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nagle_differential_equations_ISM_Part4

nagle_differential_equations_ISM_Part4 - Use the method of...

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Use the method of steps to show that the solution to the initial value problem x ( t ) = - x ( t - 1) , x ( t ) = 1 on [ - 1 , 0] , is given by x ( t ) = n k =0 ( - 1) k [ t - ( k - 1)] k k ! , for n - 1 t n , where n is a nonnegative integer. (This problem can also be solved using the Laplace transform method of Chapter 7.) (c) Use the method of steps to compute the solution to the initial value problem given in (0.1) on the interval 0 t 15 for t 0 = 3. Extrapolation When precise information about the form of the error in an approximation is known, a technique called extrapolation can be used to improve the rate of convergence. Suppose the approximation method converges with rate O ( h p ) as h 0 (cf. Section 3.6). From theoretical considerations, assume we know, more precisely, that y ( x ; h ) = φ ( x ) + h p a p ( x ) + O ( h p +1 ) , (0.3) where y ( x ; h ) is the approximation to φ ( x ) using step size h and a p ( x ) is some function that is independent of h (typically, we do not know a formula for a p ( x ), only that it exists). Our goal is to obtain approximations that converge at the faster rate O ( h p +1 ). We start by replacing h by h/ 2 in (0.3) to get y x ; h 2 = φ ( x ) + h p 2 p a p ( x ) + O ( h p +1 ) . If we multiply both sides by 2 p and subtract equation (0.3), we find 2 p y x ; h 2 - y ( x ; h ) = (2 p - 1) φ ( x ) + O ( h p +1 ) . Solving for φ ( x ) yields φ ( x ) = 2 p y ( x ; h/ 2) - y ( x ; h ) 2 p - 1 + O ( h p +1 ) . Hence, y * x ; h 2 := 2 p y ( x ; h/ 2) - y ( x ; h ) 2 p - 1 has a rate of convergence of O ( h p +1 ). 10

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(a) Assuming y * x ; h 2 = φ ( x ) + h p +1 a p +1 ( x ) + O ( h p +2 ) , show that y ** x ; h 4 := 2 p +1 y * ( x ; h/ 4) - y * ( x ; h/ 2) 2 p +1 - 1 has a rate of convergence of O ( h p +2 ). (b) Assuming y ** x ; h 4 = φ ( x ) + h p +2 a p +2 ( x ) + O ( h p +3 ) , show that y *** x ; h 8 := 2 p +2 y ** ( x ; h/ 8) - y ** ( x ; h/ 4) 2 p +2 - 1 has a rate of convergence of O ( h p +3 ). (c) The results of using Euler’s method (with h = 1, 1 / 2, 1 / 4, 1 / 8) to approximate the solution to the initial value problem y = y, y (0) = 1 at x = 1 are given in Table 1.2, page 27. For Euler’s method, the extrapolation procedure applies with p = 1. Use the results in Table 1.2 to find an approximation to e = y (1) by computing y *** (1; 1 / 8). [Hint: Compute
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nagle_differential_equations_ISM_Part4 - Use the method of...

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