Artigos de revistas: "Direct and inverse-Problem solving"

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Sorokin, S. B. "Direct method for solving the inverse coefficient problem". Sibirskii zhurnal industrial'noi matematiki 24, n.º 2 (18 de junho de 2021): 134–47. http://dx.doi.org/10.33048/sibjim.2021.24.211.

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Nikitin, A. V., L. V. Mikhaylov, A. V. Mikhaylov, Yu L. Gobov, V. N. Kostin e Ya G. Smorodinskii. "Reconstruction of the shape of a flaw in ferromagnetic plate by solving inverse problem of magnetostatics and series of direct problems". Defektoskopiâ, n.º 9 (2 de outubro de 2024): 67–72. http://dx.doi.org/10.31857/s0130308224090086.

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The article presents a verification technique for solving the inverse geometric problem of magnetostatics in a soft magnetic ferromagnet plate. The technique involves solving a number of direct problems, in which the shape of the defect obtained by solving the inverse geometric problem of magnetostatics is used as a first approximation, and then increasing or decreasing the depth of the defect without changing the shape of the boundary surface — comparing the topographies of the magnetic field components obtained during measurements above the plate surface and calculated (as a result of solving the direct problem) at the same points of the components of the magnetic stray field from the reconstructed three-dimensional defect. As a result of applying the technique, the geometric parameters of the defect under study can also be refined. Obtaining the initial conditions for solving the inverse problem and solving direct problems of magnetostatics is carried out using the finite element method in the ELMER program. The technique works with one-sided access to any surface of the plate (a defect-free surface or a surface with a defect).

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Xue, Qi Wen, Xiu Yun Du e Ga Ping Wang. "Solving the Inverse Heat Conduction Problem with Multi-Variables". Advanced Materials Research 168-170 (dezembro de 2010): 195–99. http://dx.doi.org/10.4028/www.scientific.net/amr.168-170.195.

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This paper presents a general numerical model to solve non-linear inverse heat conduction problems with multi-variables which include thermal parameters and boundary conditions, and can be identified singly or simultaneously. The direct problems are numerically modeled via FEM, facilitating to sensitivity analysis that is required in solving inverse problems via a least-square based CGM (Conjugate Gradient Method). Inhomogeneous distribution of parameters is considered, and a number of numerical examples are given to illustrate the work proposed.

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Kravchenko, Vladislav V., e Lady Estefania Murcia-Lozano. "An Approach to Solving Direct and Inverse Scattering Problems for Non-Selfadjoint Schrödinger Operators on a Half-Line". Mathematics 11, n.º 16 (16 de agosto de 2023): 3544. http://dx.doi.org/10.3390/math11163544.

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In this paper, an approach to solving direct and inverse scattering problems on the half-line for a one-dimensional Schrödinger equation with a complex-valued potential that is exponentially decreasing at infinity is developed. It is based on a power series representation of the Jost solution in a unit disk of a complex variable related to the spectral parameter by a Möbius transformation. This representation leads to an efficient method of solving the corresponding direct scattering problem for a given potential, while the solution to the inverse problem is reduced to the computation of the first coefficient of the power series from a system of linear algebraic equations. The approach to solving these direct and inverse scattering problems is illustrated by several explicit examples and numerical testing.

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Askerbekova, J. A. "NUMERICAL ALGORITHM FOR SOLVING THE CONTINUATION PROBLEM FOR THE ACOUSTIC EQUATION". BULLETIN Series of Physics & Mathematical Sciences 70, n.º 2 (30 de junho de 2020): 7–13. http://dx.doi.org/10.51889/2020-2.1728-7901.01.

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In this paper we consider the initial-boundary value problem for the acoustics equation in the temporal-triangular domain. We reduce the original ill-posed problem to an equivalent inverse problem with respect to some direct problem. This direct problem is well-posed. The inverse problem is replaced by a minimization problem. An algorithm for solving the inverse problem by the Landweber iteration method is constructed. We apply the method of successive approximations to the equation, we obtain a natural extension to nonlinear problems. This method leads to optimal convergence rate in certain cases. An analysis of the iterative Landweber method for nonlinear problems depends on the source conditions and additional conditions. Convergence analysis and error estimates are usually made with many assumptions, which are very difficult to verify from a practical point of view. This method leads to optimal convergence rate under certain conditions. Theoretical analysis is confirmed by numerical results. Visual examples are processed numerically.

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Oyama, Eimei, Taro Maeda e Susumu Tachi. "A human system learning model for solving the inverse kinematics problem by direct inverse modeling". Systems and Computers in Japan 27, n.º 8 (1996): 53–68. http://dx.doi.org/10.1002/scj.4690270805.

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Chmielowska, Agata, Rafał Brociek e Damian Słota. "Reconstructing the Heat Transfer Coefficient in the Inverse Fractional Stefan Problem". Fractal and Fractional 9, n.º 1 (16 de janeiro de 2025): 43. https://doi.org/10.3390/fractalfract9010043.

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This paper presents an algorithm for solving the inverse fractional Stefan problem. The considered inverse problem consists of determining the heat transfer coefficient at one of the boundaries of the considered region. The additional information necessary for solving the inverse problem is the set of temperature values in selected points of the region. The fractional derivative with respect to time used in the considered Stefan problem is of the Caputo type. The direct problem was solved by using the alternating phase truncation method adapted to the model with the fractional derivative. To solve the inverse problem, the ant colony algorithm was used. This paper contains an example illustrating the accuracy and stability of the presented algorithm.

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Temirbekov, N. М., S. I. Kabanikhin, L. N. Тemirbekova e Zh E. Demeubayeva. "Gelfand-Levitan integral equation for solving coefficient inverse problem". Bulletin of the National Engineering Academy of the Republic of Kazakhstan 85, n.º 3 (15 de setembro de 2022): 158–67. http://dx.doi.org/10.47533/2020.1606-146x.184.

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In this paper, numerical methods for solving multidimensional equations of hyperbolic type by the Gelfand-Levitan method are proposed and implemented. The Gelfand-Levitan method is one of the most widely used in the theory of inverse problems and consists in reducing a nonlinear inverse problem to a one-parameter family of linear Fredholm integral equations of the first and second kind. In the class of generalized functions, the initial-boundary value problem for a multidimensional hyperbolic equation is reduced to the Goursat problem. Discretization and numerical implementation of the direct Goursat problem are obtained to obtain additional information for solving a multidimensional inverse problem of hyperbolic type. For the numerical solution, a sequence of Goursat problems is used for each giveny. A comparative analysis of numerical experiments of the two-dimensional Gelfand-Levitan equation is performed. Numerical experiments are presented in the form of tables and figures for various continuous functions q(x, y).

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Shishlenin, M. A., N. S. Novikov e D. V. Klyuchinskiy. "On the recovering of acoustic attenuation in 2D acoustic tomography". Journal of Physics: Conference Series 2099, n.º 1 (1 de novembro de 2021): 012046. http://dx.doi.org/10.1088/1742-6596/2099/1/012046.

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Abstract The inverse problem of recovering the acoustic attenuation in the inclusions inside the human tissue is considered. The coefficient inverse problem is formulated for the first-order system of PDE. We reduce the inverse problem to the optimization of the cost functional by gradient method. The gradient of the functional is determined by solving a direct and conjugate problem. Numerical results are presented.

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Durdiev, D. K., e J. Z. Nuriddinov. "On investigation of the inverse problem for a parabolic integro-differential equation with a variable coefficient of thermal conductivity". Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 30, n.º 4 (dezembro de 2020): 572–84. http://dx.doi.org/10.35634/vm200403.

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The inverse problem of determining a multidimensional kernel of an integral term depending on a time variable $t$ and $ (n-1)$-dimensional spatial variable $x'=\left(x_1,\ldots, x_ {n-1}\right)$ in the $n$-dimensional heat equation with a variable coefficient of thermal conductivity is investigated. The direct problem is the Cauchy problem for this equation. The integral term has the time convolution form of kernel and direct problem solution. As additional information for solving the inverse problem, the solution of the direct problem on the hyperplane $x_n = 0$ is given. At the beginning, the properties of the solution to the direct problem are studied. For this, the problem is reduced to solving an integral equation of the second kind of Volterra-type and the method of successive approximations is applied to it. Further the stated inverse problem is reduced to two auxiliary problems, in the second one of them an unknown kernel is included in an additional condition outside integral. Then the auxiliary problems are replaced by an equivalent closed system of Volterra-type integral equations with respect to unknown functions. Applying the method of contraction mappings to this system in the Hölder class of functions, we prove the main result of the article, which is a local existence and uniqueness theorem of the inverse problem solution.

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Maksimov, M. A., e I. V. Surodina. "THREE-DIMENSIONAL FORWARD AND INVERSE MODELLING OF MULTI-HEIGHT MAGNETIC DATA USING THE RELIEF MAPS". Russian Journal of geophysical technologies, n.º 2 (5 de fevereiro de 2020): 4–11. http://dx.doi.org/10.18303/2619-1563-2019-2-4.

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We have developed the high-performance software and algorithmic software for solving the direct and inverse problems of modeling and inversion of spatially distributed magnetic, taking into account the relief. We show the results of solving the inverse problem of magnetic survey taking into account the relief for synthetic noisy data.

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Akimova, Elena N., Murat A. Sultanov, Vladimir E. Misilov e Yerkebulan Nurlanuly. "Parallel Algorithm for Solving the Inverse Two-Dimensional Fractional Diffusion Problem of Identifying the Source Term". Fractal and Fractional 7, n.º 11 (2 de novembro de 2023): 801. http://dx.doi.org/10.3390/fractalfract7110801.

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This paper is devoted to the development of a parallel algorithm for solving the inverse problem of identifying the space-dependent source term in the two-dimensional fractional diffusion equation. For solving the inverse problem, the regularized iterative conjugate gradient method is used. At each iteration of the method, we need to solve the auxilliary direct initial-boundary value problem. By using the finite difference scheme, this problem is reduced to solving a large system of a linear algebraic equation with a block-tridiagonal matrix at each time step. Solving the system takes almost the entire computation time. To solve this system, we construct and implement the direct parallel matrix sweep algorithm. We establish stability and correctness for this algorithm. The parallel implementations are developed for the multicore CPU using the OpenMP technology. The numerical experiments are performed to study the performance of parallel implementations.

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Lukyanenko, Dmitry V., Igor V. Prigorniy e Maxim A. Shishlenin. "Some features of solving an inverse backward problem for a generalized Burgers’ equation". Journal of Inverse and Ill-posed Problems 28, n.º 5 (1 de novembro de 2020): 641–49. http://dx.doi.org/10.1515/jiip-2020-0078.

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AbstractIn this paper, we consider an inverse backward problem for a nonlinear singularly perturbed parabolic equation of the Burgers’ type. We demonstrate how a method of asymptotic analysis of the direct problem allows developing a rather simple algorithm for solving the inverse problem in comparison with minimization of the cost functional. Numerical experiments demonstrate the effectiveness of this approach.

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Wu, Dongqing, e Yunong Zhang. "From Penrose Equations to Zhang Neural Network, Getz–Marsden Dynamic System, and DDD (Direct Derivative Dynamics) Using Substitution Technique". Discrete Dynamics in Nature and Society 2021 (20 de novembro de 2021): 1–21. http://dx.doi.org/10.1155/2021/4227512.

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The time-variant matrix inversion (TVMI) problem solving is the hotspot of current research because of its frequent appearance and application in scientific research and industrial production. The generalized inverse problem of singular square matrix and nonsquare matrix can be related to Penrose equations (PEs). The PEs implicitly define the generalized inverse of a known matrix, which is of fundamental theoretical significance. Therefore, the in-depth study of PEs might enlighten problem solving of TVMI in a foreseeable way. For the first time, we construct three different matrix error-monitoring functions based on PEs and propose the corresponding models for TVMI problem solving by using the substitution technique and ZNN design formula. In order to facilitate computer simulation, the obtained continuous-time models are discretized by using ZTD (Zhang time discretization) formulas. Furthermore, the feasibility and effectiveness of the novel Zhang neural network (ZNN) multiple-multiplication model for matrix inverse (ZMMMI) and the PEs-based Getz–Marsden dynamic system (PGMDS) model in solving the problem of TVMI are investigated and shown via theoretical derivation and computer simulation. Computer experiment results also illustrate that the direct derivative dynamics model for TVMI is less effective and feasible.

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Hou, Songming, Yihong Jiang e Yuan Cheng. "Direct and Inverse Scattering Problems for Domains with Multiple Corners". International Journal of Partial Differential Equations 2015 (26 de janeiro de 2015): 1–9. http://dx.doi.org/10.1155/2015/968529.

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We proposed numerical methods for solving the direct and inverse scattering problems for domains with multiple corners. Both the near field and far field cases are considered. For the forward problem, the challenges of logarithmic singularity from Green’s functions and corner singularity are both taken care of. For the inverse problem, an efficient and robust direct imaging method is proposed. Multiple frequency data are combined to capture details while not losing robustness.

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Serat, Amel. "Innovative Solutions for IK: PROA and Clonal Selection Algorithms Unveiled". WSEAS TRANSACTIONS ON INFORMATION SCIENCE AND APPLICATIONS 21 (5 de novembro de 2024): 514–23. http://dx.doi.org/10.37394/23209.2024.21.47.

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Calculating joint angles for sequential manipulators consists of studying the correlation between Cartesian and joint variables. The problem-solving technique encounters two main hurdles described as direct and inverse kinematics. Matrix multiplications usually simplify the direct kinematic problem. However, inverse kinematic problems are harder as they require solving many nonlinear equations and eliminating variables a lot. In our work, we introduce two new methods of handling the complicated inverse kinematic problem for robotic manipulator arms; Poor and Rich Optimization Algorithm and Clonal Selection Algorithm (CSA). These advanced techniques enhance greatly the estimation of various joints in the arm which makes the solution more precise and efficient. To demonstrate the effectiveness, robustness, and potential benefits of these approaches for complicated kinematic problems we present extensive simulation results thereby enabling better performance of robots.

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Weglein, Arthur B. "A direct inverse method for subsurface properties: The conceptual and practical benefit and added value in comparison with all current indirect methods, for example, amplitude-variation-with-offset and full-waveform inversion". Interpretation 5, n.º 3 (31 de agosto de 2017): SL89—SL107. http://dx.doi.org/10.1190/int-2016-0198.1.

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Direct inverse methods solve the problem of interest; in addition, they communicate whether the problem of interest is the problem that we (the seismic industry) need to be interested in. When a direct solution does not result in an improved drill success rate, we know that the problem we have chosen to solve is not the right problem — because the solution is direct and cannot be the issue. On the other hand, with an indirect method, if the result is not an improved drill success rate, then the issue can be either the chosen problem, or the particular choice within the plethora of indirect solution methods, or both. The inverse scattering series (ISS) is the only direct inversion method for a multidimensional subsurface. Solving a forward problem in an inverse sense is not equivalent to a direct inverse solution. All current methods for parameter estimation, e.g., amplitude-variation-with-offset and full-waveform inversion, are solving a forward problem in an inverse sense and are indirect inversion methods. The direct ISS method for determining earth material properties defines the precise data required and the algorithms that directly output earth mechanical properties. For an elastic model of the subsurface, the required data are a matrix of multicomponent data, and a complete set of shot records, with only primaries. With indirect methods, any data can be matched: one trace, one or several shot records, one component, multicomponent, with primaries only or primaries and multiples. Added to that are the innumerable choices of cost functions, generalized inverses, and local and global search engines. Direct and indirect parameter inversion are compared. The direct ISS method has more rapid convergence and a broader region of convergence. The difference in effectiveness increases as subsurface circumstances become more realistic and complex, in particular with band-limited noisy data.

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Гусев, О. В. "SOLVING THE DIRECT KINEMATIC PROBLEM FOR A SIX-UNIT ROBOT MANIPULATOR". Proceedings in Cybernetics 23, n.º 2 (2024): 39–48. http://dx.doi.org/10.35266/1999-7604-2024-2-5.

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The study describes steps to solve the direct kinematic problem for a six-unit robot manipulator, the FANUC Robot M-20iA/35M. The problem solving is based on modern solid CAD modeling technologies combined with a physical modeling environment, as well as Simulink’s SimMechanics multi-unit spatial mechanisms. Simulink’s SimMechanics environment is used for visualizing the dynamics of the manipulator’s operating component. The manipulator’s matrix equation can then be used for solving the inverse kinematic problem.

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Litvinov, V. L., e K. V. Litvinova. "On one solution of the problem of vibrations of mechanical systems with moving boundaries". Vestnik of Samara University. Natural Science Series 30, n.º 1 (24 de abril de 2024): 40–49. http://dx.doi.org/10.18287/2541-7525-2024-30-1-40-49.

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An analytical method of solving the wave equation describing the oscillations of systems with moving boundaries is considered. By changing the variables that stop the boundaries and leave the equation invariant, the original boundary value problem is reduced to a system of functional-difference equations, which can be solved using direct and inverse methods. An inverse method is described that makes it possible to approximate quite diverse laws of boundary motion by laws obtained from solving the inverse problem. New particular solutions are obtained for a fairly wide range of laws of boundary motion. A direct asymptotic method for the approximate solution of a functional equation is considered. An estimate of the errors of the approximatemethod was made depending on the speed of the boundary movement.

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Argun, Raul, Alexandr Gorbachev, Natalia Levashova e Dmitry Lukyanenko. "Inverse Problem for an Equation of the Reaction-Diffusion-Advection Type with Data on the Position of a Reaction Front: Features of the Solution in the Case of a Nonlinear Integral Equation in a Reduced Statement". Mathematics 9, n.º 18 (21 de setembro de 2021): 2342. http://dx.doi.org/10.3390/math9182342.

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The paper considers the features of numerical reconstruction of the advection coefficient when solving the coefficient inverse problem for a nonlinear singularly perturbed equation of the reaction-diffusion-advection type. Information on the position of a reaction front is used as data of the inverse problem. An important question arises: is it possible to obtain a mathematical connection between the unknown coefficient and the data of the inverse problem? The methods of asymptotic analysis of the direct problem help to solve this question. But the reduced statement of the inverse problem obtained by the methods of asymptotic analysis contains a nonlinear integral equation for the unknown coefficient. The features of its solution are discussed. Numerical experiments demonstrate the possibility of solving problems of such class using the proposed methods.

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Durdiev, Durdimurod, e Zhonibek Zhumaev. "ON DETERMINATION OF THE COEFFICIENT AND KERNEL IN AN INTEGRO -DIFFERENTIAL EQUATION OF PARABOLIC TYPE". Eurasian Journal of Mathematical and Computer Applications 11, n.º 1 (março de 2023): 49–65. http://dx.doi.org/10.32523/2306-6172-2023-11-1-49-65.

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The inverse problem of determination of x-dependent coefficient a(x) at u and the kernel k(t) functions in the one-dimensional integro–differential parabolic equation is investigated. The direct problem is the initial-boundary problem for this equation. Firstly, we studied the solvability of the direct problem, by used to the Fourier method and approximation series methods. As additional information for solving inverse problem, the solution of the direct problem by over determining condition is given. The problem is reduced to an equivalent closed system of Volterra-type integral equations with respect to unknown functions. Applying the method of contraction mappings to this system in the continuous class of functions, we prove the main result of the article, which is a local existence and uniqueness theorem of inverse problem solution.

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Pellegrini, Sergio P., Flavio C. Trigo e Raul G. Lima. "Solving the electrical impedance tomography inverse problem for logarithmic conductivity". COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 37, n.º 2 (5 de março de 2018): 681–90. http://dx.doi.org/10.1108/compel-11-2016-0501.

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PurposeIn the context of electrical impedance tomography (EIT), this paper aims to evaluate limitations of estimating conductivity or resistivity, as well as the improvements achieved with the use of an alternate description of the solution space, the logarithmic conductivity. Design/methodology/approachA quantitative analysis is performed, solving the inverse EIT problem by using the Gauss–Newton and non-linear conjugate gradient methods for a numerical phantom of 15 elements. A property of symmetry is studied for the direct EIT problem for a phantom of 385,601 elements. FindingsSolving the inverse EIT problem in logarithmic conductivity is more robust to the initial guess, as solutions are kept within physical bounds (conductivity positiveness). Also, convergence is faster and less dependent on the final values of the estimates. Research limitations/implicationsLogarithmic conductivity provides an advantageous description of the solution space for the EIT inverse problem. Similar estimation problems might be subject to analogous conclusions. Originality/valueThis study provides a novel analysis, quantitatively comparing the effect of different variables to solve the inverse EIT problem.

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Bezgachev, M. V., M. A. Shishlenin e A. V. Sokolov. "Identification of a Mathematical Model of Economic Development of Two Regions of the World". Bulletin of Irkutsk State University. Series Mathematics 47 (2024): 12–30. http://dx.doi.org/10.26516/1997-7670.2024.47.12.

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This paper is devoted to solving the inverse problem (determining the parameters of a system of ordinary differential equations based on additional information determined at discrete points in time) and analyzing its solution for a mathematical model describing the dynamics of changes in the population and capital of two regions of the world. The inverse problem is reduced to the problem of minimizing the target functional and is solved by the method of differential evolution. A numerical method for solving direct and inverse problems is implemented. The developed method was tested on model and real data for countries such as Russia, China, India and the USA.

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Mathis, R., e Y. Re´mond. "A New Approach to Solving the Inverse Problem for Compound Gear Trains". Journal of Mechanical Design 121, n.º 1 (1 de março de 1999): 98–106. http://dx.doi.org/10.1115/1.2829436.

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The calculation of a gear train, if the concept is given (resolution of the direct problem) never poses fundamental problems, but is no help in its invention. The inverse problem (i.e., conception of a gear train if the kinematics are fixed) having never been approached in an analytical way, we propose, from the general formula of the ratios, one equation which allows us to resolve the inverse problem for every elementary or two-set nested gear train.

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Kasenov, Syrym E., Aigerim M. Tleulesova, Ainur E. Sarsenbayeva e Almas N. Temirbekov. "Numerical Solution of the Cauchy Problem for the Helmholtz Equation Using Nesterov’s Accelerated Method". Mathematics 12, n.º 17 (23 de agosto de 2024): 2618. http://dx.doi.org/10.3390/math12172618.

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In this paper, the Cauchy problem for the Helmholtz equation, also known as the continuation problem, is considered. The continuation problem is reduced to a boundary inverse problem for a well-posed direct problem. A generalized solution to the direct problem is obtained and an estimate of its stability is given. The inverse problem is reduced to an optimization problem solved using the gradient method. The convergence of the Landweber method with respect to the functionals is compared with the convergence of the Nesterov method. The calculation of the gradient in discrete form, which is often used in the numerical solutions of the inverse problem, is described. The formulation of the conjugate problem in discrete form is presented. After calculating the gradient, an algorithm for solving the inverse problem using the Nesterov method is constructed. A computational experiment for the boundary inverse problem is carried out, and the results of the comparative analysis of the Landweber and Nesterov methods in a graphical form are presented.

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Huang, J., A. V. Grigorev e D. Kh Ivanov. "Numerical methods for identifying the diffusion coefficient in a nonlinear elliptic equation". Журнал «Математические заметки СВФУ», n.º 1(109) (30 de março de 2021): 78–92. http://dx.doi.org/10.25587/svfu.2021.81.41.007.

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Two different approaches for solving a nonlinear coefficient inverse problem are investigated in this paper. As a classical approach, we use the finite element method to discretize the direct and inverse problems and solve the inverse problem by the conjugate gradient method. Meanwhile, we also apply the neural network approach to recover the coefficient of the inverse problem, which is to map measurements at some fixed points and the unknown coefficient. According to the results of applying the two approaches, our methods are shown to solve the nonlinear coefficient inverse problem efficiently, even with perturbed data.

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Lapich, A. O., e M. Y. Medvedik. "Algorithm for Searching Inhomogeneities in Inverse Nonlinear Diﬀraction Problems". Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki 166, n.º 3 (6 de outubro de 2024): 395–406. http://dx.doi.org/10.26907/2541-7746.2024.3.395-406.

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This study aims to solve the inverse problem for determining the heterogeneity of an object. The scattered ﬁeld was measured outside its boundaries at a set of observation points. Both the radiation source and observation points were assumed to be located outside the object. The scattered ﬁeld was modeled by solving the direct problem. The inverse problem was solved using a two-step method. Nonlinearities of various types were considered. When introducing the computational grid, the generalized grid method was applied. A numerical method for solving the problem was proposed and implemented. The numerical results obtained illustrate how the problem is solved for speciﬁed experimental data.

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Roitberg, Inna, e Alexander Sakhnovich. "The Discrete Self-Adjoint Dirac Systems of General Type: Explicit Solutions of Direct and Inverse Problems, Asymptotics of Verblunsky-Type Coefficients and the Stability of Solving of the Inverse Problem". Zurnal matematiceskoj fiziki, analiza, geometrii 14, n.º 4 (25 de dezembro de 2018): 532–48. http://dx.doi.org/10.15407/mag14.04.532.

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Novikov, N. S., D. V. Klyuchinskiy, M. A. Shishlenin e S. I. Kabanikhin. "On the modeling of ultrasound wave propagation in the frame of inverse problem solution". Journal of Physics: Conference Series 2099, n.º 1 (1 de novembro de 2021): 012044. http://dx.doi.org/10.1088/1742-6596/2099/1/012044.

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Abstract In this paper we consider the inverse problem of detecting the inclusions inside the human tissue by using the acoustic sounding wave. The problem is considered in the form of coefficient inverse problem for first-order system of PDE and we use the gradient descent approach to recover the coefficients of that system. The important part of the sceme is the solution of the direct and adjoint problem on each iteration of the descent. We consider two finite-volume methods of solving the direct problem and study their Influence on the performance of recovering the coefficients.

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Durdiev, D. K. "Inverse coefficient problem for the time-fractional diffusion equation". EURASIAN JOURNAL OF MATHEMATICAL AND COMPUTER APPLICATIONS 9, n.º 1 (2021): 44–54. http://dx.doi.org/10.32523/2306-6172-2021-9-1-44-54.

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We study the inverse problem of determining the time depending reaction diffu- sion coefficient in the Cauchy problem for the time-fractional diffusion equation by a single observation at the point x = 0 of the diffusion process. To represent the solution of the direct problem, the fundamental solution of the time-fractional diffusion equation is used and properties of this solution are investigated. The fundamental solution contains the Fox’s H− functions widely used in fractional calculus. In particular, using estimates of the fundamental solution and its derivatives, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown coefficient which will be used in study inverse problem. The inverse problem is reduced to the equivalent integral equation. For solving this equation the contracted mapping principle is applied. The local existence and global uniqueness results are proven. Also the stability estimate is obtained.

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Gessese, Alelign, e Mathieu Sellier. "A Direct Solution Approach to the Inverse Shallow-Water Problem". Mathematical Problems in Engineering 2012 (2012): 1–18. http://dx.doi.org/10.1155/2012/417950.

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The study of open channel flow modelling often requires an accurate representation of the channel bed topography to accurately predict the flow hydrodynamics. Experimental techniques are the most widely used approaches to measure the bed topographic elevation of open channels. However, they are usually cost and time consuming. Free surface measurement is, on the other hand, relatively easy to obtain using airborne photographic techniques. We present in this work an easy to implement and fast to solve numerical technique to identify the underlying bedrock topography from given free surface elevation data in shallow open channel flows. The main underlying idea is to derive explicit partial differential equations which govern this inverse reconstruction problem. The technique described here is a “one-shot technique” in the sense that the solution of the partial differential equation provides the solution to the inverse problem directly. The idea is tested on a set of artificial data obtained by first solving the forward problem governed by the shallow-water equations. Numerical results show that the channel bed topographic elevation can be reconstructed with a level of accuracy less than 3%. The method is also shown to be robust when noise is present in the input data.

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Vagin, Denis. "The structure and features of the software for geophysical geometrical 3D inversions". Analysis and data processing systems, n.º 2 (18 de junho de 2021): 35–46. http://dx.doi.org/10.17212/2782-2001-2021-2-35-46.

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The structure and features of a software package for 3D inversion of geophysical data are considered. The presented software package is focused on solving direct and inverse problems of electrical exploration and engineering geophysics. In addition to the parameters that determine physical properties of the medium, the software package allows you to restore the geometry parameters of the geophysical model, namely layer reliefs and boundaries of three-dimensional inclusions. The inclusions can be in the form of arbitrary hexagons or prisms with a polygonal base. The software package consists of four main subsystems: an interface, subsystems for solving direct and inverse problems, and a client-server part for performing calculations on remote computing nodes. The graphical interface consists of geophysicist-oriented pre- and postprocessor modules that allow you to describe the problem and present the results of its solution in user-friendly terms. To solve direct problems, the finite element method and the technology for dividing the field into normal and anomalous components are used. At the same time, special methods of discretization of the computational domain are used, which make it possible to take into account both the complex three-dimensional structure of the environment and the presence of man-made objects (wells) in the computational domain. To increase the efficiency of solving direct problems, nonconforming grids with cells in the form of arbitrary hexahedrons are used. Methods for efficient calculation of derivatives (with respect to these parameters) necessary for solving inverse problems by the Gauss-Newton method are also described for the geometry parameters. The main idea for efficient derivatives computation is to identify the effect of changing the value of the parameter (used to compute the value of the generalized derivative) on the problem. The main actions performed by the subsystem for solving inverse problems and the features associated with the processing of geometry parameters are described.

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Foddis, Maria Laura, Philippe Ackerer, Augusto Montisci e Gabriele Uras. "ANN-based approach for the estimation of aquifer pollutant source behaviour". Water Supply 15, n.º 6 (30 de junho de 2015): 1285–94. http://dx.doi.org/10.2166/ws.2015.087.

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The problem of identifying an unknown pollution source in polluted aquifers, based on known contaminant concentration measurements, is part of the broader group of issues called inverse problems. This paper investigates the feasibility of solving the groundwater pollution inverse problem by using artificial neural networks (ANNs). The approach consists first in training an ANN to solve the direct problem, in which the pollutant concentration in a set of monitoring wells is calculated for a known pollutant source. Successively, the trained ANN is frozen and is used to solve the inverse problem, where the pollutant source is calculated which corresponds to a set of concentrations in the monitoring wells. The approach has been applied for a real case which deals with the contamination of the Rhine aquifer by carbon tetrachloride (CCl4) due to a tanker accident. The obtained results are compared with the solution obtained with a different approach retrieved from literature. The results show the suitability of ANN-based methods for solving inverse non-linear problems.

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Melničenko, Grigorijus. "Direct and inverse factorization algorithms of numbers". Lietuvos matematikos rinkinys 60 (5 de dezembro de 2019): 39–45. http://dx.doi.org/10.15388/lmr.b.2019.15234.

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The factoring natural numbers into factors is a complex computational task. The complexity of solving this problem lies at the heart of RSA security, one of the most famous cryptographic methods. The classical trial division algorithm divides a given number N into all divisors, starting from 2 and to integer part of √N. Therefore, this algorithm can be called the direct trial division algorithm. We present the inverse trial division algorithm, which divides a given number N into all divisors,starting from the integer part of √N to 2.

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Weglein, Arthur B., Haiyan Zhang, Adriana C. Ramírez, Fang Liu e Jose Eduardo Lira. "Clarifying the underlying and fundamental meaning of the approximate linear inversion of seismic data". GEOPHYSICS 74, n.º 6 (novembro de 2009): WCD1—WCD13. http://dx.doi.org/10.1190/1.3256286.

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Linear inversion is defined as the linear approximation of a direct-inverse solution. This definition leads to data requirements and specific direct-inverse algorithms, which differ with all current linear and nonlinear approaches, and is immediately relevant for target identification and inversion in an elastic earth. Common practice typically starts with a direct forward or modeling expression and seeks to solve a forward equation in an inverse sense. Attempting to solve a direct forward problem in an inverse sense is not the same as solving an inverse problem directly. Distinctions include differences in algorithms, in the need for a priori information, and in data requirements. The simplest and most accessible examples are the direct-inversion tasks, derived from the inverse scattering series (ISS), for the removal of free-surface and internal multiples. The ISS multiple-removal algorithms require no subsurface information, and they are independent of earth model type. A direct forward method solved in an inverse sense, for modeling and subtracting multiples, would require accurate knowledge of every detail of the subsurface the multiple has experienced. In addition, it requires a different modeling and subtraction algorithm for each different earth-model type. The ISS methods for direct removal of multiples are not a forward problem solved in an inverse sense. Similarly, the direct elastic inversion provided by the ISS is not a modeling formula for PP data solved in an inverse sense. Direct elastic inversion calls for [Formula: see text], [Formula: see text], [Formula: see text], … data, for direct linear and nonlinear estimates of changes in mechanical properties. In practice, a judicious combination of direct and indirect methods are called upon for effective field data application.

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Gurevich, Garold, Sergey Lutmanov e Oleg Penskiy. "Mathematical models of centers of equal pressure in stellar systems". Вестник Пермского университета. Математика. Механика. Информатика, n.º 3(54) (2021): 25–30. http://dx.doi.org/10.17072/1993-0550-2021-3-25-30.

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Mathematical models are proposed that allow calculating the coordinates of the centers of equal pressure in stellar systems and solving the inverse problem of determining the radiation sources of a material substance during the formation of macro-bodies. It is shown that the solutions of direct and inverse problems are not unique.

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Lukyanenko, Dmitry V., Maxim A. Shishlenin e Vladimir T. Volkov. "Asymptotic analysis of solving an inverse boundary value problem for a nonlinear singularly perturbed time-periodic reaction-diffusion-advection equation". Journal of Inverse and Ill-posed Problems 27, n.º 5 (1 de outubro de 2019): 745–58. http://dx.doi.org/10.1515/jiip-2017-0074.

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Abstract In this paper, a new asymptotic-numerical approach to solving an inverse boundary value problem for a nonlinear singularly perturbed parabolic equation with time-periodic coefficients is proposed. An unknown boundary condition is reconstructed by using known additional information about the location of a moving front. An asymptotic analysis of the direct problem allows us to reduce the original inverse problem to that with a simpler numerical solution. Numerical examples demonstrate the efficiency of the method.

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Vityaev, Evgeny E., Sergey S. Goncharov e Dmitry I. Sviridenko. "On the task approach in artificial intelligence". Siberian Journal of Philosophy 17, n.º 4 (2019): 5–25. http://dx.doi.org/10.25205/2541-7517-2019-17-4-5-25.

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The authors discuss the problem of the integration approach to artificial intelligence, analyzing the content and positive aspects of the integration agent approach. It is noted that this approach implicitly follows the task approach. The paper gives answers to the questions that make up the essence of the task approach - where do the tasks come from, what is the task, what should be considered a solution to the problem. It also discusses the classification of intellectual problems into direct, inverse, and hybrid. It is noted that modern artificial intelligence focuses mainly on solving direct and inverse problems, leaving a huge and important class of hybrid problems outside its scope of attention. The paper describes the theoretical model approach to solving the whole variety of intellectual problems, called semantic modeling. It analyzes the advantages of the proposed conception, including the possibility of a flexible combination when solving hybrid problems of tools already created in artificial intelligence. It also discusses the problem of creating a “strong” / “general” artificial intelligence (AGI) in the framework of the task approach.

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Bakanov, G. B., e S. K. Meldebekova. "Discrete inverse problem for a hyperbolic equation, properties of the solution to a discrete direct and auxiliary discrete problems". Q A Iasaýı atyndaǵy Halyqaralyq qazaq-túrіk ýnıversıtetіnіń habarlary (fızıka matematıka ınformatıka serııasy) 29, n.º 2 (30 de junho de 2024): 7–18. http://dx.doi.org/10.47526/2024-2/2524-0080.01.

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This paper considers the formulation of a discrete inverse problem for a hyperbolic equation. First, the continuous inverse problem is reduced to a convenient form for research. In the inverse problem, the required function is considered even. Since the Dirac delta function is present in the problem data, the structure of a generalized solution to the Cauchy problem for a hyperbolic equation is determined. The solution to the Cauchy problem for a hyperbolic equation is determined only for positive values in time, therefore the solution to the Cauchy problem for negative values in time is determined using odd continuation. After some transformations, the formulation of the continuous inverse problem is reduced to a form convenient for research. A grid domain is introduced, and for all functions in the problem statement the corresponding grid functions and a discrete analogue of the Dirac delta function are determined. Differential operators, initial conditions and additional data of the inverse problem are approximated by finite differences. Assuming that a solution to the discrete inverse problem exists, we prove the data lemma of the discrete inverse problem. In order to study the discrete inverse problem for a hyperbolic equation, a theorem on the existence and uniqueness of the discrete direct problem is proved, as well as on the properties of the solution to this discrete problem. In the course of proving the theorem, a discrete analogue of d'Alembert's formula for solving the Cauchy problem for a hyperbolic equation was obtained. The theorem on the existence of a unique solution to the auxiliary discrete problem and its properties is proved.

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Abgarian, K. K., R. G. Noskov e D. L. Reviznikov. "The inverse coefficient problem of heat transfer in layered nanostructures". Izvestiya Vysshikh Uchebnykh Zavedenii. Materialy Elektronnoi Tekhniki = Materials of Electronics Engineering 20, n.º 3 (9 de dezembro de 2017): 213–19. http://dx.doi.org/10.17073/1609-3577-2017-3-213-219.

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The rapid development of electronics leads to the creation and use of electronic components of small dimensions, including nanoelements of complex, layered structure. The search for effective methods for cooling electronic systems dictates the need for the development of methods for the numerical analysis of heat transfer in nanostructures. A characteristic feature of energy transfer in such systems is the dominant role of contact thermal resistance at interlayer interfaces. Since the contact resistance depends on a number of factors associated with the technology of heterostructures manufacturing, it is of great importance to determine the corresponding coefficients from the results of temperature measurements.The purpose of this paper is to evaluate the possibility of reconstructing the thermal resistance coefficients at the interfaces between layers by solving the inverse problem of heat transfer.The complex of algorithms includes two major blocks — a block for solving the direct heat transfer problem in a layered nanostructure and an optimization block for solving the inverse problem. The direct problem was formulated in an algebraic (finite difference) form under the assumption of a constant temperature within each layer, which is due to the small thickness of the layers. The inverse problem was solved in the extreme formulation, the optimization was carried out using zero-order methods that do not require the calculation of the derivatives of the optimized function. As a basic optimization algorithm, the Nelder—Mead method was used in combination with random restarts to search for a global minimum.The results of the identification of the contact thermal resistance coefficients obtained in the framework of a quasi-real experiment are presented. The accuracy of the identification problem solution is estimated as a function of the number of layers in the heterostructure and the «measurements» error.The obtained results are planned to be used in the new technique of multiscale modeling of thermal regimes of the electronic component base of the microwave range, when identifying the coefficients of thermal conductivity of heterostructure.

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Ablabekov, B. S., e A. T. Mukanbetova. "ON SOLVABILITY OF THE BOUNDARY INVERSE PROBLEM FOR THE HEAT CONDUCTIVITY EQUATION". Heralds of KSUCTA, №4, 2021, n.º 4-2021 (27 de dezembro de 2021): 670–76. http://dx.doi.org/10.35803/1694-5298.2021.4.670-676.

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In applied problems related to the study of non-stationary thermal processes, quite often a situation arises when it is impossible to carry out direct measurements of the required physical quantity and its characteristics are restored from the results of indirect measurements. In this case, the only way to find the required values is related to the solution of the inverse problem of heat conduction with the initial data known only on a part of the boundary. Problems of this kind arise not only in the study of thermal processes, but also in the study of diffusion processes, the study of the properties of materials associated with thermal characteristics. The article is devoted to solving the boundary inverse problem for solving the heat equation.

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Nubiola, Albert, e Ilian A. Bonev. "GEOMETRIC APPROACH TO SOLVING THE INVERSE DISPLACEMENT PROBLEM OF CALIBRATED DECOUPLED 6R SERIAL ROBOTS". Transactions of the Canadian Society for Mechanical Engineering 38, n.º 1 (março de 2014): 31–44. http://dx.doi.org/10.1139/tcsme-2014-0003.

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This paper presents a simple but efficient way to numerically calculate the inverse displacement problem of calibrated decoupled 6R serial robots. The method is iterative and works with any type of calibrated robot model, such as level-3 models, since it requires no algebraic computation and no resolution of high-order polynomials, only the computation of the forward displacement problem of the calibrated robot model and the inverse kinematics of the nominal robot model. The method proposed can find up to eight possible solutions for a given end-effector pose. A numerical example is presented, with one million arbitrary end-effector poses of a level-3 calibrated ABB IRB 120 robot. The computation time for solving the inverse problem is analyzed, and in most cases is found to be only four times the time needed to calculate the nominal inverse kinematics and the calibrated direct kinematics. Furthermore, the method is fast enough to be implemented directly into the robot controller using the RAPID programming language.

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Yermekkyzy, L. "VARIATIVE SOLUTION OF THE COEFFICIENT INVERSE PROBLEM FOR THE HEAT EQUATIONS". BULLETIN Series of Physics & Mathematical Sciences 72, n.º 4 (29 de dezembro de 2020): 23–27. http://dx.doi.org/10.51889/2020-4.1728-7901.03.

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One of the main types of inverse problems for partial differential equations are problems in which the coefficients of the equations or the quantities included in them must be determined using some additional information. Such problems are called coefficient inverse problems for partial differential equations. Coefficient inverse problems (identification problems) have become the subject of close study, especially in recent years. Interest in them is caused primarily by their important applied values. They find applications in solving problems of planning the development of oil fields (determining the filtration parameters of fields), in creating new types of measuring equipment, in solving problems of environmental monitoring, etc. The standard formulation of the coefficient inverse problem contains a functional (discrepancy), physics. When formulating the statements of inverse problems, the statements of direct problems are assumed to be known. The solution to the problem is sought from the condition of its minimum. Inverse problems for partial differential equations can be posed in variational form, i.e., as optimal control problems for the corresponding systems. A variational statement of one coefficient inverse problem for a one-dimensional heat equation is considered. By the solution of the boundary value problem for each fixed control coefficient we mean a generalized solution from the Sobolev space. The questions of correctness of the considered coefficient inverse problem in the variational setting are investigated.

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Kamyab, Shima, Zohreh Azimifar, Rasool Sabzi e Paul Fieguth. "Deep learning methods for inverse problems". PeerJ Computer Science 8 (2 de maio de 2022): e951. http://dx.doi.org/10.7717/peerj-cs.951.

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In this paper we investigate a variety of deep learning strategies for solving inverse problems. We classify existing deep learning solutions for inverse problems into three categories of Direct Mapping, Data Consistency Optimizer, and Deep Regularizer. We choose a sample of each inverse problem type, so as to compare the robustness of the three categories, and report a statistical analysis of their differences. We perform extensive experiments on the classic problem of linear regression and three well-known inverse problems in computer vision, namely image denoising, 3D human face inverse rendering, and object tracking, in presence of noise and outliers, are selected as representative prototypes for each class of inverse problems. The overall results and the statistical analyses show that the solution categories have a robustness behaviour dependent on the type of inverse problem domain, and specifically dependent on whether or not the problem includes measurement outliers. Based on our experimental results, we conclude by proposing the most robust solution category for each inverse problem class.

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Szénási, Sándor. "Solving the inverse heat conduction problem using NVLink capable Power architecture". PeerJ Computer Science 3 (20 de novembro de 2017): e138. http://dx.doi.org/10.7717/peerj-cs.138.

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The accurate knowledge of Heat Transfer Coefficients is essential for the design of precise heat transfer operations. The determination of these values requires Inverse Heat Transfer Calculations, which are usually based on heuristic optimisation techniques, like Genetic Algorithms or Particle Swarm Optimisation. The main bottleneck of these heuristics is the high computational demand of the cost function calculation, which is usually based on heat transfer simulations producing the thermal history of the workpiece at given locations. This Direct Heat Transfer Calculation is a well parallelisable process, making it feasible to implement an efficient GPU kernel for this purpose. This paper presents a novel step forward: based on the special requirements of the heuristics solving the inverse problem (executing hundreds of simulations in a parallel fashion at the end of each iteration), it is possible to gain a higher level of parallelism using multiple graphics accelerators. The results show that this implementation (running on 4 GPUs) is about 120 times faster than a traditional CPU implementation using 20 cores. The latest developments of the GPU-based High Power Computations area were also analysed, like the new NVLink connection between the host and the devices, which tries to solve the long time existing data transfer handicap of GPU programming.

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Perera, Upeksha, e Christine Böckmann. "Solutions of Direct and Inverse Even-Order Sturm-Liouville Problems Using Magnus Expansion". Mathematics 7, n.º 6 (14 de junho de 2019): 544. http://dx.doi.org/10.3390/math7060544.

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In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm–Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular (and some singular) SLPs of even orders (tested for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively.

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Mohebbi, Farzad, e Mathieu Sellier. "Estimation of Functional Form of Time-Dependent Heat Transfer Coefficient Using an Accurate and Robust Parameter Estimation Approach: An Inverse Analysis". Energies 14, n.º 16 (18 de agosto de 2021): 5073. http://dx.doi.org/10.3390/en14165073.

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This paper presents a numerical method to address function estimation problems in inverse heat transfer problems using parameter estimation approach without prior information on the functional form of the variable to be estimated. Using an inverse analysis, the functional form of a time-dependent heat transfer coefficient is estimated efficiently and accurately. The functional form of the heat transfer coefficient is assumed unknown and the inverse heat transfer problem should be treated using a function estimation approach by solving sensitivity and adjoint problems during the minimization process. Based on proposing a new sensitivity matrix, however, the functional form can be estimated in an accurate and very efficient manner using a parameter estimation approach without the need for solving the sensitivity and adjoint problems and imposing extra computational cost, mathematical complexity, and implementation efforts. In the proposed sensitivity analysis scheme, all sensitivity coefficients can be computed in only one direct problem solution at each iteration. In this inverse heat transfer problem, the body shape is irregular and meshed using a body-fitted grid generation method. The direct heat conduction problem is solved using the finite-difference method. The steepest-descent method is used as a minimization algorithm to minimize the defined objective function and the termination of the minimization process is carried out based on the discrepancy principle. A test case with three different functional forms and two different measurement errors is considered to show the accuracy and efficiency of the used inverse analysis.

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Mamytov, A. O. "ON A PROBLEM OF DETERMINING THE RIGHT-HAND SIDE OF THE PARTIAL INTEGRO-DIFFERENTIAL EQUATION". Bulletin of the South Ural State University series "Mathematics. Mechanics. Physics" 13, n.º 3 (2021): 31–38. http://dx.doi.org/10.14529/mmph210304.

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As it is known, in the inverse problem, apart from the sought-for “basic” solution of the problem (i. e., the solution of the direct problem), the components of the direct problem are unknown. It is required to find these unknown components, so they will be also included in the solution of the inverse problem. To determine these components in the inverse problem, some additional information on the solution of the direct problem is added to the given equations. The additional information is called the inverse problem data. In the proposed article, the specific fourth-order partial integro-differential equation with the known initial and boundary conditions is considered. For simplicity, the homogeneous boundary conditions have been examined, since with the help of a linear transformation, the always inhomogeneous boundary conditions can be reduced to the homogeneous ones. The right-hand side of the equation contains n unknown functions: φi(t), i = 1,2,…,n.. To determine these unknown functions: φi(t), i = 1,2,…,n in the inverse problem there is additional information on the solution of the direct problem, i.e., the values of the sought-for “basic” solution to the problem in the inner segments of the investigated region are known, i. e., u(t,xi) = gi(t), t∈[0,T], xi∈ (0,1), i = 1,2,…,n. The problem is investigated in a rectangle located in the first quarter of the Cartesian coordinate system. To solve the inverse problem, an algorithm has been elaborated and sufficient conditions for the existence and the uniqueness of the solution of the inverse problem for the restoration of the right-hand side in a fourth-order partial integrodifferential equation have been found. When solving the inverse problem, the methods of transformations, Green's function, solutions of systems of linear Volterra integral equations have been used. As a result, the inverse problem has been reduced to a system of (n + 1) linear Volterra integral equations of the second kind, the solution of which for small 0 < T exists and is unique. The considered inverse problem can be called the inverse source problem.

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Tanana, V. P. "Completeness of the system of eigenfunctions of the Sturm-Liouville problem with the singularity". Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 30, n.º 1 (março de 2020): 59–63. http://dx.doi.org/10.35634/vm200105.

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Mathematical modeling of composite materials plays an important role in modern technology, and the solution and study of inverse boundary value problems of heat transfer is impossible without the use of systems of eigenfunctions of the Sturm-Liouville problem for the differential equation with discontinuous coefficients. One of the most important properties of such systems is their completeness in the corresponding spaces. This property of systems allows to prove theorems of existence and uniqueness of both direct problems and inverse boundary value problems of thermal conductivity, and also to prove numerical methods of solving such problems. In this paper, we prove the completeness of the Sturm-Liouville problem in the space $L_2[r_0,r_2]$ for a second-order differential operator with a discontinuous coefficient. This problem arises when investigating and solving the inverse boundary problem of thermal conductivity for a hollow ball consisting of two balls with different temperature conductivity coefficients. Self-conjugacy, injectivity, and positive definiteness of this operator are proved.

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Durdiev, D. K., e J. J. Jumaev. "Inverse problem of determining the kernel of integro-differential fractional diffusion equation in bounded domain". Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, n.º 10 (26 de outubro de 2023): 22–35. http://dx.doi.org/10.26907/0021-3446-2023-10-22-35.

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In this paper, an inverse problem of determining a kernel in a one-dimensional integro-differential time-fractional diffusion equation with initial-boundary and overdetermination conditions is investigated. An auxiliary problem equivalent to the problem is introduced first. By Fourier method this auxilary problem is reduced to equivalent integral equations. Then, using estimates of the Mittag-Leffler function and successive aproximation method, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown kernel which will be used in study of inverse problem. The inverse problem is reduced to the equivalent integral equation. For solving this equation the contracted mapping principle is applied. The local existence and global uniqueness results are proven.

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