Wave Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 103

Remark 1. The two terms in the right-hand side of Eq. (2.93) clearly have the same
unit. The unit of the first term is, by viewing u as the displacement,

⎡
⎣

⎤
⎦ [

]= a − 1 ∂
∂

1

[ u [ ϕ ] d ξ d η ]

t

2 2 2

( at ) − ( ξ − x ) − ( η − y )

T
L ·

1
T ·

1
L ·

L 2 = L · = L ,

which shows the correctness of the unit in Eq. (2.93).

Remark 2. In a polar coordinate system, Eq. (2.93) reads

⎡
⎣ ∂
∂

at

r d r 2π
0

1 ϕ ( x + r cos θ , y + r sin θ )

u ( x , y , t )= d θ

2 π a t 2

0

( ) −

r 2 at

⎤
⎦ .

at

r d r 2π
0

ψ ( x + r cos θ , y + r sin θ )

+ d θ (2.94)

0 2

r 2

( at ) −

Example 1 . Find the solution of
u tt =

a 2

Δ u , − ∞ < x , y < + ∞ , 0 < t ,

x 2 u ( x , y , 0 )= ( x + y ) , u t ( x , y , 0 )= 0 .

Solution. By Eq. (2.94), we obtain

⎡
⎣

⎤
⎦

at

r d r 2π
0

2

1

a ∂

( x + r cos θ ) ( x + y + r cos θ + r sin θ )

u ( x , y , t )= d θ

2 π ∂ t

0 2

r 2 ( at ) −

⎧
⎨

⎫
⎬

2π

at 2π

1

a ∂

r d r

x 2 r 2 cos 2 = ( x + y ) d θ + ( 3 x + y ) θ d θ

⎩ ⎭

2 π ∂ t

0 2 0 0

( at ) − r 2

x 2 a 2 t 2

= ( x + y )+( 3 x + y ) .

thus has unit L 3 . From the solution, we If x and y have the unit of length, u ( x , y , 0 )

L 3

have [ u ( x , y , t )] = , which agrees with the unit of u ( x , y , 0 ) .Otherwise,theremust

exist some errors in derivation.

Remark 3. By a similar approach we can also obtain, from Eq. (2.85), the solution
of

u tt =

a 2 Δ u + f ( x , y , t ) , − ∞ < x , y < + ∞ , 0 < t ,

u ( x , y , 0 )= 0 , u t ( x , y , 0 )= 0 .