By J. D. Lambert

ISBN-10: 0471511943

ISBN-13: 9780471511946

**Read or Download Computational Methods in Ordinary Differential Equations (Introductory mathematics for scientists & engineers) PDF**

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**Extra info for Computational Methods in Ordinary Differential Equations (Introductory mathematics for scientists & engineers) **

**Sample text**

Moreover. if Co = C, = ... = C, = O. then. from (22). +,. + l = C,+2 - tC,+,. Hence the first non-vanishing coefficient in the expansion (21) is independent of the choice for t. but subsequent coefficients are functions of t. The formulae giving the constants Dq. defined by (21). in terms of the coefficients a). PJ are Do = ao + a l + a, + ... , D, + (1 - t)

K - I? (ii) How do we choose a suitable value for the steplength h? H? To these we can add a fourth question which we ought always to ask after the calculation has been completed. (iv) How accurate is the numerical solution we have obtained? As we shall see, problem (i) presents little difficulty. Problem (ii), which is closely linked with (iv), constitutes the major problem in the application of linear multistep methods, and one to which a completely satisfactory answer does not exist at the present time.

The word 'zero' is chosen here since the stability phenomenon under consideration at the moment is allied to the notion of convergence in the limit as h tends to zero. Zero-stability ensures that those solutions of the difference equation for Yn which arise because the first-order differential equation is being replaced bya higher order difference equation (frequently called parasitic solutions) are damped out in the limit as h ... O. For a one-step method, the polynomial has degree one, and if is + 1.

### Computational Methods in Ordinary Differential Equations (Introductory mathematics for scientists & engineers) by J. D. Lambert

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