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On the stability of finite-difference schemes for parabolic equations subject to integral conditions with applications to thermoelasticity. (English) Zbl 1161.65072
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.

65M12Stability and convergence of numerical methods (IVP of PDE)
65M06Finite difference methods (IVP of PDE)
74F05Thermal effects in solid mechanics
35K05Heat equation
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