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By J. D. Lambert

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

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0, and this is the first of the conditions (26). It should be observed that we have not made any use of the differential equation so far. } tends to some function y(x). Linear multistep methods I: basic theory 31 The second condition of (26) ensures that the function y(x) does in fact satisfy the differential equation. For, under the limiting process, (y,+ j - y,)/jh = jhy'(x) --+ j = 1, 2, ... ,(h), j = 1,2, ... ,(h) = 0. ,(h). Since L~=o et j = 0, we have, on dividing through by h, • L PJ,+ j=O j = • • y'(x) L jetj + L jetjcPj,,(h).

L = th(3/"+1 - . 2/,,) to compute a numerical solution/or the initial value problem of e"Xample 4. For this method, L:• aj = 0, J-O so that the first of the consistency conditions (26) is satisfied, but the second is not. } will <:onverge to a function y(x)as the steplength tends to zero, but this function y(x) will not be the solution of the initial value problem_ We observe that the failure of this inconsistent method is less dramatic than that of the zero-unstable method of example 4. This is because our inconsistent method is zero-stable and thus does not propagate errors in an explosive manner.

Since P2 = 0 for Euler's rule, the validity of (24) is verified. 2! 3! Y + ... (q-I)! y + .. 2! + (~! )h'A' + .. I I 1h4 A • + ... 3! J h2A2 = [exp (nhA)] { exp (hA) - I - hA - ""2 - thA[exp(hA) - I - hAJ} = [exp(nhA)][(1 - 1hA) exp (hA) - (I Also Y•• I - Y. = 1hA(y•• I + yJ + 1hA)]. , / 30 Computational methods in ordinary differential equations + thA) exp (nhA), or (1 - 4hA)y = (1. • "+ 1 Y. + 1] = by the localizing assumption (l - thA) exp [In = [exp(nhA)][(! '. = + I)hA]- that + thA) exp (nhA) thA)exp(hA) - (1 + 1M)] (1 -2'[y{x,); h], verifying (24).

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