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[answered] MMAE 350 - Review Problems Problem #1 t y t 0.60 - 0.97 (a)


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MMAE 350 ? Review Problems

 

Problem #1 t

 

y t 0.45

 

1.46 0.60

 

2.35 1.00

 

0.45 1.60

 

- 0.72 2.20

 

0.97 (a) Using the data in the above table, write down the MATLAB statements needed to

 

compute (i) the first derivative

 

, and (ii) the second derivative

 

using the built-in

 

2

 

dy

 

d y

 

dt

 

dt2

 

MATLAB command ?diff?. (b) Derive the equations for the coefficients C1, C2 , and C3 , in the equation y C1 f1 x C2 f2 x C3 f3 x

 

that best fits a given set of data points xi , yi . Recall that this is called i 1,2, 3,?,n the least-squares method. The first step is to write down the expression for the ?squared error?.

 

The second step is to write down the necessary conditions to minimize the

 

?squared error?, which is a function of three variables; C1 , C2 , and C3 . The third step is to rearrange the equations so they can be solved for C1, C2 , and C3 . (c) Write down the MATLAB commands needed to interpolate the value of the dependent

 

variable at

 

using a cubic spline and the built-in MATLAB function y ?interp1?. x 4.57 x

 

y 1.39

 

6300 2.39

 

6375 3.93

 

6525 5.40

 

6600 Problem #2

 

The following boundary value problem is to be solved using the MATLAB

 

function?bvp4c?. The independent variable is and the dependent variable is r u r . u r1 9

 

d2u 1 du

 

35 6u 2

 

dr

 

r dr

 

r 1 r 2 2 du u r2 26 dr r2 (a) Transform the above second order ODE and boundary conditions into a system of first

 

order ODEs and boundary conditions.

 

(b) Write the MATLAB function m-file for the system of first order ODEs.

 

(c) Write the MATLAB function m-file for the boundary conditions.

 

(d) Write the MATLAB script m-file to solve the BVP using ?bvp4c?.

 

(e) Run the problem on MATLAB and plot for

 

. Write down the

 

1 r 2

 

du

 

vs. r

 

dr

 

MATLAB statements you used to do this. To check your results, compare them with the

 

plot given below.

 

0 du/dr ?5 ?10 ?15 1 1.1 1.2 1.3 1.4 1.5 r 1.6 1.7 1.8 1.9 2 Problem #3

 

(a) Construct an integration formula in the following form:

 

h I f f x dx c1 f h c2 f 0 c3 f h

 

h Derive the coefficients c1 , c2 , and c3 by making the integration rule exact when f x 1, x, and x2 . (b) Three-point Gauss quadrature for estimating 1 I f f x dx is given by: 1 I f f x dx C1 f x1 C2 f x2 C3 f x3 1 1 where . 5

 

8

 

5

 

3

 

3

 

C1 , C2 , C3 , x1 , x2 0, and x3 9

 

9

 

9

 

5

 

5

 

Explain in words how the six constants,

 

system of equations that C1, C2 , C3 , x1, x2 , and x3 , are derived. Include the C1 , C2 , C3 , x1, x2 , and x3 must satisfy (do not solve them). See pages 356-357 of the text, and/or ?Gauss Quadrature? on Blackboard.

 

The documents say that in order to determine the unknowns C1, C2, C3, x1, x2, and

 

x3, you make the integration formula exact when f(x) = 1, x, x2, x3, x4, and x5. From

 

that statement you can write out the six nonlinear equations for the six unknowns

 

C1, C2, C3, x1, x2, and x3. The solution of this system of equations (which you don't

 

have to do) is how the coefficients are derived.

 

(c) Review the MATLAB commands ?trapz? and ?integral?. Then complete the

 

following statement:

 

The _____________ command is used to integrate tabulated data, while the

 

_____________ command is used to integrate functions.

 

(d) In MATLAB, use the built-in function ?trapz? to integrate the function

 

over the interval

 

. Write the statements below.

 

f x 8 4.5(sinh x x4 ) 2,2 >> __________________________________________________

 

>> __________________________________________________ Problem #4

 

For the system of second order differential equations:

 

2 2 dx dy

 

d2 x 0.3 510x 0.3 100y 50

 

2

 

dt dt dt 2 2 dy dx

 

d2 y 0.86 28.6 y 0.86 28.6x 0

 

2

 

dt dt dt with zero initial conditions, i.e., x 0 0 dx 0 0 y 0 0

 

dt dy 0 0

 

dt (a) Transform the above into a system of first order differential equations, including the

 

initial conditions.

 

(b) Write down the statements for the function m-file for this problem.

 

(c) Obtain the numerical solution using ?ode45? and plot vs. . The time interval

 

y

 

dy

 

dt

 

is 0 to 2 seconds. Write down the MATLAB statements you used to do this. To check

 

your results, compare them with the plot given below. 1.5 1 dydt 0.5 0 ?0.5 ?1

 

?0.05 0 0.05 0.1 0.15 y 0.2 0.25 0.3 0.35 0.4 Problem #5 (This problem is very similar to Problem 11.10) Consider the following boundary value problem (BVP): 3y 0 dy 0 6

 

dx d2 y 3y x

 

dx2 0 x 1 y 1 1

 

(a) Discretize the BVP using finite difference formulas for the derivatives that have

 

truncation error

 

. Assume the interval

 

is divided into 5 equal parts. Use O h2 0 x 1 the table of finite difference formulas on the next page.

 

(b) Put the equations from part (a) in matrix form.

 

(c) Obtain the numerical solution of the system of equations from part (b) using left

 

division and plot

 

. Write down the MATLAB statements you used to do this. To y vs. x check your results, compare them with the plot given below. 5

 

finite difference solution 4.5 4 y 3.5 3 2.5 2 1.5 1 0 0.1 0.2 0.3 0.4 0.5 x 0.6 0.7 0.8 0.9 1 FINITE DIFFERENCE FORMULAS

 

First Derivative

 

Method

 

Two-point forward

 

difference

 

Three-point forward

 

difference

 

Two-point backward

 

difference

 

Three-point

 

backward difference

 

Two-point central

 

difference Formula f xi f xi 1 f xi h 3 f xi 4 f xi 1 f xi 2 2h

 

f xi f xi 1 f xi h

 

f xi 2 4 f xi 1 3 f xi f xi 2h

 

f xi 1 f xi 1 f xi 2h

 

f xi Truncation

 

Error O h O h2 O h O h2 O h2 Second Derivative

 

Method

 

Three-point forward

 

difference

 

Three-point

 

backward difference

 

Three-point central

 

difference

 

Four-point forward

 

difference

 

Four-point backward

 

difference Formula f xi 2 f xi1 f xi2 h2

 

f xi 2 2 f xi 1 f xi f xi h2

 

f xi 1 2 f xi f xi 1 f xi h2

 

2 f xi 5 f xi1 4 f xi2 f xi3 f xi h2 f xi3 4 f xi2 5 f xi1 2 f xi f xi h2

 

f xi Truncation

 

Error O h

 

O h O h2 O h2 O h2 Problem 6

 

You are given the following MATLAB code:

 

function [dudt] = Example(t,u)

 

dudt = [u(2);?(5/2)*u(1)?(0.5/2)*u(2)+(1/2)*12*sin(7*t)];

 

Multiple-choice question

 

Circle the correct answer.

 

This is a

 

(a) script file that defines a function

 

(b) built-in function

 

(c) user-defined function

 

(d) (a) and (b)

 

(e) (a) and (c)

 

Complete the following statements:

 

(a) The name of this function is _____________________________________________.

 

(b) The input arguments are ________________________________________________.

 

(c) The output arguments are _______________________________________________.

 

(d) The independent variables are ___________________________________________.

 

(e) The dependent variables are _____________________________________________. Questions on input and output arguments

 

(1) Describe the input arguments, i.e., are they scalars? vectors? matrices? If the input

 

arguments are vectors and/or matrices, give their dimensions (number of rows and

 

number of columns). (2) Describe the output arguments, i.e., are they scalars? vectors? matrices? If the output

 

arguments are vectors and/or matrices, give their dimensions (number of rows and

 

number of columns). Questions on the system of ODEs

 

(1) In terms of , what system of first order ODEs does this MATLAB

 

du

 

dv

 

u,

 

, v, and

 

dt

 

dt

 

function represent? Use the information given in the table.

 

Table of Notation

 

OD

 

E x

 

dx

 

dt x, and

 

(2) In terms of

 

represent? System of

 

ODEs

 

du

 

u

 

dt

 

dv

 

v

 

dt MATLAB ODEs

 

u(1

 

) dudt(1

 

) u(2

 

) dudt(2

 

) dx

 

dt , what second order ODE does this system of first order ODEs Questions about ?ode45?

 

Given the following MATLAB code:

 

function [dudt] = Example(t,u)

 

dudt = [u(2);?(5/2)*u(1)?(0.5/2)*u(2)+(1/2)*12*sin(7*t)]; Write down the MATLAB statements needed for solving this system of ODEs

 

using ?ode45?if the solution is to be computed from

 

until

 

with t 0 zero initial conditions.

 

Write down the MATLAB statements needed for plotting t 20 v vs. u . v u

 


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