arXiv subjects

Alemdar Hasanov

Publications and source records attributed to Alemdar Hasanov.

8 recordsLinked to original sources

arXiv2026

The continuity properties of the solutions of the wave equation at different boundary conditions across the characteristic line

It is well-known the continuity properties of solutions to the wave equation depend strongly on both the initial data and the boundary conditions, particularly across the characteristic lines, where information propagates. Moreover, the continuity properties of solutions to the wave equation depend strongly on the type of boundary condition imposed and on the interaction of the characteristics with the boundary. Across the characteristic lines, singularities and discontinuities propagate according to the geometry of the problem, while the boundary conditions determine how these singularities are re?ected or transmitted. In this study, we will examine the behavior of solutions of the wave equation, corresponding to different boundary conditions, on its characteristic line, using the formulas obtained as a result of d'Alembert's formula, for all the three, Dirichlet, Neumann and Robin, types of boundary conditions.

math.AP↗

arXiv2026

Simultaneous determination of an unknown bending moment and shear force in the Euler Bernoulli cantilever beam from measured boundary defection and slope

It is known that the study of vibration characteristics at the tip of the micro-cantilever and its relationship to the sample plays a very important role in improving the resolution of an Atomic Force Microscopy (AFM). In this paper, within the Euler Bernoulli beam model, a mathematical model, defined as a model with two unknown inputs and two measured outputs, is considered for the simultaneous determination of the unknown bending moment and the shear force at the tip of the micro-cantilever from two feasible measured outputs at the same tip: the deflection and the slope. This model leads to the following inverse problem: find $M(t)$ and $g(t)$ in $\rho_A(x)u_{tt}+\mu(x) u_{t}+(r(x)u_{xx})_{xx}=0$, $(x,t)\in \Omega_T:=(0,\ell)\times (0,T)$ subject to the boundary conditions $u(0,t)=u_{x}(0,t)=0$, $\left(r(x)u_{xx}\right)_{x=\ell}=M(t)$, $\left((r(x)u_{xx})_x\right)_{x=\ell}=g(t)$, and the homogenous initial conditions, from the measured outputs $w_{\ell}(t):=u(\ell,t)$ and $\theta_{\ell}(t):=u_x(\ell,t)$. It is proved that the vector-form input-output map $\mathcal{P}:=\left (\Phi, \Psi \right )$, with $\left (\Phi q \right )(t):=u(0,t;q)$ and $\left (\Psi q\right )(t):=u_x(0,t;q)$, where $q(t):=\left (M(t),g(t)\right )$, corresponding to the inverse problem, is compact and Lipschitz continuous. This result allows us to prove the existence of a solution of the minimization problem for the Tikhonov functional. As a consequence, the existence of a quasi-solution to the inverse problem is established. Furthermore, a vector form expression for the Frechet gradient of the Tikhonov functional is derived, and the Lipschitz continuity of the Frechet gradient is rigorously proven. This crucial property ensures the monotonic behavior of iterative gradient-based numerical methods.

math.AP↗

arXiv2025

An accurate approach to determining the spatiotemporal vehicle load on bridges based on measured boundary slopes

In this paper, a novel mathematical model is developed to evaluate the spatiotemporal vehicle loads on long bridges from slope measurements made at the ends of a bridge based on Euler-Bernoulli beam model with internal and external damping. The mathematical modelling of this phenomena leads to the inverse source problem of determining the spatiotemporal vehicle load $F(x,t)$ in the variable coefficient Euler-Bernoulli equation $ρ_A(x)u_{tt}+μ(x) u_{t}+(r(x)u_{xx})_{xx}+(κ(x)u_{xxt})_{xx}=F(x,t)$, $(x,t)\in Ω_T:=(0,\ell)\times (0,T)$ subject to the "simply supported" boundary conditions $u(0,t)=(r(x)u_{xx}+(κ(x)u_{xxt})_{x=0}=0$, $u(\ell,t)=(r(x)u_{xx}+(κ(x)u_{xxt})_{x=\ell}=0$, from the both measured outputs: $θ_1(t):=u_x(0,t)$ and $θ_2(t):=u_x(\ell,t)$, that is, the measured boundary slopes. It is shown that the input-output maps $(ΦF)(t):=u_x(0,t;F)$, $(ΨF)(t):=u_x(\ell,t;F)$, $F \in \mathcal{F}\subset L^2(Ω_T)$, corresponding to the inverse problem, are compact and Lipschitz continuous. Then Tikhonov functional $J(F)=\Vert ΦF-θ_1 \Vert_{L^2(0,T)}^2+\Vert ΨF-θ_2 \Vert_{L^2(0,T)}^2$ is introduced to prove the existence of a quasi-solution to the inverse problem. An explicit gradient formula for the Fréchet derivative of the Tikhonov functional is derived. The Lipschitz continuity of the Fréchet gradient, which guarantees the monotonicity of iterations in gradient methods, has been proven.

math.AP↗

arXiv2024

Identification of source terms in the Schrödinger equation with dynamic boundary conditions from final data

In this paper, we study an inverse problem of identifying two spatial-temporal source terms in the Schrödinger equation with dynamic boundary conditions from the final time overdetermination. We adopt a weak solution approach to solve the inverse source problem. By analyzing the associated Tikhonov functional, we prove a gradient formula of the functional in terms of the solution to a suitable adjoint system, allowing us to obtain the Lipschitz continuity of the gradient. Next, the existence and uniqueness of a quasi-solution are also investigated. Finally, our theoretical results are validated by numerical experiments in one dimension using the Landweber iteration method.

math.AP↗

arXiv2023

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.

math.AP↗

arXiv2023

Exponential stability of damped Euler-Bernoulli beam controlled by boundary springs and dampers

In this paper, the vibration model of an elastic beam, governed by the damped Euler-Bernoulli equation $ρ(x)u_{tt}+μ(x)u_{t}$$+\left(r(x)u_{xx}\right)_{xx}=0$, subject to the clamped boundary conditions $u(0,t)=u_x(0,t)=0$ at $x=0$, and the boundary conditions $\left(-r(x)u_{xx}\right)_{x=\ell}=k_r u_x(\ell,t)+k_a u_{xt}(\ell,t)$, $\left(-\left(r(x)u_{xx}\right)_{x}\right )_{x=\ell}$$=- k_d u(\ell,t)-k_v u_{t}(\ell,t)$ at $x=\ell$, is analyzed. The boundary conditions at $x=\ell$ correspond to linear combinations of damping moments caused by rotation and angular velocity and also, of forces caused by displacement and velocity, respectively. The system stability analysis based on well-known Lyapunov approach is developed. Under the natural assumptions guaranteeing the existence of a regular weak solution, uniform exponential decay estimate for the energy of the system is derived. The decay rate constant in this estimate depends only on the physical and geometric parameters of the beam, including the viscous external damping coefficient $μ(x) \ge 0$, and the boundary springs $k_r,k_d \ge 0$ and dampers $k_a,k_v \ge 0$. Some numerical examples are given to illustrate the role of the damping coefficient and the boundary dampers.

math.AP↗

arXiv2023

Exponential stability of Euler-Bernoulli beam under boundary controls in rotation and angular velocity

This paper addresses the analysis of a boundary feedback system involving a non-homogeneous Euler-Bernoulli beam governed by the equation $m(x)u_{tt}+μ(x)u_{t}$$+\left(r(x)u_{xx}\right)_{xx}=0$, subject to the initial $u(x,0)=u_0(x)$, $u_t(x,0)=v_0(x)$ and boundary conditions $u(0,t)=0$, $\left (-r(x)u_{xx}(x,t)\right )_{x=0}=-k^{-}_r u_{x}(0,t)-k^{-}_a u_{xt}(0,t)$, $u(\ell,t)=0$, $\left (-r(x)u_{xx}(x,t)\right )_{x=\ell}=-k^{+}_r u_{x}(\ell,t)-k^{+}_a u_{xt}(\ell,t)$, with boundary control at both ends resulting from the rotation and angular velocity. The approach proposed in this study relies on the utilization of regular weak solutions, energy identity, and a physically motivated Lyapunov function. By imposing natural assumptions concerning physical parameters and other inputs, which ensure the existence of a regular weak solution, we successfully derive a uniform exponential decay estimate for the system's energy. The decay rate constant featured in this estimate is solely dependent on the physical and geometric properties of the beam. These properties encompass crucial parameters such as the viscous external damping coefficient $μ(x)$, as well as the boundary springs $k^{-}_r,k^+_r $ and dampers $k^{-}_a,k^+_a$. To illustrate the practical effectiveness of our theoretical findings, numerical examples are provided. These examples serve to demonstrate the applicability and relevance of our derived results in real-world scenarios.

math.OC↗

arXiv2023

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

In this paper, a dynamic model of reconstruction of the shear force $g(t)$ in the Atomic Force Microscopy (AFM) cantilever tip-sample interaction is proposed. The interaction of the cone-shaped cantilever tip with the surface of the specimen (sample) is modeled by the damped Euler-Bernoulli beam equation $ρ_A(x)u_{tt}$ $+μ(x)u_{t}+(r(x)u_{xx}+κ(x)u_{xxt})_{xx}=0$, $(x,t)\in (0,\ell)\times (0,T)$, subject to the following initial, $u(x,0)=0$, $u_t(x,0)=0$ and boundary, $u(0,t)=0$, $u_{x}(0,t)=0$, $\left (r(x)u_{xx}(x,t)+κ(x)u_{xxt} \right )_{x=\ell}=M(t)$, $\left (-(r(x)u_{xx}+κ(x)u_{xxt})_x\right )_{x=\ell}=g(t)$ conditions, where $M(t):=2h\cos θ\,g(t)/π$ is the momentum generated by the transverse shear force $g(t)$. For the reconstruction of $g(t)$ the measured displacement $ν(t):=u(\ell,t)$ is used as an additional data. The least square functional $J(F)=\frac{1}{2}\Vert u(\ell,\cdot)-ν\Vert_{L^2(0,T)}^2$ is introduced and an explicit gradient formula for the Fréchet derivative through the solution of the adjoint problem is derived. This allows 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 $ν(t)$. Computational experiments show that the proposed algorithm is very fast and robust. This allows to develop a numerical "gadget" for computational experiments of generic AFMs.

math-ph↗