216 in subsequent sections problems 1 problem 1 derive

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Unformatted text preview: e rst multistep formulas that we consider are explicit methods called Adams-Bashforth methods. These are derived by integrating the ODE y0 = f (t y) on the interval (tn;1 tn) to obtain Z or tn tn;1 y0dt = Z tn tn;1 y(tn) = y(tn;1) + Z (5.3.1) f (t y(t))dt tn tn;1 f (t y(t))dt: (5.3.2) Speci c numerical techniques are obtained by approximating f (t y(t)) in (5.3.2) by an interpolating polynomial and integrating the result. If, for example, f (t y(t)) were approximated by (the constant interpolating polynomial) f (t y(tn;1)) then (5.3.2) would produce Euler's method yn = yn;1 + hf (tn;1 yn;1): More generally, we'll interpolate f (t y(t)) by a k ; 1 st degree polynomial passing through tn;1, tn;2, : : : , tn;k . We'll use the Newton backward-di erence form of the interplating polynomial. For this application, we identify interpolation point tk in (5.2.16) with tn;1 , tk;1 with tn;2, : : : , and t1 with tn;k . Then, (5.2.16) becomes i;1 k ;1 X ri fn;1 Y (t ; tn;1;j ): Pk;1(t) = i i=0 i!h j =0 (5.3.3a) Using (5.2.11) with k replaced by k ; 1 and re-indexing the interpolation points as described above, we determine the error of this interpolation as Y f (k) ( y( )) k;1(t ; t Ek;1(t) = f (t y(t)) ; Pk;1(t) = n;1;j ) k! j =0 2 (tn;k tn;1): Let's expand both (5.3.3a) and (5.3.3b) to reveal their structure 2 n n Pk;1(t) = fn;1 + rfh ;1 (t ; tn;1) + r2!fh2;1 (t ; tn;1)(t ; tn;2 ) 13 (5.3.3b) k ;1 + : : : + r fn;1 1 (t ; tn;1 )(t ; tn;2) : : : (t ; tn;k+1) (k ; 1)!hk; and (k) Ek;1(t) = f ( k!y( )) (t ; tn;1)(t ; tn;2) : : : (t ; tn;k ): Using (5.2.14, 5.2.15), the rst few divided di erences are r0 fn;1 = fn;1 rfn;1 = fn;1 ; fn;2 r2 fn;1 = rfn;1 ; rfn;2 = fn;1 ; 2fn;2 + fn;3: The integration (5.3.2) can be simpli ed by the change of variables = t ; tn;1 h (5.3.4a) Since tn;1;j = tn;1 ; jh, we have t ; tn;1;j = ( + j )h and i;1 i;1 Y 1 Y(t ; t n;1;j ) = ( + j ) = ( + 1) : : : ( + i ; 1): hi j=0 j =0 Recall the combination symbol i = = ( ; 1) : : : ( ; i + 1) i!( ; i)! i! ! 0 = 1: (5.3.4b) This formula makes sense when is negative, i.e., ; i = ; (; ; 1) : : : (; ; i + 1) = (;1)i ( + 1) : : : ( + i ; 1) : i! i! Thus, i;1 1 Y(t ; t i; n;1;j ) = (;1) i i i!h j=0 : (5....
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.

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