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HW_07 - 1 and 3 over the interval[0 2 using the trapezoidal...

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ChE 348 Homework # 7 _______________________________________________________________________ _ 1. Solve ( by hand ) the following problems using Euler’s method with a stepsize of h = 0.1. Compute the error using the true answer Y( x ). (a) [ ] ) ( tan ) ( 0 ) 0 ( , 5 . 0 0 , )) ( cos( ) ( 1 2 / x x Y Y x x Y x Y - = = = (b) [ ] 2 2 2 / 1 ) ( 0 ) 0 ( , 5 . 0 0 , ) ( 2 1 1 ) ( x x x Y Y x x Y x x Y + = = - + = 2. Using the approximation h x y x y x y x y n n n n 2 ) ( 3 ) ( 4 ) ( ) ( ' 1 1 1 - + - - + - derive the following numerical method for solving initial value problems. ) , ( 2 3 4 1 1 1 1 - - - + - - = n n n n n y x hf y y y 3. Use the trapezoidal rule predictor-corrector with h = 0.1 to compute (by hand) approximate value of y( 0.3) for the following initial value problems. ( 29 ( 29 2 2 2 1 ) ( : 0 ) 0 ( , 0 2 1 1 ' x x x y solution true y y x y + = = = + + - 4. Write a computer program that solves each of the initial value problems in Problems
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Unformatted text preview: 1 and 3 over the interval [0, 2], using the trapezoidal rule predictor-corrector with a sequence of grids h = 1/4, 1/8, and 1/16. Produce an orderly table of ( x n , y n ) pairs and discuss the errors. 5. For the model of the tumor growth, take λ = μ = 1, a = 0.25, S i = 0.8 and σ = 0.25. Use the trapezoidal predictor-corrector to solve the differential equation, using h = 1/16, and show that the tumor radius approaches a limiting value as t → ∞ . σ μ λσ 4 2 3 1 ) ( ' 2 2 + + +-= R R R S t R i a R = ) ( ,...
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