scholarly journals Conditional Stability for an Inverse Problem of Determining a Space-Dependent Source Coefficient in the Advection-Dispersion Equation with Robin’s Boundary Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Shunqin Wang ◽  
Chunlong Sun ◽  
Gongsheng Li
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Dispersion Equation ◽  
Integral Identity ◽  
Adjoint Problem ◽  
Conditional Stability ◽  
Admissible Set ◽  
Lipschitz Stability ◽  
One Dimensional ◽  
Advection Dispersion

This paper deals with an inverse problem of determining the space-dependent source coefficient in one-dimensional advection-dispersion equation with Robin’s boundary condition. Data compatibility for the inverse problem is analyzed by which an admissible set for the unknown is set forth. Furthermore, with the help of an integral identity, a conditional Lipschitz stability is established by suitably controlling the solution of an adjoint problem.

A conditional Lipschitz stability for determining a space-dependent source coefficient in the 2D/3D advection-dispersion equation

2017 ◽  
Vol 25 (2) ◽  
Author(s):  
Gongsheng Li ◽  
Xianzheng Jia ◽  
Chunlong Sun
Keyword(s):  
Inverse Problem ◽  
Dispersion Equation ◽  
Adjoint Method ◽  
Noisy Data ◽  
Adjoint Problem ◽  
Admissible Set ◽  
Lipschitz Stability ◽  
Optimal Perturbation ◽  
Advection Dispersion ◽  
Bilinear Functional

Abstract This paper deals with an inverse problem of determining a space-dependent source coefficient in the 2D/3D advection-dispersion equation with final observations using the variational adjoint method. Data compatibility for the inverse problem is analyzed by which an admissible set for the unknowns is induced. With the aid of an adjoint problem, a bilinear functional based on the variational identity is set forth with which a norm for the unknown is well-defined under suitable conditions, and then a conditional Lipschitz stability for the inverse problem is established. Furthermore, numerical inversions with random noisy data are performed using the optimal perturbation algorithm, and the inversion solutions give good approximations to the exact solution as the noise level goes to small.

scholarly journals Determination of a nonlinear source term in a diffusion equation with final observations

2004 ◽  
Vol 2004 (14) ◽  
pp. 741-753 ◽  
Author(s):  
Gongsheng Li ◽  
Yichen Ma ◽  
Kaitai Li
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Integral Identity ◽  
Source Term ◽  
Conditional Stability ◽  
Inverse Source Problem ◽  
Source Terms ◽  
Nonlinear Source ◽  
Quasilinear Diffusion

This paper deals with an inverse problem of determining a nonlinear source term in a quasilinear diffusion equation with overposed final observations. Applying integral identity methods, data compatibilities are deduced by which the inverse source problem here is proved to be reasonable and solvable. Furthermore, with the aid of an integral identity that connects the unknown source terms with the known data, a conditional stability is established.

scholarly journals Analytic adjoint solutions for the quasi-one-dimensional Euler equations

2001 ◽  
Vol 426 ◽  
pp. 327-345 ◽  
Author(s):  
MICHAEL B. GILES ◽  
NILES A. PIERCE
Keyword(s):  
Boundary Condition ◽  
Euler Equations ◽  
Arbitrary Shape ◽  
Logarithmic Singularity ◽  
Adjoint Problem ◽  
Function Approach ◽  
Transonic Flows ◽  
One Dimensional ◽  
Adjoint Variables ◽  
Adjoint Solutions

The analytic properties of adjoint solutions are examined for the quasi-one-dimensional Euler equations. For shocked flow, the derivation of the adjoint problem reveals that the adjoint variables are continuous with zero gradient at the shock, and that an internal adjoint boundary condition is required at the shock. A Green's function approach is used to derive the analytic adjoint solutions corresponding to supersonic, subsonic, isentropic and shocked transonic flows in a converging–diverging duct of arbitrary shape. This analysis reveals a logarithmic singularity at the sonic throat and confirms the expected properties at the shock.

ONE-DIMENSIONAL TEMPORALLY DEPENDENT ADVECTION-DISPERSION EQUATION IN POROUS MEDIA: ANALYTICAL SOLUTION

2010 ◽  
Vol 23 (4) ◽  
pp. 521-539 ◽  
Author(s):  
R. R. YADAV ◽  
DILIP KUMAR JAISWAL ◽  
HAREESH KUMAR YADAV ◽  
GUL RANA
Keyword(s):  
Porous Media ◽  
Analytical Solution ◽  
Dispersion Equation ◽  
One Dimensional ◽  
Advection Dispersion

Advection-dispersion modelling tools: what about numerical dispersion?

2005 ◽  
Vol 52 (3) ◽  
pp. 19-27 ◽  
Author(s):  
R. Bouteligier ◽  
G. Vaes ◽  
J. Berlamont ◽  
C. Flamink ◽  
J.G. Langeveld ◽  
...  
Keyword(s):  
Dispersion Equation ◽  
Model Calibration ◽  
Suspended Particles ◽  
Numerical Dispersion ◽  
Model Parameters ◽  
One Dimensional ◽  
Advection Dispersion ◽  
The One ◽  
Calibration Tool ◽  
Tracer Experiments

In general the transport of dissolved substances and fine suspended particles is governed by the one-dimensional advection-dispersion equation. In order to model the transport of dissolved substances and fine suspended particles, the advection-dispersion equation is incorporated into commonly used urban drainage modelling tools such as InfoWorks CS (Wallingford Software, United Kingdom) and MOUSE (DHI Software, Denmark). Two examples show the use of InfoWorks CS and MOUSE using standard model settings. Modelling results using tracer experiments show that numerical model parameters need to be altered in order to calibrate the model. Using tracer experiments as a model calibration tool, it is shown that a non-negligible amount of dispersion is generated by InfoWorks CS and MOUSE and that it is in fact the numerical dispersion that is calibrated.

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