scholarly journals Discretization of equations Gelfand-Levitan-Krein and regularization algorithms

2021 ◽  
Vol 2092 (1) ◽  
pp. 012015
Author(s):  
Bektemessov Maktagali ◽  
Temirbekova Laura
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Initial Boundary ◽  
Iterative Regularization ◽  
Fredholm Equations ◽  
One Dimensional ◽  
Regularizing Algorithms ◽  
The Difference ◽  
The One ◽  
Initial Boundary Value

Abstract The paper considers the initial-boundary-value inverse problem of acoustics for onedimensional and multidimensional cases. The inverse problems are to reconstruct the coefficients using one-dimensional and multidimensional analogues of the Gelfand-Levitan-Krein integral equations. It is known that such equations are linear integral Fredholm equations of the first kind, which are ill-posed. The aim of the work is to find a numerical solution of the Gelfand-Levitan-Krein equation using iterative regularizing algorithms. Using the specifics of these equations (the kernel of the equation depends on the difference of arguments) it is possible to create highly efficient iterative regularizing algorithms. The implemented algorithms can be successfully applied in solving such problems as reconstruction of blurred and defocused images, inverse problem of gravimetric, linear programming problem with inaccurately given matrix of constraints, inverse problem of Geophysics, inverse problems of computed tomography, etc. The main results of the work are the discretization of the one-dimensional and multidimensional Gelfand-Levitan-Krein equation and the construction of iterative regularization algorithms.

Initial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay

2020 ◽  
Vol 75 (8) ◽  
pp. 713-725 ◽  
Author(s):  
Guenbo Hwang
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
One Dimensional ◽  
Advection Dispersion ◽  
Asymptotically Periodic ◽  
The One ◽  
Initial Boundary Value

AbstractInitial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay (LAD) are studied by utilizing a unified method, known as the Fokas method. The method takes advantage of the spectral analysis of both parts of Lax pair and the global algebraic relation coupling all initial and boundary values. We present the explicit analytical solution of the LAD equation posed on the half line and a finite interval with general initial and boundary conditions. In addition, for the case of periodic boundary conditions, we show that the solution of the LAD equation is asymptotically t-periodic for large t if the Dirichlet boundary datum is periodic in t. Furthermore, it can be shown that if the Dirichlet boundary value is asymptotically periodic for large t, then so is the unknown Neumann boundary value, which is uniquely characterized in terms of the given asymptotically periodic Dirichlet boundary datum. The analytical predictions for large t are compared with numerical results showing the excellent agreement.

scholarly journals Micromechanical modeling of sulphate corrosion in concrete: Influence of ettringite forming reaction

2008 ◽  
Vol 35 (1-3) ◽  
pp. 29-52 ◽  
Author(s):  
M. Basista ◽  
W. Weglewski
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Sodium Sulphate ◽  
Micromechanical Modeling ◽  
Hardened Cement ◽  
Ettringite Formation ◽  
Equivalent Inclusion Method ◽  
The Difference ◽  
The One ◽  
Initial Boundary Value

Two micromechanical models are developed to simulate the expansion of cementitious composites exposed to external sulphate attack. The difference between the two models lies in the form of chemical reaction of the ettringite formation (through-solution vs. topochemical). In both models the Fick's second law with reaction term is assumed to govern the transport of the sulphate ions. The Eshelby solution and the equivalent inclusion method are used to determine the eigenstrain of the expanding ettringite crystals in microcracked hardened cement paste. The degradation of transport properties is studied in the effective medium and the percolation regime. An initial-boundary value problem (2D) of expansion of a mortar specimen immersed in a sodium sulphate solution is solved and compared with available test data. The obtained results indicate that the topochemical mechanism is the one capable of producing the experimentally observed amount of expansion.

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