1550078481-Ordinary_Differential_Equations__Roberts_

The Initial Value Problem y' = f(x, y); y(c) = d 83 Just as we found that the IVP (1) is equivalent to the integral equation (2) ...

84 Ordinary Differential Equations equation (12) is an upper bound and not a realistic error estimate. However, equation (12) do ...

The Initial Value Problem y' = f(x, y); y(c) = d 85 Table 2.1 Euler's approximation to the IVP (7) y' = y + x; y(O) = 1 on [O, 1 ...

86 Ordinary Differential Equations 2.4.1.2 Runge-Kutta Methods Among the more popular single-step numerical approximation method ...

The Initial Value Problem y' = f(x, y) ; y(c) = d 87 method, the second order Runge-Kutta method, or the Heun method for produci ...

88 Ordinary Differential Equations Table 2.3 Improved Euler's approximation to the IVP (7) y' = y + x; y(O) = 1 on [O, 1] with s ...

The Initial Value Problem y' = l(x, y); y(e) = d 89 (18) kz = l(xn + ehn, Yn + dk1hn) = l(xn, Yn) + ehnlx + dk1hnly+ ezh2 d2h2 k ...

90 Ordinary Differential Equations The choice of constants which leads to the classical fourth order Runge-Kutta recursion is an ...

The Initial Value Problem y' = f(x, y); y(c) = d 91 SOLUTION Table 2.4 contains the fourth order Runge-Kutta approximation to th ...

92 Ordinary Differential Equations Table 2.5 Approximation to the solution of the IVP (7) y' = y + x; y(O) = 1 on [O, l] with st ...

The Initial Value Problem y' = f(x, y); y(c) = d 93 These results illustrate that it is not only the number off function evaluat ...

94 Ordinary Differential Equations his procedure, two Runge-Kutta methods of different order are run simultane- ously. The user ...

The Initial Value Problem y' = f(x, y); y(c) = d 95 with(DEtools): de:=diff(y(x), x) = y(x) + x: sol:=dsolve( {de, y(O) = 1 }, n ...

96 Ordinary Differential Equations c. Estimate the maximum discretization error p er step on the interval [O, l] for the stepsiz ...

The Initial Value Problem y' = f(x, y); y(c) = d 97 e. Find the expli cit solution of the IVP y' = y/x + 2; y(l) = l. On what in ...

98 Ordinary Differential Equations /Approximate solution Slopefn-m+1-....._. ~- Slope[, X ' n-p I' (xn-m+11Yn-m+1) ''\/... Slope ...

m 0 1 2 3 The Initial Value Problem y' = f(x, y); y(c) = d Adams-Bashforth formulas Yn+i = Yn +hf n h(3J n - fn-1) Yn+l = Yn + 2 ...

100 Ordinary Differential Equations Table 2 .7 Adams-Bashforth approximations to the solution of the IVP (7) y' = y + x; y(O) = ...

The Initial Value Problem y' = f(x, y); y(c) = d 101 form= 1- is known as the midpoint rule and has a local discretization error ...

102 Ordinary Differential Equations c. How small must the stepsize be to ensure six decimal place accuracy per step? 2. a. Use t ...