Asymptotic solution of the inverse problem for restoring the modular type source in Burgers’ equation with modular advection

2020 ◽  
Vol 28 (5) ◽  
pp. 633-639
Author(s):  
Nikolay Nikolaevich Nefedov ◽  
V. T. Volkov
Keyword(s):  
Inverse Problem ◽  
Direct Problem ◽  
Burgers Equation ◽  
Type Equation ◽  
Singularly Perturbed ◽  
Time Interval ◽  
Internal Transition Layer ◽  
Internal Transition ◽  
Time Periodic ◽  
Time Periodic Solution

AbstractFor a singularly perturbed Burgers’ type equation with modular advection that has a time-periodic solution with an internal transition layer, asymptotic analysis is applied to solve the inverse problem for restoring the function of the source using known information about the observed solution of a direct problem at a given time interval (period).

Some features of solving an inverse backward problem for a generalized Burgers’ equation

2020 ◽  
Vol 28 (5) ◽  
pp. 641-649
Author(s):  
Dmitry V. Lukyanenko ◽  
Igor V. Prigorniy ◽  
Maxim A. Shishlenin
Keyword(s):  
Inverse Problem ◽  
Direct Problem ◽  
Numerical Experiments ◽  
Burgers Equation ◽  
Simple Algorithm ◽  
Singularly Perturbed ◽  
Generalized Burgers Equation ◽  
Backward Problem ◽  
Singularly Perturbed Parabolic Equation ◽  
The Cost

AbstractIn this paper, we consider an inverse backward problem for a nonlinear singularly perturbed parabolic equation of the Burgers’ type. We demonstrate how a method of asymptotic analysis of the direct problem allows developing a rather simple algorithm for solving the inverse problem in comparison with minimization of the cost functional. Numerical experiments demonstrate the effectiveness of this approach.

scholarly journals Stable transition layer induced by degeneracy of the spatial inhomogeneities in the Allen-Cahn problem

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Maicon Sônego ◽  
Arnaldo Simal do Nascimento
Keyword(s):  
Boundary Condition ◽  
Transition Layer ◽  
Stationary Solutions ◽  
Singularly Perturbed ◽  
Flux Boundary Condition ◽  
Internal Transition Layer ◽  
Internal Transition ◽  
Stable Stationary Solutions ◽  
Spatial Inhomogeneities

<p style='text-indent:20px;'>In this article we consider a singularly perturbed Allen-Cahn problem <inline-formula><tex-math id="M1">\begin{document}$ u_t = \epsilon^2(a^2u_x)_x+b^2(u-u^3) $\end{document}</tex-math></inline-formula>, for <inline-formula><tex-math id="M2">\begin{document}$ (x,t)\in (0,1)\times\mathbb{R}^+ $\end{document}</tex-math></inline-formula>, supplied with no-flux boundary condition. The novelty here lies in the fact that the nonnegative spatial inhomogeneities <inline-formula><tex-math id="M3">\begin{document}$ a(\cdot) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ b(\cdot) $\end{document}</tex-math></inline-formula> are allowed to vanish at some points in <inline-formula><tex-math id="M5">\begin{document}$ (0,1) $\end{document}</tex-math></inline-formula>. Using the variational concept of <inline-formula><tex-math id="M6">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-convergence we prove that, for <inline-formula><tex-math id="M7">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> small, such degeneracy of <inline-formula><tex-math id="M8">\begin{document}$ a(\cdot) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ b(\cdot) $\end{document}</tex-math></inline-formula> induces the existence of stable stationary solutions which develop internal transition layer as <inline-formula><tex-math id="M10">\begin{document}$ \epsilon\to 0 $\end{document}</tex-math></inline-formula>.</p>

QUADRANT-SEARCHING: A NEW TECHNIQUE TO REDUCE THE SEARCH SPACE OF THE INVERSE PROBLEM OF EIT USING BEM TO THE DIRECT PROBLEM

2011 ◽  
Vol 22 (08) ◽  
pp. 825-839 ◽  
Author(s):  
OLAVO H. MENIN ◽  
VANESSA ROLNIK
Keyword(s):  
Inverse Problem ◽  
Iterative Method ◽  
Direct Problem ◽  
Search Space ◽  
Initial Solution ◽  
New Technique ◽  
Time Interval ◽  
Functional Surface ◽  
Error Functional ◽  
The Error Functional

The image reconstruction using the EIT (Electrical Impedance Tomography) technique is a nonlinear and ill-posed inverse problem which demands a powerful direct or iterative method. A typical approach for solving the problem is to minimize an error functional using an iterative method. In this case, an initial solution close enough to the global minimum is mandatory to ensure the convergence to the correct minimum in an appropriate time interval. The aim of this paper is to present a new, simple and low cost technique (quadrant-searching) to reduce the search space and consequently to obtain an initial solution of the inverse problem of EIT. This technique calculates the error functional for four different contrast distributions placing a large prospective inclusion in the four quadrants of the domain. Comparing the four values of the error functional it is possible to get conclusions about the internal electric contrast. For this purpose, initially we performed tests to assess the accuracy of the BEM (Boundary Element Method) when applied to the direct problem of the EIT and to verify the behavior of error functional surface in the search space. Finally, numerical tests have been performed to verify the new technique.

NUMERICAL SOLUTION OF THE INVERSE PROBLEM FOR A SYSTEM OF DIFFERENTIAL EQUATIONS

2020 ◽  
pp. 106-110
Author(s):  
S.E. Kasenov ◽  
◽  
G.E. Kasenova ◽  
A.A. Sultangazin ◽  
B.D. Bakytbekova ◽  
...  
Keyword(s):  
Inverse Problem ◽  
Differential Equations ◽  
Numerical Solution ◽  
Direct Problem ◽  
Nonlinear Differential Equations ◽  
Direct And Inverse Problems ◽  
Additional Information ◽  
Several Variables ◽  
Gradient Of The Functional ◽  
Objective Functional

The article considers direct and inverse problems of a system of nonlinear differential equations. Such problems are often found in various fields of science, especially in medicine, chemistry and economics. One of the main methods for solving nonlinear differential equations is the numerical method. The initial direct problem is solved by the Rune-Kutta method with second accuracy and graphs of the numerical solution are shown. The inverse problem of finding the coefficients of a system of nonlinear differential equations with additional information on solving the direct problem is posed. The numerical solution of this inverse problem is reduced to minimizing the objective functional. One of the methods that is applicable to nonsmooth and noisy functionals, unconditional optimization of the functional of several variables, which does not use the gradient of the functional, is the Nelder-Mead method. The article presents the NellerMead algorithm. And also a numerical solution of the inverse problem is shown.

Sign in / Sign up

Export Citation Format

Share Document