Wave Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 57

Tabl e 2 . 1 Eigenfunctions
x = 0 x = l Eigenvalues Eigenfunctions Normal square M m Notes

m l 2

sin m π x
l

l
2

1) X = 0 m = 1 , 2 , ···

( 2 m + 1 ) π
2 l

2

sin ( 2 m + 1 ) π x
2 l

l
2

X = 2) X = 0 0 m = 0 , 1 , ···

1 −

μ 2

sin μ

l
2

sin2 μ

m m

l

m

X + h 2 X = 0
h 2 > 0

3) x μ m ( m = 1 , 2 , ··· ) are

positive zero-points of
f

l 2 μ m

x
lh 2

( x )= tan x +

( 2

2 m + 1 ) π

cos (

2 m + 1 ) π x l

2

4) X = 0 m = 0 , 1 , ···

2 l 2 l

m 2

l

cos m

π x l

2

5) X = 0 X = 0 l , m = 0 , 1 , ···

l 1

μ 2

cos μ

l
2

sin2 μ m m m

X + 6) h 2 X = 0 x + are

positive zero-points of
g

μ ( m = 1 , 2 , ··· ) m

l l 2μ m

h 2 > 0

x
lh 2

( x )= cot x −

1

μ m
l

2

sin μ m
l

+ ϕ m
tanϕ m = μ m / lh 1

l
2

sin μ m
μ m ·

7) X = 0 x − cos ( μ m + 2 ϕ m ) are

positive zero-points of
f ( x )= tan x +

μ m ( m = 1 , 2 , ··· )

x
lh 1

1

μ m
l

2

sin μ m
l

m

l
2

sin μ m

μ m ·

8) X − X = h 1 X = 0 0 x + ϕ − cos ( μ + 2 ϕ ) are

positive zero-points of
g

μ ( m = 1 , 2 , ··· )

m m m

h 1 > 0

tan ϕ m = μ m / lh 1

x
lh 1

( x )= cot x −

1

μ m
l

2

sin μ m

l x + ϕ m
tanϕ m = μ m / lh 1

l
2

sin μ m
μ m ·

X + h 2 X =

− ( μ m +

ϕ m ) μ m ( m =

, , ··· )

9) 0 cos 2 are

positive zero-points of
F ( x )= cot x −

1 2

h 2 > 0

1
l ( h 1 + h 1 )

x −

l 2 h 1 h 2