0

Now, suppose that w is also a polynomial of the order n:

w( x) = b0 + b1 x + · · · + bn xn .

This may also be written in an alternative format:

w( x) = bT p ,

∼ ∼

257 Exercises

with

⎡ ¤ ⎡ ¤

b0 1

⎢ ⎥ ⎢ ⎥

b1 x

⎢ ⎥ ⎢ ⎥

b=⎢ p=⎢

⎥, ⎥.

. .

⎣ ¦ ⎣ ¦

∼ . .

∼

. .

xn

bn

Likewise h( x) can be written as

h( x) = aT p ,

∼ ∼

with

⎡ ¤

a0

⎢ ⎥

a1

⎢ ⎥

a=⎢ ⎥.

.

⎣ ¦

∼ .

.

an

Use of these expressions yields

1

w( x) [h( x) ’f ( x) ] dx

0

1 1

= b pp a dx ’ bT p f ( x) dx = 0.

T T

∼ ∼ ∼

∼∼ ∼

0 0

Notice that both a and b are arrays with polynomial coef¬cients indepen-

∼ ∼

dent of x, while p is an array with known functions of the coordinate x.

∼

Therefore, this equation may also be written as

1 1

pp dx a = b

T T T

b p f ( x) dx .

∼ ∼ ∼

∼∼ ∼

0 0

The integral expression on the left-hand side is a matrix:

1

K= ppT dx,

∼∼

0

while the integral on the right-hand side is a column:

1

f= p f ( x) dx.

∼ ∼

0

The equation must be satis¬ed for all b hence, with the use of the above

∼

matrix-array notation it follows that:

K a = f.

∼ ∼

Numerical solution of one-dimensional diffusion equation

258

(a) Suppose that w( x) and h( x) are both ¬rst-order polynomials.

“ Show that in that case

1

1 2

K= .

1 1

2 3

If f ( x) = 3, show that

“

3

f= .

3

∼

2

“ Compute the coef¬cients of the polynomial h( x) collected in a.

∼

Explain the results. (Hint: it is recommended to use MATLAB

for this purpose.)

(b) If w( x) and h( x) are both polynomials of the order n, show that

⎡ ¤

· · · n+1

1 1

1 2

⎢1 ⎥

⎢2 · · · n+2 ⎥

1 1

⎢ ⎥

3

K=⎢ . . ⎥.

. .

⎢. .⎥

. .

⎣. .¦

. .

· · · 2n+1

1 1 1

n+1 n+2

(c) Let f ( x) be such that

f ( x) = 1 for 0 ¤ x ¤ 0.5,

f ( x) = 0 for 0.5 < x ¤ 1.

“ Show that in this case

⎡ ¤

1

2

⎢ ⎥

⎢ ⎥

1 12

1

2( 2)

p f ( x) dx = ⎢ ⎥.

f= ⎢ ⎥

.

.

∼ ∼

⎣ ¦

.

0

1 1 n+1

n+1 ( 2 )

“ Use MATLAB to ¬nd the polynomial approximation h( x) of f ( x)

for n = 2, n = 3, etc. up to n = 10. Plot the original function

f ( x) as well as the polynomial approximation h( x). Hint: use the

function polyval. If the MATLAB array a represents a and n ∼

denotes the order of the polynomial, then to plot the function h( x)

you may use

x=0:0.01:1; plot(x,polyval(a(n+1:-1:1),x))

Investigate the condition number of the matrix K with increasing

“

n. Hint: use the MATLAB function cond. What does this condi-

tion number mean and what does this imply with respect to the

coef¬cients of h( x), collected in a?

∼

259 Exercises

14.2 Consider the differential equation

du d du

u+ + + f = 0,

c

dx dx dx

on the domain a ¤ x ¤ b.

Derive the weak form of this differential equation, and explain what steps

are taken.

14.3 Let f ( x) be a function on the domain 0 ¤ x ¤ 1. Let f ( x) be known at

n points, denoted by xi , homogeneously distributed on the above domain.

Hence the distance x between two subsequent points equals

1

x= .

n’1

A polynomial fh ( x) of order n ’ 1 can be constructed through these points,

which generally will form an approximation of f ( x):

fh ( x) = a0 + a1 x + · · · + an’1 xn’1 .

(a) Show that the coef¬cients of ai can be found by solving

⎡ ¤⎡ ¤⎡ ¤

n’1

1 x1 x1 · · · x1

2 f1

a0

⎢ 1 x x2 · · · xn’1 ⎥ ⎢ a ⎥ ⎢ f ⎥

⎢ ⎥⎢ 1⎥ ⎢2 ⎥

2

⎢ n’1 ⎥ ⎢ ⎥⎢ ⎥

2 2

⎢ 1 x3 x3 · · · x3 ⎥ ⎢ a2 ⎥ = ⎢ f3 ⎥,

2

⎢ ⎥⎢ ⎥⎢ ⎥

⎢. . . ⎥⎢ . ⎥ ⎢ . ⎥

. . . ¦⎣ . ¦ ⎣ .

.. . .

⎣. . ¦

. .

. . .

···

2 n’1 an’1 fn

1 xn xn xn

where fi = f ( xi ).

(b) Use this to ¬nd a polynomial approximation for different values of n

to the function:

f ( x) = 1 for 0 ¤ x ¤ 0.5,

f ( x) = 0 for 0.5 < x ¤ 1.

Compare the results to those obtained using the weighted residuals

formulation in Exercise 14.1(c). Explain the differences.

14.4 Consider the domain ’1 ¤ x ¤ 1. Assume that the function u is known

at x1 = ’1, x2 = 0 and x3 = 1, say u1 , u2 and u3 respectively. The

polynomial approximation of u, denoted by uh is written as

uh = a0 + a1 x + a2 x2 .

(a) Determine the coef¬cients a0 , a1 and a2 to be expressed as a function

of u1 , u2 and u3 .

Numerical solution of one-dimensional diffusion equation

260

(b) If the polynomial uh is written in terms of the shape functions Ni :

n

uh = Ni ( x) ui ,

i=1

then determine Ni , i = 1, 2, 3 as a function of x.

(c) Sketch the shape functions Ni .

(d) Is it possible that the shape function Ni = 1 at x = xi ? Explain.

(e) Is it possible that the shape function Ni = 0 at x = xj with j = i?

Explain.

14.5 Consider the differential equation

d du

=1

c

dx dx

on the domain 0 ¤ x ¤ h1 + h2 . At both ends of this domain u is set to

zero. Consider the element distribution as depicted in the ¬gure below.

u1 u3 u2

©1 ©2

(a) Derive the weak formulation of this problem.

(b) The elements employed have linear shape functions stored in array

N ( x). Express these shape functions in terms of x and the element

∼

lengths h1 and h2 .

(c) Show that the coef¬cient matrix of an element is given by

dN dN T

Ke = ∼

c ∼ dx.

dx dx

e

(d) Demonstrate that the coef¬cient matrix of element e is given by

’1

c 1

Ke = ,

’1 1

he

if c is a constant.