirichlet condition at the left end of the rod (at x = 0) corresponding to the heating of this end and the homogeneous condition of the first kind at the right end (at x = 1) corresponding to cooling during interaction with the environment as boundary conditions. At the point of discontinuity of the thermophysical properties (at x = x0), the conditions for the continuity of temperature and heat flux are set. In the inverse problem, the boundary condition at the left end is assumed to be unknown. To find it, the value of the direct problem solution at the point x0, i.e., the point of separation of the rod into two homogeneous sections, is set. In this work, we carried out an analytical study of the direct problem, which allowed us to apply the time Fourier transform to the inverse boundary value problem. The projection-regularization method is used to solve the inverse boundary value problem for the heat equation and obtain error estimates of this solution correct to the order.

A time-nonlocal inverse problem for a hyperbolic equation with an integral overdetermination condition

This article is concerned with the study of the unique solvability of a timenonlocal inverse boundary value problem for second-order hyperbolic equation with an integral overdetermination condition. To study the solvability of the inverse problem, we first reduce the considered problem to an auxiliary system with trivial data and prove its equivalence (in a certain sense) to the original problem. Then using the Banach fixed point principle, the existence and uniqueness of a solution to this system is shown. Further, on the basis of the equivalency of these problems the existence and uniqueness theorem for the classical solution of the inverse coefficient problem is proved for the smaller value of time.

An Inverse Parameter Problem with Generalized Impedance Boundary Condition for Two-Dimensional Linear Viscoelasticity

We analyze an inverse boundary value problem in two-dimensional viscoelastic media with a generalized impedance boundary condition on the inclusion via boundary integral equation methods. The model...

Solving of an Inverse Boundary Value Problem for the Heat Conduction Equation by Using Lavrentiev Regularization Method

In this paper, the inverse problem for the boundary value of the heat equation is posed and solved. It is well known that this problem classified as an ill-posed problem. The boundary value problem can be represented as an integral equation of the first kind by using the separation of variables method. The discretization of the integral equation allowed us to reduce the integral equation to a system of linear algebraic equations or a linear operator equation of the first kind on Hilbert spaces. In order to find an approximation solution, we need to apply a regularization algorithm. In this type of equation and through the regularization step we faced a non-injective operator problem. The Lavrentev regularization method was used to obtain the solution instead of the Tikhonov regularization method.

Solution of the inverse boundary value problem of heat transfer for an inhomogeneous ball

Solution of the inverse boundary value problem of heat transfer for an inhomogeneous ball

МОДЕЛИРОВАНИЕ НЕСТАЦИОНАРНОГО ТЕЧЕНИЯ НЕСЖИМАЕМОЙ ЖИДКОСТИ В ПЕРФОРИРОВАННОМ ТРУБОПРОВОДЕ

A mathematical model of the unsteady flow of an incompressible viscous fluid through a perforated pipeline is proposed, which is described by a system of nonlinear partial differential equations. In the framework of the model, the purpose is to determine the pressure and the flow rate of the fluid at the pipeline inlet, providing the flow rate and the pressure required at the pipeline outlet. By combining the system of the equations, the original problem is reduced to a boundary-value inverse problem for a nonlinear parabolic equation with respect to fluid flow rate. To solve the boundary inverse problem, the method of nonlocal perturbation of boundary conditions is proposed. A discrete analog of the inverse problem is obtained using the finitedifference approximation, and a special approach is suggested for solving the resulting system of difference equations. As a result, the difference problem for each discrete value of the time variable splits into two second-order difference problems and a linear equation with respect to an approximate value of the desired flow rate at the pipeline inlet. The absolutely stable Thomas method is used to numerically solve the obtained difference problems. After determining the flow rate distribution along the entire pipeline, the pressure at the pipeline inlet is also calculated using an explicit formula. Based on the proposed computational algorithm, the numerical experiments are performed for benchmark problems.

Methodology of non-linear robust estimation for the solutions synthesis of inverse and direct multidisciplinary problems in engineering dimensional chains calculation based on discrete analog data

This paper analyses the definition of inverse and direct problems in engineering dimensional chains calculation based on discrete analogue data and the methodologies for solving these problems. It is shown that the direct dimensional chains calculation, which belongs to the class of inverse boundary value problems in a stochastic formulation, can be transformed into multi-criteria problems of stochastic optimization with mixed conditions. The new multi-step solutions search methodology for these problems is based on non-linear robust estimation methods. It can be achieved through hierarchical two-level decisions synthesis scheme development. At the first step, this scheme includes identification of surrogate models (in the form of regression equations). At the second step, the effective robust estimates are computed to determine unknown values; estimations of unknown quantities are carried out under a priori and parametric data uncertainties. Results of calculations of inverse and direct problems in engineering dimensional chains for two-stage axial compressors are presented. They were obtained using interactive computer systems for decision-making support “ROD&IDS”.

Direct and inverse problems for the nonlinear time-harmonic Maxwell equations in Kerr-type media

In the current paper we consider an inverse boundary value problem of electromagnetism in a nonlinear Kerr medium. We show the unique determination of the electromagnetic material parameters and the nonlinear susceptibility parameters of the medium by making electromagnetic measurements on the boundary. We are interested in the case of the time-harmonic Maxwell equations.

Recovery of coefficients for a weighted p-Laplacian perturbed by a linear second order term

This paper considers the inverse boundary value problem for the equation ∇ ⋅ (σ∇u + a|∇u| p−2∇u) = 0. We give a procedure for the recovery of the coefficients σ and a from the corresponding Dirichlet-to-Neumann map, under suitable regularity and ellipticity assumptions.

Inverse initial boundary value problem for a non-linear hyperbolic partial differential equation

In this article we are concerned with an inverse initial boundary value problem for a non-linear wave equation in space dimension n ⩾ 2. In particular we consider the so called interior determination problem. This non-linear wave equation has a trivial solution, i.e. zero solution. By linearizing this equation at the trivial solution, we have the usual linear wave equation with a time independent potential. For any small solution u = u(t, x) of our non-linear wave equation which is the perturbation of linear wave equation with time-independent potential perturbed by a divergence with respect to (t, x) of a vector whose components are quadratics with respect to ∇ t,x u(t, x). By ignoring the terms with smallness O(|∇ t,x u(t, x)|3), we will show that we can uniquely determine the potential and the coefficients of these quadratics by many boundary measurements at the boundary of the spacial domain over finite time interval and the final overdetermination at t = T. In other words, our measurement is given by the so-called the input-output map (see ()).

Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations

We study various partial data inverse boundary value problems for the semilinear elliptic equation \Delta u+ a(x,u)=0 in a domain in \mathbb{R}^n by using the higher order linearization technique introduced by Lassas–Liimatainen–Lin–Salo and Feizmohammadi–Oksanen. We show that the Dirichlet-to-Neumann map of the above equation determines the Taylor series of a(x,z) at z = 0 under general assumptions on a(x,z) . The determination of the Taylor series can be done in parallel with the detection of an unknown cavity inside the domain or an unknown part of the boundary of the domain. The method relies on the solution of the linearized partial data Calderón problem by Ferreira–Kenig–Sjöstrand–Uhlmann, and implies the solution of partial data problems for certain semilinear equations \Delta u+ a(x,u) = 0 also proved by Krupchyk–Uhlmann. The results that we prove are in contrast to the analogous inverse problems for the linear Schrödinger equation. There recovering an unknown cavity (or part of the boundary) and the potential simultaneously are longstanding open problems, and the solution to the Calderón problem with partial data is known only in special cases when n \geq 3 .

Inverse Problem of Pipeline Transport of Weakly-Compressible Fluids

Nonstationary one-dimensional flow of a weakly-compressible fluid in a pipeline is considered. The flow is described by a nonlinear system of two partial differential equations for the fluid flow rate and pressure in the pipeline. An inverse problem on determination of the fluid pressure and flow rate at the beginning of the pipeline needed for the passage of the assigned quantity of fluid in the pipeline at a certain pressure at the pipeline end was posed and solved. To solve the above problem, a method of nonlocal perturbation of boundary conditions has been developed, according to which the initial problem is split at each discrete moment into two successively solvable problems: a boundary-value inverse problem for a differential-difference equation of second order for the fluid flow rate and a direct differential-difference problem for pressure. A computational algorithm was suggested for solving a system of difference equations, and a formula was obtained for approximate determination of the fluid flow rate at the beginning of the pipeline. Based on this algorithm, numerical experiments for model problems were carried out.