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Zbl 1161.65072
Jesevičiūtė, Ž.; Sapagovas, M.
On the stability of finite-difference schemes for parabolic equations subject to integral conditions with applications to thermoelasticity. (English)
[J] Comput. Methods Appl. Math. 8, No. 4, 360-373 (2008). ISSN 1609-4840; ISSN 1609-9389/e

The paper deals with the one-dimensional, linear, constant coefficient heat diffusion equation $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + f(x,t), \quad a_1<x<a_2, \;\; 0\leq t \leq T,$$ subject to nonlocal boundary conditions -- namely, integral conditions -- $$u(a_i,t)=\int_{a_1}^{a_2} \alpha_i(x)u(x,t)dx + \beta_i(t),\; \;i=1,2,$$ as well as the initial condition $$ u(x,0) =\phi(x).$$ An implicit finite difference approximation scheme is proposed, for which stability criteria are deduced, the main point being that the system matrix is never symmetric.
[Carlos A. De Moura (Rio de Janeiro)]

MSC 2000:: *65M12 Stability and convergence of numerical methods (IVP of PDE)
65M06 Finite difference methods (IVP of PDE)
74F05 Thermal effects
35K05 Heat equation

Keywords: one-dimensional parabolic equations; nonlocal integral conditions; finite-difference scheme; stability; thermoelasticity; heat diffusion equation; nonlocal boundary conditions

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