Two dimensional determination of source terms in linear parabolic equation from the final overdetermination

AbstractIn this study, we develop and implement numerical schemes to solve classes of two-dimensional fourth-order partial differential equations. These methods are fourth-order accurate in space and second-order accurate in time and require only nine spatial grid points of a single compact cell. The proposed discretizations allow the use of Dirichlet boundary conditions only without the need to discretize the derivative boundary conditions and thus avoids the use of ghost points. No transformation or linearization technique is used to handle nonlinearity and the obtained block tri-diagonal nonlinear system has been solved by Newton’s block iteration method. It is discussed how our formulation is able to tackle linear singular problems and it is ensured that the methods retain their orders and accuracy everywhere in the solution region. The proposed two-level method is shown to be unconditionally stable for a class of two-dimensional fourth-order linear parabolic equation. We also discuss the alternating direction implicit (ADI) method for solving two-dimensional fourth-order linear parabolic equation. The proposed difference methods has been successfully tested on the two-dimensional vibration problem, Boussinesq equation, extended Fisher–Kolmogorov equation and Kuramoto–Sivashinsky equation. Numerical results demonstrate that the schemes are highly accurate in solving a large class of physical problems.

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Determination of the coefficient of u x in a linear parabolic equation

Inverse Problems ◽

10.1088/0266-5611/10/3/002 ◽

1994 ◽

Vol 10 (3) ◽

pp. 521-531 ◽

Cited By ~ 7

Author(s):

J R Cannon ◽

S Perez-Esteva

Keyword(s):

Parabolic Equation ◽

Linear Parabolic Equation

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An inverse backward problem for degenerate two-dimensional parabolic equation

Opuscula Mathematica ◽

10.7494/opmath.2020.40.4.427 ◽

2020 ◽

Vol 40 (4) ◽

pp. 427-449

Author(s):

Khalid Atifi ◽

El-Hassan Essoufi ◽

Bouchra Khouiti

Keyword(s):

Parabolic Equation ◽

Lipschitz Continuity ◽

Descent Method ◽

Initial Condition ◽

Two Dimensional ◽

Backward Problem ◽

Output Least Squares ◽

Open Bounded Subset ◽

Least Squares Approach

This paper deals with the determination of an initial condition in the degenerate two-dimensional parabolic equation \[\partial_{t}u-\mathrm{div}\left(a(x,y)I_2\nabla u\right)=f,\quad (x,y)\in\Omega,\; t\in(0,T),\] where \(\Omega\) is an open, bounded subset of \(\mathbb{R}^2\), \(a \in C^1(\bar{\Omega})\) with \(a\geqslant 0\) everywhere, and \(f\in L^{2}(\Omega \times (0,T))\), with initial and boundary conditions \[u(x,y,0)=u_0(x,y), \quad u\mid_{\partial\Omega}=0,\] from final observations. This inverse problem is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. To show the convergence of the descent method, we prove the Lipschitz continuity of the gradient of the Tikhonov functional. Also we present some numerical experiments to show the performance and stability of the proposed approach.

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