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.

scholarly journals Uniqueness and numerical reconstruction for inverse problems dealing with interval size search

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jone Apraiz ◽  
Jin Cheng ◽  
Anna Doubova ◽  
Enrique Fernández-Cara ◽  
Masahiro Yamamoto
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Inverse Problems ◽  
Heat Equation ◽  
Numerical Experiments ◽  
Spatial Dimension ◽  
Numerical Reconstruction ◽  
Spatial Interval ◽  
Boundary Information ◽  
Size Estimates

<p style='text-indent:20px;'>We consider a heat equation and a wave equation in one spatial dimension. This article deals with the inverse problem of determining the size of the spatial interval from some extra boundary information on the solution. Under several different circumstances, we prove uniqueness, non-uniqueness and some size estimates. Moreover, we numerically solve the inverse problems and compute accurate approximations of the size. This is illustrated with several satisfactory numerical experiments.</p>

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.

scholarly journals Integrating Semilinear Wave Problems with Time-Dependent Boundary Values Using Arbitrarily High-Order Splitting Methods

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1113
Author(s):  
Isaías Alonso-Mallo ◽  
Ana M. Portillo
Keyword(s):  
Numerical Experiments ◽  
Splitting Method ◽  
Method Of Lines ◽  
Time Integration ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
High Order ◽  
Time Dependent ◽  
Boundary Values ◽  
Lipschitz Continuous

The initial boundary-value problem associated to a semilinear wave equation with time-dependent boundary values was approximated by using the method of lines. Time integration is achieved by means of an explicit time method obtained from an arbitrarily high-order splitting scheme. We propose a technique to incorporate the boundary values that is more accurate than the one obtained in the standard way, which is clearly seen in the numerical experiments. We prove the consistency and convergence, with the same order of the splitting method, of the full discretization carried out with this technique. Although we performed mathematical analysis under the hypothesis that the source term was Lipschitz-continuous, numerical experiments show that this technique works in more general cases.

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