scholarly journals A Note on the Inverse Problem for a Fractional Parabolic Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Abdullah Said Erdogan ◽  
Hulya Uygun
Keyword(s):  
Inverse Problem ◽  
Parabolic Equation ◽  
Operator Theory ◽  
Source Term ◽  
Approximate Solutions ◽  
Time Dependent ◽  
Stability Estimates ◽  
Theory Approach ◽  
One Dimensional ◽  
Term Stability

For a fractional inverse problem with an unknown time-dependent source term, stability estimates are obtained by using operator theory approach. For the approximate solutions of the problem, the stable difference schemes which have first and second orders of accuracy are presented. The algorithm is tested in a one-dimensional fractional inverse problem.

Identification of thermal conductivities by temperature gradient profiles: One‐dimensional steady flow

Geophysics ◽  
1989 ◽  
Vol 54 (5) ◽  
pp. 643-653 ◽  
Author(s):  
G. Ponzini ◽  
G. Crosta ◽  
M. Giudici
Keyword(s):  
Inverse Problem ◽  
Temperature Gradient ◽  
Heat Source ◽  
Multiplicative Noise ◽  
Central Italy ◽  
Admissible Solution ◽  
Stability Estimates ◽  
Source Terms ◽  
One Dimensional ◽  
Thermal Conductivities

The comparison model method (CMM) is applied to the identification of spatially varying thermal conductivity in a one‐dimensional domain. This method deals with the discretized steady‐state heat equation written at the nodes of a lattice, a lattice which models a stack of plane parallel layers. The required data are temperature gradient and heat source (or sink) values. The unknowns of this inverse problem are not nodal values but internode thermal conductivities, which appear in the node heat balance equation. The conductivities, e.g., the solutions to the inverse problem obtained by the CMM, are a one‐parameter family. In order to achieve uniqueness (which coincides with identifiability in this case), a suitable value of this parameter must be found. To this end we consider (1) parameterization, i.e., introducing equality constraints between the unknown coefficients, (2) the use of two data sets at least in a subdomain, and (3) self‐identifiability. Each of these items formally translates the available a priori information about the system, e.g., geophysical properties of the layers. Under the assumption that an admissible solution exists, we evaluate the effects on the solution of two types of data noise: additive noise affecting temperature and a kind of multiplicative noise affecting heat‐source terms. In the numerical examples we provide, parameterization is combined with the CMM in order to obtain the unique solution to a test inverse problem, the geophysical data of which come from a well drilled across Tertiary layers in central Italy. Finally, we consider data perturbed by pseudorandom noise. More precisely, we add noise to temperature gradients and obtain stability estimates which correctly predict the numerical results. In particular, if the layers where the parameterization constraint applies contain a strong source term (e.g., due to flowing water), the solution is relatively insensitive to noise. Also, for multiplicative noise affecting heat‐source terms, theoretical stability predictions are confirmed numerically. The main advantage of the CMM is its simple algebraic formulation. Its implementation in the field by means of a pocket calculator allows both a consistency check on the data being collected and estimates of the unknown values of conductivity.

scholarly journals Recovery of dipolar sources and stability estimates

Filomat ◽  
2019 ◽  
Vol 33 (13) ◽  
pp. 4095-4114
Author(s):  
Ridha Mdimagh
Keyword(s):  
Heat Flux ◽  
Inverse Problem ◽  
Heat Equation ◽  
Numerical Experiments ◽  
Time Dependent ◽  
Stability Estimates ◽  
Lateral Boundary ◽  
Lipschitz Stability ◽  
Spatial Locations ◽  
Stability Results

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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