By W. J. Thron
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Extra resources for Analytic Theory of Continued Fractions II
Can the same be said if the equation is nonlinear as it is for the van der Pol equation y + a(1 − y 2 )y + y = 0? 22. This problem examines diﬀerent ways to derive, and express, the velocity Verlet method. 74) by using Taylor’s theorem on y(tj + k). 71). What is the resulting ﬁnite diﬀerence approximation? (c) Show that velocity Verlet can be reduced to the Verlet method derived in (b). 23. 71) is yj+1 = yj + kvj , k vj+1 = vj + F (yj+1 ). m (a) Derive this method and show that it is ﬁrst order.
11). 13. This problem concerns the IVP involving the Bernoulli equation y + y3 = y , a+t for t > 0, Exercises 37 where y(0) = 1. You are to solve this problem using the backward Euler, trapezoidal, and RK4 methods. (a) Verify that the exact solution is y= a+t β + 23 (a + t)3 . 01, on the same axes plot the exact and the three numerical solutions for 0 ≤ t ≤ 3 in the case M = 80. (c) Redo (b) for M = 20, M = 40, and M = 160. If one or more of the methods is unstable you can exclude it from the plot (for that value of M ) but make sure to state this in your write-up.
A) Show that the centered diﬀerence formula for the ﬁrst derivative becomes f (xi ) = h2i+1 f (xi+1 ) − f (xi−1 ) +O hi+1 + hi hi+1 + hi +O h2i hi+1 + hi . 42 1 Initial Value Problems (b) Show that the centered diﬀerence formula for the second derivative becomes f (xi ) = 2 hi f (xi+1 ) − (hi+1 + hi )f (xi ) + hi+1 f (xi−1 ) hi+1 hi (hi+1 + hi ) 2 hi+1 h2i +O +O . 31. Consider the second-order diﬀerence equation yj+1 + 2αyj + βyj−1 = 0, where α, β are known constants with β = α2 . (a) By assuming a solution of the form yj = rj , show that the general solution j j of the equation has the form yj = Ar+ + Br− , where A, B are arbitrary constants and r± = −α ± α2 − β.
Analytic Theory of Continued Fractions II by W. J. Thron