EGR 599 ADVANCED ENGINEERING MATHEMATICS II ______

LAST NAME, FIRST (3 pts)

Quiz #5

Note: Your answers must be correct to 4 significant figures.

The maximum score is 25 pts.

I. (1) Use the Rayleigh-Ritz method to approximate the solution of

y” = 2x + 1, y(0) = 0, y(1) = 0,

using a quadratic in x as the approximating function.

u = ___x(x - 1)____

u(x) = cx(x - 1) Þ = x2 - x; = 2cx - c; = 2x

= 2dx + 2 dx = 0

2dx + 2dx = 0

4dx + 2dx = 0

4 + 2 = 0 Þ c = 1

(2) Solve problem (1) by Galerkin’s method.

u = ____x(x - 1)____

R(x, c) = u”- (2x + 1) = 2c - (2x + 1)

dx = 0

dx = 0

- - - c + + = 0 Þ c = 1

(3) Solve the following problem by collocation, using a quadratic in x as the approximating function and setting the residual to zero at x = 0.5.

y” + y = 2x + 1, y(0) = 0, y(1) = 0,

u = x(x - 1)

u(x) = cx(x - 1) Þ = 2cx - c; = 2c

R(x, c) = u”+ u - (2x + 1) = 2c + cx(x - 1) - (2x + 1) = 0

x = 0.5 Þ 2c - 0.25c - 2 = 0 Þ c = = = 1.1429

II. (4) The equation y = x + 2exp(x) is used to fit the data

x / 1 / 2 / 3
Y / 6 / 16 / 40

The correlation coefficient for the equation is ____0.9924____

> x=[1 2 3];Y=[6 16 40];

> y=x+2*exp(x)

y =

6.4366 16.7781 43.1711

> s=sum((y-Y).^2)

s =

10.8518

> st=sum((y-mean(y)).^2)

st =

717.6534

> r=sqrt(1-s/st)

r =

0.9924


III. Use Newton’s method with x(0) = [1 2] to compute x(1) for the following nonlinear system

x12 + x2 - 37 = 0; x1 - x22 - 5 = 0

(5) x1(1) = ___17_____ (6). x2(1) = ____4_____

f = x12 + x2 - 37 Þ = 2x1; = 1

g =x1 - x22 - 5 Þ = 1; = - 2x2

= Þ = Þ =


IV. The steady-state temperature (oC) associated with selected nodal points of a two-dimensional system having a thermal conductivity of 2.0 W/m×oK are shown on the right. The ambient fluid is at 30oC with a heat transfer coefficient of 30 W/m2×oK. The isothermal surface is at 210oC.Note: For this problem, the temperature at the node must be determined from the known information. For example: the temperature at node 3 cannot be obtained from the temperature at node 2.

(7) The temperature at node 1 is __167.15oC_

T1 = +

72.2222T1 = +

T1 = 167.15oC

(8) The temperature at node 2 is ___98.61oC__

72.2222T2 = + Þ T2 = 98.61oC

(9) The temperature at node 3 is __47.15 oC_

kDy + k = k + hDy(T3 - 30)

+ = + hT3 - h´30

T3 = = 47.15 oC

(10) If the temperature at node 3 is 56oC, calculate the heat transfer rate per unit thickness normal to the page from the right surface to the fluid.

______

q’conv = 30´0.2{0.5(210 - 30) + (67 - 30) + (56 - 30) + 0.5(45.8 - 30)} = 965.4 W/m

V. Obtain the Euler equation for

(11) I = dx ______

The Euler equation is given as

- = 0

For F = xy’2 - yy’ + y we have = - y’ + 1 and = 2xy’ - y

The Euler equation is then

- y’ + 1 - (2xy’ - y) = 0 Þ (2xy’) = 1

(12) I = dx ______

For F = y’2 + k2cos(y) we have = - k2sin(y) and = 2y’

The Euler equation is then

2y’’ + k2sin(y) = 0

VI. Use the formula lf = and the trial function f(x) = c1x(1 - x) + c2x2(1 - x) to estimate the smallest eigenvalue in equation

= - lu with u(0) = u(1) = 0. We obtain

(13) S12 = ______

f1(x) = x(1 - x) and f2(x) = x2(1 - x)

S12 = = = B(4, 3) = =