Formulation and analysis of a parabolic transmission problem on disjoint intervals

Jovanović, Boško S.; Vulkov, Lubin G.

doi:10.2298/pim1205111j

Cited by 6 publications

(11 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A motivated heat-mass transfer process that leads to the one-dimensional version of the present forward problem is described in [42].…”

Section: Formulation Of the Problemmentioning

confidence: 99%

Numerical Reconstruction of Time-Dependent Boundary Conditions to 2D Heat Equation on Disjoint Rectangles at Integral Observations

Koleva,

Vulkov

2024

Mathematics

View full text Add to dashboard Cite

In this paper, two-dimensional (2D) heat equations on disjoint rectangles are considered. The solutions are connected by interface Robin’s-type internal conditions. The problem has external Dirichlet boundary conditions that, in the forward (direct) formulation, are given functions. In the inverse problem formulation, the Dirichlet conditions are unknown functions, and the aim is to be reconstructed upon integral observations. Well-posedness both for direct and inverse problems is established. Using the given 2D integrals of the unknown solution on each of the domains and the specific interface boundary conditions, we reduce the 2D inverse problem to a forward heat 1D one. The resulting 1D problem is solved using the explicit Saul’yev finite difference method. Numerical test examples are discussed to illustrate the efficiency of the approach.

show abstract

“…A motivated heat-mass transfer process that leads to the one-dimensional version of the present forward problem is described in [42].…”

Section: Formulation Of the Problemmentioning

confidence: 99%

Numerical Reconstruction of Time-Dependent Boundary Conditions to 2D Heat Equation on Disjoint Rectangles at Integral Observations

Koleva,

Vulkov

2024

Mathematics

View full text Add to dashboard Cite

show abstract

“…[10,11,12,17,21]. Despite of the special structure of the equation ( 14), it can be investigated for existence, uniqueness and continuous dependence from input data of the BVP (1)-( 6) and ( 7), following the methodology in [9].…”

Section: Ip[1]mentioning

confidence: 99%

“…[1,4,6,18]. Numerical methods for such type interface problems are studied in [7,8,9,16] Differential equations, involving values of the unknown function at domain points very often are called loaded differential equations, see e.g. [2,3,19,21].…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of time-dependent right-hand side in parabolic equations on disjoint domains

Koleva,

Vulkov

2023

J. Phys.: Conf. Ser.

View full text Add to dashboard Cite

We study inverse problems of reconstructing the time-dependent right-hand side from point observation in a one dimensional parabolic equation on disjoint intervals. These problems are ill-possed, i.e. very slight errors in the additional input may cause relatively significant errors in the output of the left and right internal right-hand side. In this work, we construct computational algorithms, using the loaded equation method. First, we perform a decomposition with respect to the unknown source of the inverse problem solutions. Then the inverse problems are reduced to a loaded parabolic equation problems. The well-posedness of the inverse problems is studied on the base of loaded equation ones. The numerical performance of the approach is realized by finite difference schemes, solved with decomposition algorithms. Computational experiments show the efficiency of the method.

show abstract

“…In this work, we consider a parabolic transmission problem on disjoint domains with exact interface conditions. The derivation of this conditions is discussed in [6,7].…”

Section: Introductionmentioning

confidence: 99%

“…The direct parabolic transmission problem in disjoint domains is well studied in the literature. The existence and uniqueness of the weak and strong solutions and the a priori estimate in an appropriate Sobolev-like space were proven in [6,[8][9][10][11]. Numerical methods for solving such problems were constructed and analyzed in [8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of the Retrospective Inverse Parabolic Problem on Disjoint Intervals

Koleva,

Vulkov

2023

Computation

View full text Add to dashboard Cite

The retrospective inverse problem for evolution equations is formulated as the reconstruction of unknown initial data by a given solution at the final time. We consider the inverse retrospective problem for a one-dimensional parabolic equation in two disconnected intervals with weak solutions in weighted Sobolev spaces. The two solutions are connected with nonstandard interface conditions, and thus this problem is solved in the whole spatial region. Such a problem, as with other inverse problems, is ill-posed, and for its numerical solution, specific techniques have to be used. The direct problem is first discretized by a difference scheme which provides a second order of approximation in space. For the resulting ordinary differential equation system, the positive coerciveness is established. Next, we develop an iterative conjugate gradient method to solve the ill-posed systems of the difference equations, which are obtained after weighted time discretization, of the inverse problem. Test examples with noisy input data are discussed.

show abstract

Formulation and analysis of a parabolic transmission problem on disjoint intervals

Abstract: We investigate an initial-boundary-value problem for one dimensional parabolic equations in disjoint intervals. Under some natural assumptions on the input data we proved the well-posedness of the problem. Nonnegativity and energy stability of its weak solutions are also studied.

Cited by 6 publications

References 23 publications

Numerical Reconstruction of Time-Dependent Boundary Conditions to 2D Heat Equation on Disjoint Rectangles at Integral Observations

Numerical Reconstruction of Time-Dependent Boundary Conditions to 2D Heat Equation on Disjoint Rectangles at Integral Observations

Reconstruction of time-dependent right-hand side in parabolic equations on disjoint domains

Numerical Solution of the Retrospective Inverse Parabolic Problem on Disjoint Intervals

Contact Info

Product

Resources

About