Asymptotic expansion of solutions to an inverse problem of parabolic type with non-homogeneous boundary conditions

2011 ◽  
Vol 141 (4) ◽  
pp. 777-817 ◽  
Author(s):  
Davide Guidetti
Keyword(s):  
Inverse Problem ◽  
Asymptotic Expansion ◽  
Boundary Conditions ◽  
General Result ◽  
Parabolic Problem ◽  
Source Function ◽  
Parabolic Type ◽  
Time Dependent ◽  
Asymptotic Expansion Of Solutions ◽  
Inverse Parabolic Problem

We consider an inverse parabolic problem of reconstruction of the source function, together with the traditional solution. In contrast with older literature, we consider non-homogeneous and time-dependent boundary conditions. We are able to prove a general result of convergence to a stationary state, and of asymptotic expansion as t → ∞.

Identification of a space- and time-dependent source in a variable coefficient advection-diffusion equation from Dirichlet and Neumann boundary measured outputs

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Cristiana Sebu
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Variable Coefficient ◽  
Source Function ◽  
Practical Interest ◽  
Time Dependent ◽  
Input Output ◽  
Space And Time ◽  
Advection Diffusion Equation ◽  
Advection Diffusion

AbstractThis paper considers the inverse problem of identifying an unknown space- and time-dependent source function F(x,t) in the variable coefficient advection-diffusion equationu_{t}=(D(x)u_{x})_{x}-(V(x)u)_{x}+F(x,t)from the Dirichlet \nu(t):=u(\ell,t) and Neumann f(t):=-D(0)u_{x}(0,t), t\in(0,T], boundary measured outputs. This problem was motivated by several important real-world applications in the field of contaminant hydrogeology, and the novel analysis presented here is highly relevant to problems of practical interest. The input-output operators corresponding to the Dirichlet and Neumann measured boundary data are introduced. The inverse problem is then formulated as a system of operator equations consisting of these operators and the measured outputs. The compactness and Lipschitz continuity of the input-output operators are proved in the relevant classes of admissible source functions ℱ and \mathcal{F}_{r}. These results together with the derived trace estimates allow us to show the existence of a quasi-solution of the inverse source problem as a minimum of the Tikhonov functional, under minimal regularity assumptions with respect to the source function and other inputs. An explicit gradient formula for the Fréchet gradient of the Tikhonov functional is also derived by means of an appropriate adjoint problem.

Identification of a Time Dependent Source Function in a Parabolic Inverse Problem Via Finite Element Approach

2020 ◽  
Vol 51 (4) ◽  
pp. 1587-1602
Author(s):  
J. Damirchi ◽  
R. Pourgholi ◽  
T. R. Shamami ◽  
H. Zeidabadi ◽  
A. Janmohammadi
Keyword(s):  
Inverse Problem ◽  
Finite Element ◽  
Source Function ◽  
Time Dependent ◽  
Finite Element Approach ◽  
Element Approach

scholarly journals On the inverse problem for a heat-like equation

1987 ◽  
Vol 1 (2) ◽  
pp. 81-97 ◽  
Author(s):  
Igor Malyshev
Keyword(s):  
Inverse Problem ◽  
Boundary Data ◽  
Mixed Type ◽  
Parabolic Type ◽  
Time Dependent ◽  
Nonlinear Volterra Integral Equation ◽  
Parabolic Case ◽  
Special Cases ◽  
Uniqueness Of The Solution ◽  
Explicit Formulae

Using the integral representation of the solution of the boundary value problem for the equation with one time-dependent coefficient at the highest space-derivative three inverse problems are solved. Depending on the property of the coefficient we consider cases when the equation is of the parabolic type and two special cases of the degenerate/mixed type. In the parabolic case the corresponding inverse problem is reduced to the nonlinear Volterra integral equation for which the uniqueness of the solution is proved. For the special cases explicit formulae are derived. Both “minimal” and overspecified boundary data are considered.

The inverse problem of recovering the coefficients of a differential equations on a graph

2020 ◽  
Vol 28 (5) ◽  
pp. 727-738
Author(s):  
Victor Sadovnichii ◽  
Yaudat Talgatovich Sultanaev ◽  
Azamat Akhtyamov
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Characteristic Equation ◽  
Arbitrary Number ◽  
New Class ◽  
Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

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