Simultaneous identification of diffusion coefficient, spacewise dependent source and initial value for one‐dimensional heat equation

Zhao, Zhixue; Banda, Mapundi K.; Guo, Bao‐Zhu

doi:10.1002/mma.4245

Cited by 15 publications

(3 citation statements)

References 27 publications

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“…Chen et al [12] applied an inverse algorithm based on the conjugate gradient method and the discrepancy principle to solve the inverse hyperbolic heat conduction problem. Zhao and Banda [13] deal with an inverse problem of determining the diffusion coefficient, spacewise dependent source term, and initial value simultaneously for the one-dimensional heat equation. Beck and Woodbury [14] estimated the time and/or space dependence of the surface heat flux or temperature utilizing interior temperature measurements at discrete times and/or locations.…”

Section: Introductionmentioning

confidence: 99%

Estimation of Heat Flux in Two-Dimensional Nonhomogeneous Parabolic Equation Based on a Sufficient Descent Levenberg–Marquard Algorithm

Pang

et al. 2021

Journal of Mathematics

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The main work of this paper focuses on identifying the heat flux in inverse problem of two-dimensional nonhomogeneous parabolic equation, which has wide application in the industrial field such as steel-making and continuous casting. Firstly, the existence of the weak solution of the inverse problem is discussed. With the help of forward solution and dual equation, this paper proves the Lipchitz continuity of the cost function and derives the Lipchitz constant. Furthermore, in order to accelerate the convergence rate and reduce the running time, this paper presents a sufficient descent Levenberg–Marquard algorithm with adaptive parameter (SD-LMAP) to solve this inverse problem. At last, compared with other methods, the results of simulation experiment show that this algorithm can obviously reduce the running time and iterative number.

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Section: Introductionmentioning

confidence: 99%

Estimation of Heat Flux in Two-Dimensional Nonhomogeneous Parabolic Equation Based on a Sufficient Descent Levenberg–Marquard Algorithm

Pang

et al. 2021

Journal of Mathematics

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“…Coupled boundary integral equation method was developed to recover a time-dependent heat source under additional measures of temperature at interior points [31,32]. In [39] a Dirichlet series representation for the boundary observation, and a finite difference approximation method in conjunction with the truncated singular value decomposition was used for the problem inverse of determining the diffusion coefficient, spacewise dependent source term. The reciprocity gap principle [5] was used in many works for the study of the point sources identification problem via elliptic equations [10, 16-19, 23, 33, 34], and was extended in [22] for the monopolar sources identification from fractional diffusion equation.…”

Section: Introductionmentioning

confidence: 99%

Recovery of dipolar sources and stability estimates

Mdimagh

2019

Filomat

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The inverse problem of identifying dipolar sources with time-dependent moments, located in a bounded domain, via the heat equation is investigated, by applying a heat flux, and from a single lateral boundary measurement of temperature. An uniqueness, and local Lipschitz stability results for this inverse problem are established which are the main contributions of this work. A non-iterative algebraic algorithm based on the reciprocity gap concept is proposed, which permits to determine the number, the spatial locations, and the time-dependent moments of the dipolar sources, Some numerical experiments are given in order to test the efficiency and the robustness of this method.

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“…However, there is no general methodology for the identification and observation of infinite-dimensional systems. Nevertheless the works of [19], [20] and [18] have inspired us a lot for the present study.…”

Section: Introductionmentioning

confidence: 99%

Identification of water depth and velocity potential for water waves

Yang

Pei

2019

Systems & Control Letters

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The purpose of this paper is to investigate the identification of the water depth and the water velocity potential in a coastal region by using the linearized water wave equation (LWWE). Existence and uniqueness of the solutions to the partial differential equation LWWE are shown by using the semigroup theory. Moreover the analytical solution is found by the separation of variables method. We assume that the surface wave elevation is measurable. We like to recover the water depth and the water velocity potential from the measurement. This identification problem is shown to be well-posed by proving the parameters' identifiability by the surface elevation. Based on the classical gradient descent method we elaborate an identification algorithm to recover simultaneously both the water depth and the velocity potential. Numerical simulations are carried out to illustrate effectiveness of the algorithm. through the mathematical model of water waves and measuring the surface wave elevation.The water wave equation consists in describing the motion of the water waves occupying a domain delimited below by a fixed bottom and by a free surface above. To write down the water wave equation, let Ω t = {(x, z) ∈ R + × (−h, η(x, t)) } denote the water domain in 2-dimensional euclidian framework as illustrated in Fig.??, where R + = [0, ∞), η(x, t) is the surface elevation of water wave at position x and time t, and h the water depth. We assume that the water and the water waves satisfy the following assumptions: (A1) The water is incompressible ; (A2) There is no surface tension and the water is inviscid; (A3) The water particles do not cross the bottom and the surface; (A4) The external pressure is constant; (A5) The seabed is flat, so that h is positive constant; (A6) The water wave is irrotational. Since the irrotational assumption has been made, consequently there exists a flow potential φ = φ(x, z, t) such that the velocity field V is written by V = (φ x , φ z ) T , where φ x , φ z denote the partial derivative of φ with respect to x and to z, respectively. Thus, the mass conservation is expressed by the Laplace equationwith the boundary condition at the bottom φ z = 0, on z = −h.The Neumann condition (2) means that at the bottom of seabed, the normal component of the velocity is zero. The dynamical and kinematical boundary conditions on the free surface are

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Simultaneous identification of diffusion coefficient, spacewise dependent source and initial value for one‐dimensional heat equation

Cited by 15 publications

References 27 publications

Estimation of Heat Flux in Two-Dimensional Nonhomogeneous Parabolic Equation Based on a Sufficient Descent Levenberg–Marquard Algorithm

Estimation of Heat Flux in Two-Dimensional Nonhomogeneous Parabolic Equation Based on a Sufficient Descent Levenberg–Marquard Algorithm

Recovery of dipolar sources and stability estimates

Identification of water depth and velocity potential for water waves

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