Abstract
In this paper we study finite difference procedures for a class of parabolic equations with non‐local boundary condition. The semi‐implicit and fully implicit backward Euler schemes are studied. It is proved that both schemes preserve the maximum principle and monotonicity of the solution of the original equation, and fully‐implicit scheme also possesses strict monotonicity. It is also proved that finite difference solutions approach to zero as t → ∞ exponentially. The numerical results of some examples are presented, which support our theoretical justifications.
Highlights
In this paper we study finite difference approximations to the solution of the following parabolic equations with non-local boundary condition: u Au 0 in
In this article we study finite difference schemes for (1.1)
The finite difference procedures proposed below preserve monotonicity, the maximum principle and the exponential decay of the solution for equation (1.1); they are considered as good numerical approximations
Summary
In this paper we study finite difference approximations to the solution of the following parabolic equations with non-local boundary condition:. The solution u represents the entropy in a quasi-static theory of thermoelasticity [5, 6], so that Day’s results shown that the maximum modulus of the entropy is decreasing in time. In [8] Friedman extended Day’s results to a general parabolic equation in n-dimensions of the ,form with a(z,t) >_ 0 and with the initial and boundary conditions as given in (1.1). Be the entropy, one has [5] that u satisfies auzz b*ut with the boundary conditions. The finite difference procedures proposed below preserve monotonicity, the maximum principle and the exponential decay (if the kernel is non-negative) of the solution for equation (1.1); they are considered as good numerical approximations. For example the weights can be chosen by using trapezoidal rule, AxAy, m,l 1,2,...,N- 1; w,,,t 1/4 AzAy, rn, 6 {0,g}; otherwise
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