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nonlinear inverse problem for a Sixth Order Differential Equation with Two Redefinition Functions

LOBACHEVSKII JOURNAL OF MATHEMATICS 2022年第3期43卷 804-814页

作者： Yuldashev, T. K. Kilichev, O. Sh Natl Univ Uzbekistan Uzbek Israel Joint Fac High Technol & Engn Math Tashkent 100174 Uzbekistan Acad Sci Uzbek VI Romanovskii Inst Math Tashkent 100174 Uzbekistan

In this paper, we consider an inhomogeneous sixth-order partial differential equation with two redefinition functions at the end point of the given segment. These redefinition functions enter nonlinearly into partial differential equation. The Fourier method of separation of variables is applied. Absolutely and uniformly convergence of Fourier series are proved. The Cauchy-Schwartz inequalities and the Bessel inequality are used. Theorems on the unique solvability of inverse problems are proved. The method of successive approximations is used in combination with the method of contraction mappings.

关键词： nonlinear inverse problem two redefinition functions conditions at the end point of a segment Fourier series method unique solvability

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Monte Carlo simulation via a numerical algorithm for solving a nonlinear inverse problem

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COMMUNICATIONS IN nonlinear SCIENCE AND NUMERICAL SIMULATION 2010年第9期15卷 2436-2444页

作者： Farnoosh, R. Ebrahimi, M. Iran Univ Sci & Technol Narmak Sch Math Tehran 16846 Iran

This paper is intended to provide a numerical algorithm involving the combined use of the finite differences scheme and Monte Carlo method for estimating the diffusion coefficient in a one-dimensional nonlinear parabolic inverse problem. In the present study, the functional form of the diffusion coefficient is unknown a priori. The unknown diffusion coefficient is approximated by the polynomial form and the present numerical algorithm is employed to find the solution. To modify the values of estimated coefficients of this polynomial form, we introduce a random search algorithm in Monte Carlo method for global optimization. A numerical test is performed in order to show the efficiency and accuracy of the present work. (C) 2009 Elsevier B.V. All rights reserved.

关键词： nonlinear inverse problem Finite differences scheme Monte Carlo optimization Diffusion coefficient Random search algorithm

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Solving the full nonlinear inverse problem for GPR using a three step method

Solving the full nonlinear inverse problem for GPR using a t...

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4th International Workshop on Advanced Ground Penetrating Radar

作者： van Kempen, L. Thanh, N. T. Sahli, H. Hao, D. N. Vrije Univ Brussel Pl Laan 2 B-1050 Brussels Belgium Hanoi Inst Math Hanoi 10307 Vietnam

ISBN: (纸本)9781424408863

In order to extract accurate quantitative information out of Ground Penetrating Radar (GPR) measurement data, one needs to solve a nonlinear inverse problem. In this paper we formulate such problem as a nonlinear least squares problem which is non convex. Solving a non-convex optimization problem requires a good initial estimation of the optimal solution. Therefore we use a three step method to solve the above non-convex problem. In a first step the qualitative solution of the linearized problem is estimated to obtain the detection and support of the subsurface scatterers. For this first step Synthetic Aperture Radar (SAR) is proposed. The second step consists out of a qualitative solution of the linearized problem to obtain a first guess for the material parameter values of the detected objects. The method proposed for this is Algebraic Reconstruction Technique (ART), which is an iterative method, starting from the initial value, given by the first step, and improving on this until an optimum is achieved. The final step then consists out of the solution of the nonlinear inverse problem using a variational method.

关键词： Ground Penetrating Radar nonlinear inverse problem ART adjoint method

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nonlinear Optimal Control of Thermal Processes in a nonlinear inverse problem

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LOBACHEVSKII JOURNAL OF MATHEMATICS 2020年第1期41卷 124-136页

作者： Yuldashev, T. K. Natl Univ Uzbekistan Uzbek Israel Joint Fac High Technol & Engn Math Tashkent 100174 Uzbekistan

The questions of weakly generalized solvability of a nonlinear inverse problem in nonlinear optimal control of thermal processes for a parabolic differential equation are studied. The parabolic equation is considered under initial and boundary conditions. To determine the recovery function, a nonlocal integral condition is specified. Moreover, the recovery function nonlinearly enters into the differential equation. Is applied the method of variable separation based on the search for a solution to the mixed inverse problem in the form of a Fourier series. It is assumed that the recovery function and nonlinear term of the given differential equation are also expressed as a Fourier series. For fixed values of the control function, the unique solvability of the inverse problem is proved by the method of compressive mappings. The quality functional has a nonlinear form. The necessary optimality conditions for nonlinear control are formulated. The determination of the optimal control function is reduced to a complicated functional-integral equation, the process of solving which consists of solving separately taken two nonlinear functional and nonlinear integral equations. nonlinear functional and integral equations are solved by the method of successive approximations. Formulas are obtained for the approximate calculation of the state function of the controlled process, the recovery function, and the optimal control function. Is proved the absolutely and uniformly convergence of the obtained Fourier series.

关键词： parabolic equation nonlinear inverse problem necessary conditions for optimal control nonlinear control minimization of the functional

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Simulation study for cross-sectional absorption distribution in turbid medium using spatially resolved backscattered light with lateral scanning and one-dimensional solution of nonlinear inverse problem

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JAPANESE JOURNAL OF APPLIED PHYSICS 2021年第7期60卷

作者： Nishida, Kazuhiro Kato, Yuji Kudo, Nobuki Shimizu, Koichi Hokkaido Univ Grad Sch Informat Sci & Technol Kita Ku N14W9 Sapporo Hokkaido 0600814 Japan Waseda Univ Grad Sch Informat Prod & Syst Hibikino 2-7 Kitakyushu Fukuoka 8080135 Japan

For a practical technique of cross-sectional imaging of animal bodies, we developed a new method using spatially resolved backscattered light. This method is based on the solution of the one-dimensional nonlinear inverse problem, and on the lateral scan of the source-detector pair along the object surface. Using this method, unknown variables in inverse problems can be reduced more greatly than when using conventional methods. A stable solution for the inverse problem becomes possible. The possibility of using the proposed method was assessed using simulation analysis. The results verified that cross-sectional imaging from several to 10 millimeter depths is possible for animal tissue. This analysis clarified the specific spatial resolution and accuracy in the estimated absorption coefficient. Distortionless imaging was confirmed. Results suggest that the proposed method represents new options as a stable and practical method for biological cross-sectional imaging.

关键词： Biomedical application Cross-sectional imaging Optical computed tomography nonlinear inverse problem Backscattered light

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GLOBAL SOLVABILITY OF nonlinear inverse problemS WITH DZHRBASHYAN–NERSESYAN DERIVATIVES AND SECTORIAL OPERATORS

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Journal of Mathematical Sciences (United States) 2025年 1-15页

作者： Plekhanova, Marina Izhberdeeva, Elizaveta Melekhina, Darya Sagimbaeva, Angelina Department of Mathematical Analysis Mathematics Faculty Chelyabinsk State University Kashirin Brothers St. 129 Chelyabinsk 454001 Russian Federation

nonlinear inverse problems for differential equations in Banach spaces, which are solved with respect to the highest-order Dzhrbashyan–Nersesyan fractional derivative, are considered. A linear closed operator in the equation under study generates analytic resolving families for the corresponding linear homogeneous equation, and a Lipschitz continuous nonlinear operator in the equation depends on lower-order Dzhrbashyan–Nersesyan derivatives and on a time-dependent unknown parameter. The equation is endowed by the Dzhrbashyan–Nersesyan initial value conditions and an overdetermination condition, which is set by a linear bounded operator. By means of the Banach theorem on a fixed point, the existence of a unique global solution, both mild and classical, of the inverse problem is proved. The obtained general results are used for the solvability research of a nonlinear inverse problem to an equation of anomalous diffusion. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.

关键词： Dzhrbashyan–Nersesyan derivative Global solvability Initial-boundary value problem nonlinear inverse problem Sectorial operator

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Determination of nonlinear Source Term in an inverse Convection-Reaction-Diffusion problem Using Radial Basis Functions Method

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IRANIAN JOURNAL OF SCIENCE AND TECHNOLOGY TRANSACTION A-SCIENCE 2019年第A5期43卷 2239-2252页

作者： Pourgholi, Reza Saeedi, Akram Hosseini, Abbas Damghan Univ Sch Math & Comp Sci POB 36715-364 Damghan Iran

In this paper, we propose a meshless method using radial basis functions (RBFs) method based on finite-difference method to solve a nonlinear inverse convection-reaction-diffusion problem with an unknown source function. Indeed, the aim of this paper is to show that the meshless method based on the radial basis functions approach is also suitable for the treatment of the inverse nonlinear partial differential equations. Usually, the matrices obtained from the discretization of the equations are ill-conditioned, especially in higher-dimensional problems. To overcome such difficulties, we use Tikhonov regularization method. We prove that the time discrete scheme is stable using the energy method. The effectiveness of the proposed method is illustrated by numerical examples.

关键词： nonlinear inverse problem Convection-reaction-diffusion equations Radial basis functions method Meshless methods Control function Tikhonov regularization method Energy method Unconditionally stable

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An extended version of the discrete Kalman filter applied to a nonlinear inverse heat conduction problem.

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INTERNATIONAL JOURNAL OF THERMAL SCIENCES 2000年第2期39卷 191-212页

作者： Daouas, N Radhouani, MS Ecole Natl Ingn Monastir Dept Genie Energet Monastir 5019 Tunisia

A nonlinear inverse heat conduction problem is resolved by using a formulation of the Kalman filter based on a statistical approach and extended to nonlinear systems. The time evolution of a surface heat flux density is reconstructed from a numerical simulation which allowed us to analyse the influence of some parameters, that condition the running of the filter, on the estimation result. A suitable choice of these parameters, guided by the filter behaviour observations, leads to a solution that remains stable when using noisy data, but that is slightly time-lagged compared to the exact function. This time-lag depends on the location of the interior temperature measurement needed for the inversion and on the model error caused by the approximation of the heat flux with a piece-wise constant function. The application of the extended Kalman filter with real measurements recorded from an experimental set-up, shows that this technique fits the stochastic structure of experimental measurements. The provided results are validated by using the Raynaud's and Bransier's inverse method and are in good agreement with the heat flux density estimated with th is method. (C) 2000 Editions scientifiques et medicales Elsevier SAS.

关键词： nonlinear inverse problem heat conduction extended Kalman filter estimation numerical simulation experimental set-up

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Regularized numerical solution of nonlinear inverse heat-conduction problem

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Journal of Engineering Physics 1975年第1期29卷 934-938页

作者： Alifanov, O.M. Artyukhin, E.A. S. Ordzhonikidze Moscow Aviation Institute Moscow Russian Federation

The construction of an algorithm for a numerical solution of the nonlinear inverse problem is discussed for the case of a generalized one-dimensional heat-conduction equation in a region with moving boundaries. The al... 详细信息

关键词： Statistical Physic inverse problem nonlinear inverse problem

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nonlinear inverse problems for Fractional Differential Equations with Sectorial Operators

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LOBACHEVSKII JOURNAL OF MATHEMATICS 2022年第11期43卷 3125-3141页

作者： Fedorov, V. E. Ivanova, N. D. Borel, L. V. Avilovich, A. S. Chelyabinsk State Univ Math Fac Chelyabinsk 454001 Russia South Ural State Univ Inst Nat & Exact Sci Chelyabinsk 454080 Russia St Petersburg Min Univ Fac Fundamental & Humanitarian Disciplines St Petersburg 199106 Russia Transpolar State Univ Fac Elect Power Engn Econ & Management Norilsk 663310 Krasnoyarsk Ter Russia

A nonlinear inverse problem for an equation in a Banach space, resolved with respect to the oldest Gerasimov-Caputo fractional derivative, is considered. A linear closed operator in the linear part of the equation generates an analytic resolving family of operators. nonlinear operator in the equation depends on minor fractional derivatives, fractional integrals and an unknown element, which depends on time. The equation is endowed by the Cauchy conditions and an overdetermination condition with an abstract linear bounded operator. By the method of a contraction mapping in a metric space theorems on the unique solvability in the sense of generalized solution and of smooth solution are proved. Abstract results were used for research of a nonlinear inverse problem for a system of Navier-Stokes type with time-fractional Gerasimov-Caputo derivatives and integrals.

关键词： nonlinear inverse problem existence and uniqueness of solution Gerasimov-Caputo fractional derivative Gerasimov-Caputo fractional integral analytic resolving operators family

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