By Franz L. Alt (Ed.)

ISBN-10: 0120121026

ISBN-13: 9780120121021

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**Sample text**

Thus, the effect of using two possibly unstable difference equations alternately is to produce a stable equation. Since we are interested in the solution only after the double stcp, let us alter At to be the double step and introduce a n intermediate value notation for the solution at the end of one time step. 6) is unconditionally &stable. 6) represents tridingonal systems of algebraic equations, the algebraic problem is very simple. Finally, we need to investigate the accuracy of the method. This is facilitated by eliminating the intermediate values.

3) Flatt [24] has shown that stability in the uniform sense does not hold for r > R, where R depends on the length of the rod. Another method of analysis must be introduced to treat the convergence of the solution of the Crank-Nicolson equation. This procedure will be based on a combination of Duhamel's principle and harmonic analysis. Let us consider the boundary value problem. The difference equation for the error is i = l ,. . 4) ; thus, the analysis of the error is reduced to estimating t,he solution of the homogeneous initial value problem, starting from the initial data z 2 k + 1 .

The method of proof is essentially Lees’s energy approach. 1), the algebraic equations are linear. The Crank-Nicolson equation may also be generalized; however, a proof + JIM DOUGLAS, JR. 1) has the self-adjoint form a ax (p(x, t ) 2) = ~ ( xt ), $ + F (x, t, u, lfg(x, t, s, u(x, 8)) ds The difference equation becomes Note that the quadrature has been changed in order to preserve the sccond order accuracy in time. It is the author’s conjecture that any convergent difference analogue of the differential equation can be adapted to the integro-differential equation; moreover, it should be possible to maintain the same order of accuracy by a proper choice of the quadrature method.

### Advances in Computers, Vol. 2 by Franz L. Alt (Ed.)

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