Origins, analysis, numerical analysis, and numerical approximation of a forward-backward parabolic problem

Aziz, Abdul; French, Donald A.; Jensen, Sheila Ingemann; Kellogg, R. Bruce

doi:10.1051/m2an:1999125

ESAIM: M2AN

1999

DOI: 10.1051/m2an:1999125

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Origins, analysis, numerical analysis, and numerical approximation of a forward-backward parabolic problem

Abdul Aziz

Donald A. French

Sheila Ingemann Jensen

et al.

Abstract: We consider the analysis and numerical solution of a forward-backward boundary value problem. We provide some motivation, prove existence and uniqueness in a function class especially geared to the problem at hand, provide various energy estimates, prove a priori error estimates for the Galerkin method, and show the results of some numerical computations.

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Cited by 14 publications

(1 citation statement)

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“…Singular parabolic problems arise in a lot of fields of technology and natural sciences. These problems may be divided on different classes accordingly to their singularity with respect either to spatial variables [9,12,16,26], to time variable [3,14,17], or to the unknown u, or its gradient, see [1,2,4,7,10,15,18,23,24,27,28].…”

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A Mathematical Model for the Use of Energy Resources: A Singular Parabolic Equation

López

Pardo

2020

Mathematical Modelling and Analysis

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We consider a singular parabolic equation tβut − ∆u = f, for (x,t)∈ Ω × (0,T), arising in symmetric boundary layer flows. Here Ω ⊂ RN is a bounded domain with C2 boundary ∂Ω,β ≤ 1,f = f(t,x) is bounded, and T > 0 is some fixed time. We establish sufficient conditions for the existence and uniqueness of a weak solution of this singular parabolic equation with Dirichlet boundary conditions, and we investigate its regularity. There are two different cases depending on β. If β < 1, for any initial data u0 ϵ L2(Ω), there exists a unique weak solution, which in fact is a strong solution. The singularity is removable when β < 1. While if β = 1, there exists a unique solution of the singular parabolic problem tut − ∆u = f. The initial data cannot be arbitrarily chosen. In fact, if f is continuous and f(t) → f0, as t → 0, then, this solution converges, as t → 0, to the solution of the elliptic problem −∆u = f0, for x ∈ Ω, with Dirichlet boundary conditions. Hence, no initial data can be prescribed when β = 1, and the singularity in that case is strong.

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A Mathematical Model for the Use of Energy Resources: A Singular Parabolic Equation

López

Pardo

2020

Mathematical Modelling and Analysis

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A non‐overlapping domain decomposition dual reciprocity method for solving the forward–backward heat equation in two‐dimension

Shanazari

Banei

2022

Numerical Methods Partial

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We apply a boundary element dual reciprocity method (DRBEM) to the numerical solution of the forward–backward heat equation in a two‐dimensional case. The method is employed for the spatial variable via the fundamental solution of the Laplace equation and the Crank–Nicolson finite difference scheme is utilized to treat the time variable. The physical domain is divided into two non‐overlapping subdomains resulting in two standard forward and backward parabolic equations. The subproblems are then treated by the underlying method assuming a virtual boundary in the interface and starting with an initial approximate solution on this boundary followed by updating the solution by an iterative procedure. In addition, we show that the time discrete scheme is unconditionally stable and convergent using the energy method. Furthermore, some computational aspects will be suggested to efficiently deal with the formulation of the proposed method. Finally, two forward–backward problems, for which the exact solution is available, will be numerically solved for two different domains to demonstrate the efficiency of the proposed approach.

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Iterative methods for a forward-backward heat equation in two-dimension

Sun

Cheng

2010

Appl. Math. J. Chin. Univ.

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A finite difference method is introduced to solve the forward-backward heat equation in two space dimensions. In this procedure, the backward and forward difference scheme in two subdomains and a coarse-mesh second-order central difference scheme at the middle interface are used. Maximum norm error estimate for the procedure is derived. Then an iterative method based on domain decomposition is presented for the numerical scheme and the convergence of the given method is established. Then numerical experiments are presented to support the theoretical analysis. §1 IntroductionIn this paper we consider the following forward-backward heat equation in two space dimensions:where a(x) > 0 for x > 0, a(x) < 0 for x < 0 and a(0) = 0. Note that (1.1) is a forward heat equation when a(x) > 0 and is a backward heat equation when a(x) < 0. The boundary conditions are chosen to reflect this fact.The given problem (1.1) aries in a variety of applications, such as in boundary layer problems in fluid dynamic [19], in plasma physics, and in astrophysics in the study of propagation of an electron beam through the solar corona [15].

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Origins, analysis, numerical analysis, and numerical approximation of a forward-backward parabolic problem

Cited by 14 publications

References 35 publications

A Mathematical Model for the Use of Energy Resources: A Singular Parabolic Equation

A Mathematical Model for the Use of Energy Resources: A Singular Parabolic Equation

A non‐overlapping domain decomposition dual reciprocity method for solving the forward–backward heat equation in two‐dimension

Iterative methods for a forward-backward heat equation in two-dimension

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