HW_08 - 1 ) ( , 4 1 ) ( , 4 2 2 1 / 2 1 2 1 / 1-=-= = +-= y...

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ChE 348 Homework # 8 _______________________________________________________________________ _ 1. Derive the numerical method based on using Simpson’s rule to approximate the integral in + - + - = + h x h x ds s y s f h x y h x y . )) ( , ( ) ( ) ( Is this method single-step or multi-step, implicit or explicit? 2. Write a computer program that solves the following initial value problems over the interval [0, 2], using the fourth-order Runge Kutta method (RK4) with a sequence of grids h = 1/4 and 1/8. Produce an orderly table of ( x n , y n ) pairs and discuss the errors. (a) ( 29 ( 29 2 2 2 1 ) ( : 0 ) 0 ( , 0 2 1 1 ' x x x y solution true y y x y + = = = + + - (b) ( 29 1 3 4 1 ) ( : ; 1 ) 0 ( , 1 4 4 / + = = + - = - x e x y solution true y y y 3. Repeat the problem 2 using the fourth-order Adams-Bashforth (AB4) and Admas- Moulton (AM4) predictor-corrector method. Use the fourth-order Runge-Kutta method (RK4) to generate the needed starting values. 4. Using a hand calculator and h = 0.25, compute approximate solutions to the initial value problem
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Unformatted text preview: 1 ) ( , 4 1 ) ( , 4 2 2 1 / 2 1 2 1 / 1-=-= = +-= y y y y y y y y Compute out to x = 1.0, using the Trapezoidal rule predictor-corrector method. 5. Consider the second-order equation ) ( , 8 1 ) ( , sin / // = = = + y y y y ; write this as a first-order system and compute (by hand) approximate solutions using a step size of h = /10 and the Euler method. Show the first two steps. 6. Write a computer program that approximates the solution to the third order equation ; 3 ) ( , 1 ) ( , 1 ) ( , 8 10 2 2 5 4 // / 2 / // /// =-= = + + = + + + y y y x x y y y y using the following methods with h = 0.1 over the interval [0, 2]: (a) fourth-order Runge-Kutta (RK4) (b) Adams fourth-order predictor (AB4) and corrector (AM4) method with RK4 Compare the results with the true solution: 2 ) ( x e x y x + =-...
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