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Assignment_5

# Assignment_5 - 1 − = y 1 = y a Show that the exact...

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ME 303 ADVANCED ENGINEERING MATHEMATICS Assignment 5 Question 1 . For the initial-value problem, t y y sin ' ' ' = + with ( ) ( ) 1 0 ' 0 = = y y ; ( ) 0 0 ' ' = y Use the (explicit) Euler method with a step size 1 . 0 = Δ t to find ( ) 2 . 0 = t y . Question 2. Consider the following second-order initial-value problem: 1 ' ' 2 + = + x y y with ICs ( ) 1 0 = y ; ( ) 0 0 ' = y a) Find the exact solution. b) Substitute second-order central differences to develop the discretized ODE. With a step size 1 . 0 = Δ x , find the value of ( ) 3 . 0 = x y . c) Apply the second-order Runge-Kutta method with 1 . 0 = Δ x to find ( ) 2 . 0 = x y . Question 3. Consider the following second-order boundary value problem for 1 0 x : 1 ' ' 2 + = + x y y with ( )
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Unformatted text preview: ( ) 1 − = y ; ( ) 1 = y a) Show that the exact solution is the same as that in Question 2 part (a). b) Substitute second-order central differences to develop the discretized ODE. For a step size 2 . = Δ x , develop the set of simultaneous algebraic equations that governs the unknown y values in the interval 1 ≤ ≤ x . (Do not have to solve the set) c) Write the equations of part (b) in matrix form, and discuss how you would solve them. How would the process change if we used a step size 01 . = Δ x ?...
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