Identification of the unknown diffusion coefficient in a linear parabolic equation by the semigroup approach

AbstractIn this article, we study the semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(x) in the linear parabolic equation ut(x,t)=(k(x)ux(x,t))x, with Dirichlet boundary conditions u(0,t)=ψ0, u(1,t)=ψ1. Main goal of this study is to investigate the distinguishability of the input–output mappings Φ[⋅]:K→C1[0,T], Ψ[⋅]:K→C1[0,T] via semigroup theory. In this paper, we show that if the null space of the semigroup T(t) consists of only zero function, then the input–output mappings Φ[⋅] and Ψ[⋅] have the distinguishability property. Moreover, the values k(0) and k(1) of the unknown diffusion coefficient k(x) at x=0 and x=1, respectively, can be determined explicitly by making use of measured output data (boundary observations) f(t):=k(0)ux(0,t) or/and h(t):=k(1)ux(1,t). In addition to these, the values k′(0) and k′(1) of the unknown coefficient k(x) at x=0 and x=1, respectively, are also determined via the input data. Furthermore, it is shown that measured output data f(t) and h(t) can be determined analytically, by an integral representation. Hence the input–output mappings Φ[⋅]:K→C1[0,T], Ψ[⋅]:K→C1[0,T] are given explicitly in terms of the semigroup. Finally by using all these results, we construct the local representations of the unknown coefficient k(x) at the end points x=0 and x=1

Reconstruction of shear force in Atomic Force Microscopy from measured displacement of the cone-shaped cantilever tip

We present a new comprehensive mathematical model of the cone-shaped cantilever tip-sample interaction in Atomic Force Microscopy (AFM). The importance of such AFMs with cone-shaped cantilevers can be appreciated when its ability to provide high-resolution information at the nanoscale is recalled. It is an indispensable tool in a wide range of scientific and industrial fields. The interaction of the cone-shaped cantilever tip with the surface of the specimen (sample) is modeled by the damped Euler-Bernoulli beam equation $\rho_A(x)u_{tt}$ $+\mu(x)u_{t}+(r(x)u_{xx}+\kappa(x)u_{xxt})_{xx} = 0$, $(x, t)\in (0, \ell)\times (0, T)$, subject to the following initial, $u(x, 0) = 0$, $u_t(x, 0) = 0$ and boundary, $u(0, t) = 0$, $u_{x}(0, t) = 0$, $\left (r(x)u_{xx}(x, t)+\kappa(x)u_{xxt} \right)_{x = \ell} = M(t)$, $\left (-(r(x)u_{xx}+\kappa(x)u_{xxt})_x\right)_{x = \ell} = g(t)$ conditions, where $M(t): = 2h\cos \theta\, g(t)/\pi$ is the moment generated by the transverse shear force $g(t)$. Based on this model, we propose an inversion algorithm for the reconstruction of an unknown shear force in the AFM cantilever. The measured displacement $\nu(t): = u(\ell, t)$ is used as additional data for the reconstruction of the shear force $g(t)$. The least square functional $J(F) = \frac{1}{2}\Vert u(\ell, \cdot)-\nu \Vert_{L^2(0, T)}^2$ is introduced and an explicit gradient formula for the Fréchet derivative of the cost functional is derived via the weak solution of the adjoint problem. Additionally, the geometric parameters of the cone-shaped tip are explicitly contained in this formula. This enables us to construct a gradient based numerical algorithm for the reconstructions of the shear force from noise free as well as from random noisy measured output $\nu (t)$. Computational experiments show that the proposed algorithm is very fast and robust. This creates the basis for developing a numerical "gadget" for computational experiments with generic AFMs

Inverse coefficient problem for cascade system of fourth and second order partial differential equations

The study of the paper mainly focusses on recovering the dissipative parameter in a cascade system coupling a bilaplacian operator to a heat equation from final time measured data via quasi-solution based optimization. The coefficient inverse problem is expressed as a minimization problem. We proved that minimizer exists and the necessary optimality condition which plays the crucial role to prove the required stability result for the corresponding coefficient is derived. Utilising the conjugate gradient approach, numerical results are examined to show the method's effectiveness.Comment: 24 pages, 18 figure

Monotonicity of nonlinear boundary value problems related to deformation theory of plasticity

We study nonlinear boundary value problems arising in the deformation theory of plasticity. These problems include 3D mixed problems related to nonlinear Lame system, elastoplastic bending of an incompressible hardening plate, and elastoplastic torsion of a bar. For all these different problems, we present a general variational approach based on monotone potential operator theory and prove solvability and monotonicity of potentials. The obtained results are illustrated on numerical examples

Solvability of the clamped Euler–Bernoulli beam equation

Hasanov, Alemdar (Dogus Author)In this study, solvability of the initial boundary value problem for general form Euler–Bernoulli beam equation which includes also moving point-loads is investigated. The complete proof of an existence and uniqueness properties of the weak solution of the considered equation with Dirichlet type boundary conditions is derived. The method used here is based on Galerkin approximation which is the main tool for the weak solution theory of linear evolution equations as well as in derivation of a priori estimate for the approximate solutions. All steps of the proposed technique are explained in detail

Introduction to inverse problems for differential equations

This book presents a systematic exposition of the main ideas and methods in treating inverse problems for PDEs arising in basic mathematical models, though it makes no claim to being exhaustive. Mathematical models of most physical phenomena are governed by initial and boundary value problems for PDEs, and inverse problems governed by these equations arise naturally in nearly all branches of science and engineering. The book’s content, especially in the Introduction and Part I, is self-contained and is intended to also be accessible for beginning graduate students, whose mathematical background includes only basic courses in advanced calculus, PDEs and functional analysis. Further, the book can be used as the backbone for a lecture course on inverse and ill-posed problems for partial differential equations. In turn, the second part of the book consists of six nearly-independent chapters. The choice of these chapters was motivated by the fact that the inverse coefficient and source problems considered here are based on the basic and commonly used mathematical models governed by PDEs. These chapters describe not only these inverse problems, but also main inversion methods and techniques. Since the most distinctive features of any inverse problems related to PDEs are hidden in the properties of the corresponding solutions to direct problems, special attention is paid to the investigation of these properties

Foreword to the Special Issue "Inverse problems and nonlinear phenomena''

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Sharp operator based edge detection

Ahmad et al. in their paper for the first time proposed to apply sharp function for classification of images. In continuation of their work, in this paper we investigate the use of sharp function as an edge detector through well known diffusion models. Further, we discuss the formulation of weak solution of nonlinear diffusion equation and prove uniqueness of weak solution of nonlinear problem. The anisotropic generalization of sharp operator based diffusion has also been implemented and tested on various types of images