Space-dependent Coefficient Research Articles

An inverse problem for pseudoparabolic equation: existence, uniqueness, stability, and numerical analysis

In this paper, we study an inverse problem for a linear third-order pseudoparabolic equation. The investigated inverse problem consists of determining a space-dependent coefficient of the right-hand side of a pseudoparabolic equation. As an additional information a final overdetermination condition is considered. Under the suitable conditions on the data of the problem, the unique solvability of the considered inverse problem is established. The stability of solutions is also proved. The established results are also true for inverse problems for parabolic equations, which could be obtained as a regularization of the studied pseudoparabolic equation. In addition, the pseudoparabolic problem is discretized using the cubic B-spline functions and recast as a nonlinear least-squares minimization of the Tikhonov regularization function. Numerically, this is effectively solved using the MATLAB subroutine lsqnonlin. Both exact and noisy data are inverted. Numerical results for three benchmark test examples are presented and discussed. Moreover, the von Neumann stability analysis is also discussed.

An inverse problem for a transmission wave equation with a flat interface in Rn

In this paper we study a wave equation with discontinuous principal coefficient within a bounded domain of arbitrary dimension. It is obtained the stability of the inverse problem of recovering a space-dependent coefficient by observing a trace of the corresponding solution on part of the boundary. We provide a precise estimate of the minimum required time, as a function of the velocity change and domain size. The main tools are new global Carleman estimates for the transmission system with a particular weight function adapted to the interface geometry, which allows to obtain an optimal estimate of the minimum time.

Boundary controllability for variable coefficients one-dimensional wave equation with interior degeneracy

Abstract In this paper, we study boundary controllability for the linear extension problem of a wave equation with space-dependent coefficients and having an internal degeneracy. For this purpose, we mainly focus on the well-posedness and the boundary null controllability of a relaxed version of the original problem, namely, to some degenerate transmission problem. The key ingredient is to derive direct and inverse inequalities for the associated homogeneous degenerate adjoint problem. By these inequalities, we deduce that the transmission problem has a unique solution by transposition and this solution is null controllable. Moreover, we give an explicit formula of the controllability time.

Large deviations at level 2.5 and for trajectories observables of diffusion processes: the missing parts with respect to their random-walks counterparts

Behind the nice unification provided by the notion of the level 2.5 in the field of large deviations for time-averages over a long Markov trajectory, there are nevertheless very important qualitative differences between the meaning of the level 2.5 for diffusion processes on one hand, and the meaning of the level 2.5 for Markov chains either in discrete-time or in continuous-time on the other hand. In order to analyze these differences in detail, it is thus useful to consider two types of random walks converging towards a given diffusion process in dimension d involving arbitrary space-dependent forces and diffusion coefficients, namely (i) continuous-time random walks on the regular lattice of spacing b; (ii) discrete-time random walks in continuous space with a small time-step τ. One can then analyze how the large deviations at level 2.5 for these two types of random walks behave in the limits b → 0 and τ → 0 respectively, in order to describe how the fluctuations of some empirical observables of the random walks are suppressed in the limit of diffusion processes. One can then also study the limits b → 0 and τ → 0 for any trajectory observable of the random walks that can be decomposed on its empirical density and its empirical flows in order to see how it is projected on the appropriate trajectory observable of the diffusion process involving its empirical density and its empirical current.

Determining of a Space Dependent Coefficient of Fractional Diffusion Equation with the Generalized Riemann–Liouville Time Derivative

Determining of a Space Dependent Coefficient of Fractional Diffusion Equation with the Generalized Riemann–Liouville Time Derivative

Identifiability of some space dependent coefficients in a wave equation of nonlinear acoustics

Identifiability of some space dependent coefficients in a wave equation of nonlinear acoustics

Random-search efficiency in a bounded interval with spatially heterogeneous diffusion coefficient

We consider random walkers searching for a target in a bounded 1D heterogeneous environment, in the interval , where diffusion is described by a space-dependent diffusion coefficient D(x). Boundary conditions are absorbing at the position of the target (set at x = 0) and reflecting at the border x = L. We calculate and compare the estimates of efficiency and . In the Stratonovich framework of the multiplicative random process, both measures are analytically calculated for arbitrary D(x). For other interpretations of the stochastic integrals (e.g. Itô and anti-Itô), we obtain general results for ɛ 2, while ɛ 1 is obtained for particular forms of D(x). The impact of the diffusivity profile on these measures of efficiency is discussed. Symmetries and peculiar properties arise when the search starts at the border (). In particular, heterogeneity spoils the efficiency of the search within the Stratonovich framework, while for other interpretations the searcher can perform better for certain heterogeneous diffusivity profiles.

Reconstruction coefficient analysis of honeybee collapse due to pesticide contamination

In this paper we consider the inverse problems of identifying space-dependent coefficients of the mortality rate of the bees and the rate of contamination of the forager bees by pesticides. The model is described by a weakly coupled system of two reaction-diffusion equations for the spatial distribution of uncontaminated and contaminated foraging bees. Final time t = T observations of the density of uncontaminant and contaminant forager bees are used. We propose two approaches for studying the problems. The first one uses the overspecified information to transform the problems into non-linear parabolic equations involving the solution values at the final time. This allows us to prove, using fixed-point arguments, existence of solution to the inverse problems. The second study employs the concept of the quasi-solution to establish existence of solution to the inverse problems as minimizers of least-square cost functionals.

A combined BEM and Laplace transform for unsteady modified-Helmholtz equation of time–space variable coefficients for anisotropic media

PurposeTwo-dimensional (2D) problems are governed by unsteady anisotropic modified-Helmholtz equation of time–space dependent coefficients are considered. The problems are transformed into a boundary-only integral equation which can be solved numerically using a standard boundary element method (BEM). Some examples are solved to show the validity of the analysis and examine the accuracy of the numerical method.Design/methodology/approach The 2D problems which are governed by unsteady anisotropic modified-Helmholtz equation of time–space dependent coefficients are solved using a combined BEM and Laplace transform. The time–space dependent coefficient equation is reduced to a time-dependent coefficient equation using an analytical transformation. Then, the time-dependent coefficient equation is Laplace transformed to get a constant coefficient equation, which can be written as a boundary-only integral equation. By utilizing a BEM, this integral equation is solved to find numerical solutions to the problems in the frame of the Laplace transform. These solutions are then inversely transformed numerically to obtain solutions in the original time–space frame.FindingsThe main finding of this research is the derivation of a boundary-only integral equation for the solutions of initial-boundary value problems governed by a modified-Helmholtz equation of time–space dependent coefficients for anisotropic functionally graded materials with time-dependent properties.Originality/valueThe originality of the research lies on the time dependency of properties of the functionally graded material under consideration.

Shallow Moments to Capture Vertical Structure in Open Curved Shallow Flow

Shallow water models are successfully used for simulating geophysical flows like river floods, tsunamis, sediment transport, or debris flows. Depth-averaged models are in general attractive due to their low computational cost. However, information on the vertical velocity is required a priori, typically by assuming a parametrized profile using a time-independent function, e.g., a constant. A systematic generalization of depth-averaged flow models is given by the shallow moment method. This method retains transient information about the vertical flow profile by using a finite Legendre expansion for said profile with time and space-dependent coefficients. The shallow moment approach allows to include vertical information and generates a hierarchy of equations that in the limit, recover the reference system before depth-averaging. Even a low number of basis functions significantly increases the model’s capability to resolve vertical information and, whenever this is relevant for the modeling task at stake, significantly improves the predictive power compared to classical shallow water systems. In the paper, we show a two-dimensional simulation result based on the shallow moment system to predict processes unobservable in classical shallow flow, namely secondary flow and the redistribution of velocity profiles, which are of relevance in curved channelized flow as experimentally demonstrated by Steffler.

A numerical solution for unsteady anisotropic diffusion reaction equation of time–space variable coefficients

Two‐dimensional transient problems for anisotropic functionally graded materials governed by a diffusion reaction equation of time–space dependent coefficients are considered. Using a transformation, the space–time dependent coefficients equation is reduced to a time‐dependent coefficients equation. Then taking a Laplace transform will result in a constant coefficients equation, which can be written in forms of boundary‐only integral equations and solved using a standard boundary element method for the solutions. These boundary element method solutions are then numerically inverse transformed to get the solutions in the time variable. Some problems of trigonometrically, exponentially and quadratically graded materials are solved. The numerical solutions obtained are quite accurate.

On the Wave Equation with Space Dependent Coefficients: Singularities and Lower Order Terms

This paper complements the study of the wave equation with discontinuous coefficients initiated in (Discacciati et al. in J. Differ. Equ.319 (2022) 131–185) in the case of time-dependent coefficients. Here we assume that the equation coefficients are depending on space only and we formulate Levi conditions on the lower order terms to guarantee the existence of a very weak solution as defined in (Garetto and Ruzhansky in Arch. Ration. Mech. Anal.217 (2015) 113–154). As a toy model we study the wave equation in conservative form with discontinuous velocity and we provide a qualitative analysis of the corresponding very weak solution via numerical methods.

On regularization results for a two-dimensional nonlinear time-fractional inverse diffusion problem

Ill-posed inverse problems arise in a variety of practically important applications including image restoration, medicine, ecology as well as many other branches of pure and applied sciences. In this paper, we consider an ill-posed inverse problem for a two-dimensional nonlinear time-fractional inverse diffusion model involving the Caputo time-fractional derivative as follows:{Dtβu(x,y,t)=−a(x)(ux(x,y,t)+uy(x,y,t))+ℓ(x,y,t,u(x,y,t)),x>0,y>0,t>0,u(x,y,0)=0,x>0,y>0,limx→∞⁡u(x,y,t)=0,y>0,t>0,u(1,y,t)=u1(y,t),y>0,t>0, where β∈(0,1) is fixed, a is a space-dependent diffusion coefficient, ℓ is a nonlinear function satisfying a Lipschitz condition, and the data u1 is given approximately. We want to determine u(x,y,t) for 0≤x<1. There exists a vast literature on regularization results related to the considered problem. We are concerned with regularization results for the nonlinear case. The first regularization results in such a case were studied by Tuan, Hoan and Tatar in [17]. While these results apply to a constant diffusivity coefficient in the one-dimensional setting, the results of Vo and co-worker in [18], more generally, apply to the case of the space-dependent diffusion coefficient in the two-dimensional setting. In all these papers relatively strong a priori assumptions on the regularity of the exact solution are made. In this work, we weaken such a priori assumptions via two regularization strategies based on the integral equation method. We obtain several explicit error estimates including an error estimate of the Hölder-Logarithmic type for all x∈[0,1), which can be seen as the improvement and the generalization of error estimates presented by Zheng and Wei in [20,21]. Eventually, several numerical examples are presented for the purpose of illustrating the theoretical results.

Numerical Reconstruction of a Space-Dependent Reaction Coefficient and Initial Condition for a Multidimensional Wave Equation with Interior Degeneracy

A simultaneous reconstruction of the initial condition and the space-dependent reaction coefficient in a multidimensional hyperbolic partial differential equation with interior degeneracy is of concern. A temporal integral observation is utilized to achieve that purpose. The well-posedness, existence, and uniqueness of the inverse problem under consideration are discussed. The inverse problem can be reformulated as a least squares minimization and the Fréchet gradients are determined, using the adjoint and sensitivity problems. Finally, an iterative construction procedure is developed by employing the conjugate gradient algorithm while invoking the discrepancy principle as a stopping criterion. Some numerical experiments are given to ensure the performance of the reconstruction scheme in one and two dimensions.

A numerical investigation of unsteady space–time dependent coefficients anisotropic-diffusion convection reaction equation

Two dimensional transient problems for anisotropic FGMs governed by a diffusion convection reaction (DCR) equation of time–space dependent coefficients are considered. Using a transformation, the space–time dependent coefficients equation is reduced to a time dependent coefficients equation. Then taking a Laplace transform will result in a constant coefficients equation which can be written in forms of boundary-only integral equations and solved using a standard BEM for the solutions. These BEM solutions are then numerically inverse transformed to get the solutions in the time variable. Some problems of trigonometrically, exponentially and quadratically graded materials with compressible or incompressible flow are solved. The numerical solutions obtained are quite accurate.

Inverse problems of damped wave equations with Robin boundary conditions: an application to blood perfusion

Knowledge of the properties of biological tissues is essential in monitoring any abnormalities that may be forming and have a major impact on organs malfunctioning. Therefore, these disorders must be detected and treated early to save lives and improve the general health. Within the framework of thermal therapies, e.g. hyperthermia or cryoablation, the knowledge of the tissue temperature and of the blood perfusion rate are of utmost importance. Therefore, motivated by such a significant biomedical application, this paper investigates, for the first time, the uniqueness and stable reconstruction of the space-dependent (heterogeneous) perfusion coefficient in the thermal-wave hyperbolic model of bio-heat transfer from Cauchy boundary data using the powerful technique of Carleman estimates. Additional novelties consist in the consideration of Robin boundary conditions, as well as developing a mathematical analysis that leads to stronger stability estimates valid over a shorter time interval than usually reported in the literature of coefficient identification problems for hyperbolic partial differential equations. Numerically, the inverse coefficient problem is recast as a nonlinear least-squares minimization that is solved using the conjugate gradient method (CGM). Both exact and noisy data are inverted. To achieve stability, the CGM is stopped according to the discrepancy principle. Numerical results for a physical example are presented and discussed, showing the convergence, accuracy and stability of the inversion procedure.

Two Free Boundaries Problem for a Parabolic Equation

В данной работе рассматривается задача типа Стефана с двумя свободными границами для квазилинейного параболического уравнения в одномерном случае. Исследование нелинейных задач со свободными границами методом, основанным на построении априорных оценок. Поэтому сначала устанавливаются некоторые первоначальные априорные оценки для решения рассматриваемой задачи. Основной трудностью при построении теории для задач квазилинейных параболических уравнений второго порядка является получение априорной оценки модуля производной решение, а также в задачах со свободной границей требуются дополнительные рассуждения. Для этого задача сводится к задаче с фиксированной границей через замену переменных. Полученная задача имеет зависящие от времени и положения в пространстве коэффициенты с нелинейными слагаемыми. Далее построены априорных оценок типа Шаудера для решения уравнения с нелинейными слагаемыми и закрепленной границей. На основе полученных оценок доказана единственность решения задачи. Затем мы доказываем глобальное существование решения задачи с помощью теоремы Лерэ-Шаудера о неподвижной точке. This paper considers a two-free-boundary Stefan-type problem for a quasi-linear parabolic equation in one dimension. Nonlinear problems with free boundaries are studied using a method based on constructing a priori estimates. Therefore, some initial a priori estimates for the solution to the problem under consideration are first established. The main difficulty in constructing a theory for second-order quasi-linear parabolic equations is obtaining an a priori estimate for the solution’s derivative module, and additional arguments are required in problems with a free boundary. To address this, the problem is reduced to a fixed-boundary problem through a change of variables. The resulting problem has time- and spacedependent coefficients with nonlinear terms. Next, Schauder-type a priori estimates are constructed for the equation with nonlinear terms and a fixed boundary. Based on these estimates, the uniqueness of the solution to the problem is proven. Then, the global existence of the solution to the problem is demonstrated using the Leray-Schauder fixed-point theorem.

Determination of the space-dependent blood perfusion coefficient in the thermal-wave model of bio-heat transfer

PurposeWhen modeling heat propagation in biological bodies, a non-negligible relaxation time (typically between 15-30 s) is required for the thermal waves to accumulate and transfer, i.e. thermal waves propagate at a finite velocity. To accommodate for this feature that is characteristic to heat transfer in biological bodies, the classical Fourier's law has to be modified resulting in the thermal-wave model of bio-heat transfer. The purpose of the paper is to retrieve the space-dependent blood perfusion coefficient in such a thermal-wave model of bio-heat transfer from final time temperature measurements.Design/methodology/approachThe non-linear and ill-posed blood perfusion coefficient identification problem is reformulated as a non-linear minimization problem of a Tikhonov regularization functional subject to lower and upper simple bounds on the unknown coefficient. For the numerical discretization, an unconditionally stable direct solver based on the Crank–Nicolson finite difference scheme is developed. The Tikhonov regularization functional is minimized iteratively by the built-in routine lsqnonlin from the MATLAB optimization toolbox. Both exact and numerically simulated noisy input data are inverted.FindingsThe reconstruction of the unknown blood perfusion coefficient for three benchmark numerical examples is illustrated and discussed to verify the proposed numerical procedure. Moreover, the proposed algorithm is tested on a physical example which consists of identifying the blood perfusion rate of a biological tissue subjected to an external source of laser irradiation. The numerical results demonstrate that accurate and stable solutions are obtained.Originality/valueAlthough previous studies estimated the important thermo-physical blood perfusion coefficient, they neglected the wave-like nature of heat conduction present in biological tissues that are captured by the more accurate thermal-wave model of bio-heat transfer. The originalities of the present paper are to account for such a more accurate thermal-wave bio-heat model and to investigate the possibility of determining its space-dependent blood perfusion coefficient from temperature measurements at the final time.

Estimating Space-Dependent Coefficients for 1D Transport Using Gaussian Processes as State Estimator in the Frequency Domain

This letter presents a method to estimate the space-dependent transport coefficients (diffusion, convection, reaction, and source/sink) for a generic scalar transport model, e.g., heat or mass. As the problem is solved in the frequency domain, the complex valued state as a function of the spatial variable is estimated using Gaussian process regression. The resulting probability density function of the state, together with a semi-discretization of the model, and a linear parameterization of the coefficients are used to determine the maximum likelihood solution for these space-dependent coefficients. The proposed method is illustrated by simulations.

Remarks on the controllability of parabolic systems with non-diagonalizable diffusion matrix

The distributed null controllability for coupled parabolic systems with non-diagonalizable diffusion matrices with a reduced number of controls has been studied in the case of constant matrices. On the other hand, boundary controllability issues and distributed controllability with non-constant coefficients for this kind of systems is not completely understood. In this paper, we analyze the boundary controllability properties of a class of coupled parabolic systems with non-diagonalizable diffusion matrices in the constant case and the distributed controllability of a $ 2\times 2 $ non-diagonalizable parabolic system with space-dependent coefficients. For the boundary controllability problem, our strategy relies on the moment method. For the distributed controllability problem, our findings provide positive and negative control results by using the Fattorini-Hautus test and a fictitious control strategy.