An Approximate Method for Solving an Inverse Coefficient Problem for the Heat Equation

2021 ◽  
Vol 15 (2) ◽  
pp. 175-189
Author(s):  
I. V. Boykov ◽  
V. A. Ryazantsev
Keyword(s):  
Heat Equation ◽  
Approximate Method ◽  
Coefficient Problem ◽  
Inverse Coefficient Problem ◽  
Inverse Coefficient

Simultaneously identifying the thermal conductivity and radiative coefficient in heat equation from Dirichlet and Neumann boundary measured outputs

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Alemdar Hasanov
Keyword(s):  
Thermal Conductivity ◽  
Heat Equation ◽  
Explicit Formula ◽  
Weak Solutions ◽  
Neumann Boundary ◽  
Coefficient Problem ◽  
Inverse Coefficient Problem ◽  
Inverse Coefficient

AbstractThis paper deals with an inverse coefficient problem of simultaneously identifying the thermal conductivity k(x) and radiative coefficient q(x) in the 1D heat equation u_{t}=(k(x)u_{x})_{x}-q(x)u from the most available Dirichlet and Neumann boundary measured outputs. The Neumann-to-Dirichlet and Neumann-to-Neumann operators \Phi[k,q](t):=u(\ell,t;k,q), \Psi[k,q](t):=-k(0)u_{x}(0,t;k,q) are introduced, and main properties of these operators are derived. Then the Tikhonov functionalJ(k,q)=\tfrac{1}{2}\lVert\Phi[k,q]-\nu\rVert^{2}_{L^{2}(0,T)}+\tfrac{1}{2}\lVert\Psi[k,q]-\varphi\rVert^{2}_{L^{2}(0,T)}of two functions k(x) and q(x) is introduced, and an existence of a quasi-solution of the inverse coefficient problem is proved. An explicit formula for the Fréchet gradient of the Tikhonov functional is derived through the weak solutions of two appropriate adjoint problems.

scholarly journals On the approximate method for determination of heat conduction coefficient

2019 ◽  
Vol 21 (2) ◽  
pp. 149-163
Author(s):  
Ilya V. Boikov ◽  
Vladimir A. Ryazantsev
Keyword(s):  
Approximate Method ◽  
Constant Coefficient ◽  
High Efficiency ◽  
Single Point ◽  
Operator Method ◽  
Continuous Operator ◽  
Coefficient Problem ◽  
Inverse Coefficient Problem ◽  
Wide Range ◽  
Inverse Coefficient

The problem of recovering a value of the constant coefficient in heat equation for one- and two-dimensional cases is considered in the paper. This inverse coefficient problem has broad range of applications in physics and engineering, in particular, for modelling heat exchange processes and for studying properties of materials and designing of engineering constructions. In order to solve the problem an approximate method is constructed; it is based on the continuous operator method for solving nonlinear equations. The advantages of the proposed method are its simplicity and universality. The last property allows to apply the method to a wide range of problems. In particular, in constructing and justifying a continuous operator method, in contrast to the Newton–Kantorovich method, the continuous reversibility of Frechet or Gato derivatives is not required. Moreover, derivatives may not exist on sets of measure zero. The application of continuous operator method to the solution of an inverse coefficient problem with a constant coefficient makes it possible to minimize additional conditions -- there is enough information about the exact solution at a single point x∗,t∗. Solving several model problems illustrates the high efficiency of the proposed method.

Solution of inverse coefficient problem for fractional anomalous diffusion equation

2012 ◽  
Vol 9 (2) ◽  
pp. 65-70
Author(s):  
E.V. Karachurina ◽  
S.Yu. Lukashchuk
Keyword(s):  
Anomalous Diffusion ◽  
Fractional Derivatives ◽  
Descent Method ◽  
Extremum Problem ◽  
Steepest Descent Method ◽  
Coefficient Problem ◽  
Caputo Fractional Derivatives ◽  
Inverse Coefficient Problem ◽  
Residual Functional ◽  
Inverse Coefficient

An inverse coefficient problem is considered for time-fractional anomalous diffusion equations with the Riemann-Liouville and Caputo fractional derivatives. A numerical algorithm is proposed for identification of anomalous diffusivity which is considered as a function of concentration. The algorithm is based on transformation of inverse coefficient problem to extremum problem for the residual functional. The steepest descent method is used for numerical solving of this extremum problem. Necessary expressions for calculating gradient of residual functional are presented. The efficiency of the proposed algorithm is illustrated by several test examples.

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