Inverse Problems Of Mathematical Physics Research Articles

Inverse Problem of Determining the Unknown Coefficients in an Elliptic Equation

The inverse problem of determining the coefficients of an elliptic equation under different boundary conditions in a given rectangle is considered. These problems lead to the necessity of approximate solution of inverse problems of mathematical physics, which are incorrect in the classical sense. The existence, uniqueness, and stability theorems for the solution of the set inverse problem are proved and a regularizing algorithm for determining the coefficient is constructed.

A mathematical model for the formation of the mineral composition of rocks removed to the surface during a volcano eruption


 
 
 The article examines the origin of chemicals obtained as a result of cooling on the Earth’s surface during a volcanic eruption. For this, we propose a mathematical model of the formation of the mineral composition of rocks brought to the surface during a volcanic eruption, depending on the crystallization temperature. To do this, we consider a certain inverse problem for the heat equation and prove its solvability under certain conditions. The proposed method can also be used in case of an earthquake. Thus, by studying the composition of rocks on the surface, we can determine at what depth the corresponding layer of rocks is located. Mathematically, this means that you need to find the unknown functions f (x) and u (x, t), where the function f (x) characterizes the source of the formation of rocks, and the function u (x, t) characterizes the crystallization of rocks depending on tem- perature because each rock has its crystallization temperature depending on temperature. We assume that the functions h (x, t), g (x, t) are known. Unknown functions f (x) and u (x, t). Using the inverse problem, we determine the function u (x, t) – the temperature in the lower layers of the earth, and knowing the temperature of crystallization of rocks, we can determine what depth the required rock is. The main purpose of the inverse problem is to determine the composition of rocks below the earth’s surface using data obtained at the earth’s surface. Experimentally, this mathematical model means that with the help of certain sensors on the surface, it is possible to give information about the subterranean depths. This information is determined by the nature of the object under study and the experimental complex used in this study. In such situations, for the diagnosis of objects and their internal structure, mathematical processing, and interpretation of observation results are required .We are talking about those tasks in which it is required to determine the causes if the consequences obtained as a result of observation are known. For example, determining the location and power of an earthquake from the vibrations measured on the earth’s surface. Or, by the structure of minerals on the surface of the earth, determining the depth of their location underground, etc. When processing the experimental data, a conclusion is made about the internal connections of the phenomenon or process. When the mathematical model of the process under study is known, the problem of identifying the mathematical model is posed. Namely, the determination of the coefficients of differential equations, their right-hand side, the boundary of the region, boundary, and initial conditions – these problems are related to inverse problems of mathematical physics. In general, inverse problems.
 
 

Regularization of ill-posed problems involving constant-coefficient pseudo-differential operators

This paper deals with the wavelet regularization for ill-posed problems involving linear constant-coefficient pseudo-differential operators. We concentrate on solving ill-posed equations involving these operators, which are behaving badly in theory and practice. Since a wide range of ill-posed and inverse problems in mathematical physics can be described and rewritten by the language of these operators, it has gathered significant attention in the literature. Based on a general framework, we classify ill-posed problems in terms of their degree of ill-posedness into mildly, moderately, and severely ill-posed problems in a certain Sobolev scale. Using wavelet multi-resolution approximations, it is shown that wavelet regularizers can achieve order-optimal rates of convergence for pseudo-differential operators in special Sobolev space both for the a priori and the a posteriori choice rules. Our strategy, however, turns out that both schemes yield comparable convergence rates. In this setting, ultimately, we provided some prototype examples for which our theoretical results correctly predict improved rates of convergence.

A source inverse problem for the pseudo–parabolic equation with the fractional Sturm–Liouville operator

A class of inverse problems for restoring the right-hand side of the pseudo-parabolic equation with one fractional Sturm–Liouville operator is considered. In this paper, we prove the existence and uniqueness results of the solutions using by the variable separation method that is to say the Fourier method. We are especially interested in proving the existence and uniqueness of the solutions in the abstract setting of Hilbert spaces. The mentioned results are presented as well as for the Caputo time fractional pseudoparabolic equation. There are many cases in which practical needs lead to problems determining the coefficients or the right side of a differential equation from some available decision data. These are called inverse problems of mathematical physics. Inverse problems arise in various areas of human activity, such as seismology, mineral exploration, biology, medicine, industrial quality control goods, and so on. All these circumstances put the inverse problems among the important problems of modern mathematics.

The wave model of a metric space with measure and an application

Let be a complete metric space and let be a Borel measure on . Under certain fairly general assumptions about the metric and the measure, we use lattice theory to construct an isometric copy of the space , which is called its wave model. The construction is motivated by applications to inverse problems of mathematical physics. We show how the wave model solves the problem of reconstructing a Riemannian manifold with boundary from its spectral data. Bibliography: 13 titles.

On algebraic and uniqueness properties of harmonic quaternion fields on 3d manifolds

Let Ω be a smooth compact oriented 3-dimensional Riemannian manifold with boundary. A quaternion field is a pair q = {α, u} of a function α and a vector field u on Ω. A field q is harmonic if α, u are continuous in Ω and ∇α = rot u, div u = 0 holds into Ω. The space 𝒞(Ω) of harmonic fields is a subspace of the Banach algebra 𝒬 (Ω) of continuous quaternion fields with the point-wise multiplication qq′ = {αα′ − u · u ′ , αu′ + α ′u + u ∧ u ′ }. We prove a Stone-Weierstrass type theorem: the subalgebra ∨𝒞(Ω) generated by harmonic fields is dense in 𝒬 (Ω). Some results on 2-jets of harmonic functions and the uniqueness sets of harmonic fields are provided. Comprehensive study of harmonic fields is motivated by possible applications to inverse problems of mathematical physics.

Методика решения обратных задач математической физики по определению эффективных физических свойств углеродных материалов абляционной тепловой защиты

Методика решения обратных задач математической физики по определению эффективных физических свойств углеродных материалов абляционной тепловой защиты

Ray Expansions and Identities for Second Order Equations. Applications to Inverse Problems

We develop a new method for studying inverse problems of mathematical physics by using systems of equations of ray expansion of solutions to parabolic and hyperbolic equations with variable coefficients.

The philosophical aspect of learning inverse problems of mathematical physics

The article describes specific questions student learning inverse problems of mathematical physics. When teaching inverse problems of mathematical physics to the understanding of the students brought the information that the inverse problems of mathematical physics with a philosophical point of view are the problems of determining the unknown causes of known consequences, and the search for their solutions have great scientific and educational potential. The reasons are specified in the form of unknown coefficients, right side, initial conditions of the mathematical model of inverse problems, and as a consequence are functionals of the solution of this mathematical model. In the process of learning the inverse problems of mathematical physics focuses on the philosophical aspects of the phenomenon of information and identify cause-effect relations. It is emphasized that in the process of logical analysis applied and humanitarian character, students realize that information is always related to the fundamental philosophical questions that the analysis applied and the humanitarian aspects of the obtained results the inverse problem of mathematical physics allows students to make appropriate inferences about the studied process and to, ultimately, new information, to study its properties and understand its value. Philosophical understanding of the notion of information opens up to students a new methodological opportunities to comprehend the world and helps us to reinterpret existing science and philosophy of the theory related to the disclosure of the interrelationship of all phenomena of reality.

On Problems in Mathematical Physics with Variable Parameter

We consider evolution equations with variable parameters. We obtain new representations of solutions and indicate their applications to inverse problems in mathematical physics.

Boundary control and tomography of Riemannian manifolds (the BC-method)

The BC-method provides one of the approaches to inverse problems of mathematical physics. A characteristic feature of this method is the great variety of interdisciplinary relations involved: in addition to partial differential equations as a source of problems, use is made of control theory and systems theory, asymptotic methods, functional analysis, operator theory, Banach algebras, and so on. The purpose of this paper is to present the principal ideas and tools of the BC-method and to give a survey of some results. One of the main achievements of the method is chosen for presentation: the reconstruction of Riemannian manifolds from dynamical and spectral boundary data. Bibliography: 108 titles.

Inverse problems in identification and modeling of thermal processes: Russian contributions

PurposeThe main purpose of this study, reflecting mainly the content of the authors’ plenary lecture, is to make a brief overview of several approaches developed by the author and his colleagues to the solution to ill-posed inverse heat transfer problems (IHTPs) with their possible extension to a wider class of inverse problems of mathematical physics and, most importantly, to show the wide possibilities of this methodology by examples of aerospace applications. In this regard, this study can be seen as a continuation of those applications that were discussed in the lecture.Design/methodology/approachThe application of the inverse method was pre-tested with experimental investigations on a special test equipment in laboratory conditions. In these studies, the author used the solution to the nonlinear inverse problem in the conjugate (conductive and convective) statement. The corresponding iterative algorithm has been developed and tested by a numerical and experimental way.FindingsIt can be stated that the theory and methodology of solving IHTPs combined with experimental simulation of thermal conditions is an effective tool for various fundamental and applied research and development in the field of heat and mass transfer.Originality/valueWith the help of the developed methods of inverse problems, the investigation was conducted for a porous cooling with a gaseous coolant for heat protection of the re-entry vehicle in the natural environment of hypersonic flight. Moreover, the analysis showed that the inverse methods can make a useful contribution to the study of heat transfer at the surface of a solid body under the influence of the hypersonic heterogeneous (dusty) gas stream and in many other aerospace applications.

Inverse Problems in Classes of Entire Functions

We study some inverse problems of mathematical physics in classes of entire functions. We investigate some questions of construction of solutions and source functions in described classes for the Maxwell system, Lame system and for integral Volterra equation of the first kind. Bibliography: 9 titles.

Computational experiment on the numerical solution of some inverse problems of mathematical physics

In this article the computational experiment on the numerical solution of the most popular linear inverse problems for equations of mathematical physics are presented. The discretization of retrospective inverse problem for parabolic equation is performed using difference scheme with non-positive weight multiplier. Similar difference scheme is also used for the numerical solution of Cauchy problem for two-dimensional Laplace equation. The results of computational experiment, performed on model problems with exact solution, including ones with randomly perturbed input data are presented and discussed.

Algebraic-Analytic Methods for Constructing Solutions to Differential Equations and Inverse Problems

We present new algebraic-analytic methods for constructing solutions to differential equations and inverse problems. In particular, we develop a new approach based on the ray method for inverse problems in mathematical physics. Bibliography: 23 titles.

Синтез намагничивающей системы актуатора с эффектом памяти формы

The article is devoted to the application of research methodology, based on the solution of inverse problems of mathematical physics, to the synthesis of magnetizing system actuators designed for pulsed systems stepper control. A feature of these devices is to use as the active element of a ferromagnetic shape memory alloy. Algorithms of determination of ampere-turns and optimization of a form of coils of the magnetizing system are offered. The algorithms are based on the determination of the magnetic field strength using the finite element method for the solution of the direct problem of magnetostatics and the gradient method of minimization of functional. The example of realization of a technique of synthesis of the magnetizing system is given.

Applied problems of the identification and regression analysis in stochastic structures with multimodal properties

We have been elaborating an approach founded on the identification of multimodal laws of the complex structure distribution in medicine, biology, chemistry of ultrapure materials and membrane technology as well as in technical applications. The method is based on the formulation and solution of inverse problems in mathematical physics for the respective probability density functions. The verification of the used algorithmic tools is carried out on model limited-scope samples. For stochastic structures and systems under study the method is supplemented with an original option of a regression analysis taking into account the identified stochastic laws displaying numerical parameters into the binary space. The proposed approach has been tested on clinical material in practical medicine.

On some Inverse Problem for a Parabolic Equation with a Parameter

Today inverse problems of mathematical physics play an important role in science and applications [1]. Coefficient inverse problems for parabolic equations are problems of finding solutions of differential equation with one (or more) unknown coefficients. An inverse problem for a parabolic equation with a parameter is investigated. The parameter has the same dimension as the spatial variable. Inverse problems with unknown parameter arise in various problems: in studying boundaryvalue problems for mixed-type equations and equation systems [2, 3]; in solving various inverse problems [4–7]; in studying boundary-value problems for equation systems with small parameters [8, 9].

Defining Thermophysical Parameters of Electric Devices Based on Solution of Inverse Heat Transfer Problem

This paper describes application of the study methods based on the solution of inverse problems of mathematical physics to define thermophysical parameters of electric devices. The mathematical model of the device is developed based on equations of non-stationary heat conductivity. The algorithm to define thermohysical parameters is developed; this algorithm uses the finite element method to solve a direct heat transfer problem and the gradient method to minimize the objective function. Examples of the algorithm application are given. The problem to define an equivalent heat transfer coefficient of the solenoid area covered with heavy winding and a heat emission from its inside surface is considered. In the second example thermophysical parameters of electromagnetic valve actuator of an ICE (internal combustion engine) gas distribution mechanism are defined. The obtained results show that thermophysical parameters and temperature distribution in non-stationary and steady-state operating conditions of electrical devices may be evaluated with adequate efficiency based on the solution of inverse heat transfer problems.

Выравнивание напряженности электриче- ского поля в изоляции пазовых частей обмоток электрических машин

This paper describes application of the study methods based on the solution of inverse problems of mathematical physics to improve the reliability of electrical systems with multi-layer insulation. Techniques of alignment of intensity of electric field and alignment of margin of safety in isolation of angular zones of slotted area of windings of electrical machines are developed. The technique is based on the determination of the field strength using the finite element method for the solution of the direct problem of electrostatics and the gradient method for minimizing the target functional. Depending on the objectives of the research, three options are functional, taking into account the permissible maximum, the average field intensity for all layers of insulation and different maximum permissible field strength for the individual layers. Example of the techniques application is given. The results show that the alignment of intensity of electric field and alignment of margin of safety in isolation in order to increase the reliability of electrical systems can be effectively on a basis of the solution of inverse problems of electrostatics.