Equations of the Form æ
4.4.2. Equations of the Form æ
This equation describes vibration of a circular membrane in a resisting medium with velocity- ö proportional resistance coefficient.
1 ÷ . Domain: 0 ≤ ø ≤ ù . First boundary value problem. The following conditions are prescribed:
= û ( ø ) at ü =0
(initial condition),
= ÿ ( ø ) at ü =0
(initial condition),
(boundary condition). Solution:
cos( ✡ ✝ ü )+ ☛ ✝ sin( ✡ ✝ ü ) ☞ ✌ 0 ✍ ✎
Here,
where the ö
are positive zeros of the Bessel function, ✌ 0 ( ) = 0.
. For the solution of the second and third boundary value problems, see equation 4.4.2.2 (Items 3 ÷
✖✠✗ and 4 with = 0).
Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 2 ✙
. The substitution ✧ ( ★ , ü ) = exp − 2 ✂ ü ✄✪✩ ( ★ , ✫ ) leads to the equation
( , ), which is discussed in Subsection 4.2.5. ✣
+ exp 2 ✂
2 ✦ . Domain: 0 ≤ ★ ≤ ✭ . First boundary value problem. The following conditions are prescribed:
= ✮ ✰ 0 ( ★ ) at ✫ =0
(initial condition),
= ✮ 1 ( ★ ) at ✫ =0
(initial condition),
(boundary condition). Solution:
where the ✝ are positive zeros of the Bessel function, ✌ 0 ( ) = 0. The numerical values of the first
ten ✎ are specified in Paragraph 1.2.1-3.
3 ✦ . Domain: 0 ≤ ★ ≤ ✭ . Second boundary value problem. The following conditions are prescribed:
= ✮ 0 ( ★ ) at ✫ =0
(initial condition),
= ✮ 1 ( ★ ) at ✫ =0
(initial condition),
(boundary condition). Solution:
; the ✝ are positive zeros of the first-order Bessel function, ✎ 1 ( ) = 0. The numerical values of the first ten roots ✝ are specified in Paragraph 1.2.1-4.
. Domain: 0 ≤ ★ ≤ ✭ . Third boundary value problem. The following conditions are prescribed:
= ✮ 0 ( ★ ) at ✫ =0
(initial condition),
= ✮ 1 ( ★ ) at ✫ =0
(initial condition),
(boundary condition). The solution ✧ ( ★ , ✫ ) is given by the formula in Item 3 ✦ with
are positive roots of the transcendental equation
The numerical values of the first six roots ✝ can be found in Abramowitz and Stegun (1964) and Carslaw and Jaeger (1984). ✎
. The substitution ✧ ( ★ , ✫ ) = exp ✺ 1 − 2 ✴ ✫☎✻ ✩ ( ★ , ✫ ) leads to the equation
which is discussed in Subsection 4.2.6.
2 ✦ . Domain: 0 ≤ ★ ≤ ✭ . First boundary value problem. The following conditions are prescribed:
= ✮ 0 ( ★ ) at ✫ =0
(initial condition),
= ✮ 1 ( ★ ) at ✫ =0
(initial condition),
(boundary condition). Solution:
3 ✦ . Domain: 0 ≤ ★ ≤ ✭ . Second boundary value problem. The following conditions are prescribed:
= ✮ 0 ( ★ ) at ✫ =0
(initial condition),
= ✮ 1 ( ★ ) at ✫ =0
(initial condition),
(boundary condition). Solution:
sin ✍ ✎ sin ✺✽✫✵❆ ✡ ✝ ✻✪✸
are positive roots of the transcendental equation
tan ✎ − = 0. The numerical values of the first five roots ✝ are specified in Paragraph 1.2.1-5.
. Domain: 0 ≤ ★ ≤ ✭ . Third boundary value problem. The following conditions are prescribed:
= ✮ 0 ( ★ ) at ✯ ✫ ✰ =0
(initial condition),
= ✮ 1 ( ★ ) at ✯ ✫ ❀ =0
(initial condition),
(boundary condition). The solution ✧ ( ★ , ✫ ) is given by the formula in Item 3 ✦ with
− 4 ✴ and the ✝ are positive roots of the transcendental equation cot +
− 1 = 0. The numerical values of the first six roots ✎ can be found in Carslaw and Jaeger (1984).
1 1 ✣ ✦ . The substitution ✧ ( ★ , ✫ ) = exp ✺ −
2 ✴ ✫☎✻ ✩ ( ★ , ✫ ) leads to an equation of the form 4.3.1.2:
. Domain: 0 ≤ ≤ ❉ . First boundary value problem. The following conditions are prescribed:
= ( ❈ ) at
=0 (initial condition),
= ✮ 1 ( ) at ✫ ❈ =0 (initial condition),
at ❈ = ❉
(boundary condition),
=0 (boundedness condition). Solution:
sin P✽✫✿◗ ❍ ✳ ❘
+ ❯ − 4 ✴ ; the ❑ are positive zeros of the Bessel function, 0 ( ❑ ) = 0.
. Domain: 0 ≤ ≤ ❉ . Second boundary value problem. The following conditions are prescribed:
0 ( ) at ✫ =0 (initial condition),
1 ( ) at ✫ =0 (initial condition),
at ❈ = ❉
(boundary condition),
=0 (boundedness condition). Solution:
sin P✽✫✿◗ ❘
are positive zeros of the first-order Bessel function, 1 ( ✐ ) = 0. The numerical values of the first ten roots ✐
; the ✐
are specified in Paragraph 1.2.1-5.
. Domain: 0 ≤ ≤ ❉ . Third boundary value problem. The following conditions are prescribed:
0 ( ) at ✫ =0 (initial condition),
1 ( ) at ✫ =0 (initial condition),
= ❈ ❉ (boundary condition).
at
The solution ❙ ( , ✫ ) is given by the formula in Item 3 ❣
with ❣ ▼
sin P✽✫✿◗ ✳ ❘
+ ❯ − 4 ✴ , and the ✐ are positive roots of the transcendental equation
The numerical values of the first six roots ✐ can be found in Abramowitz and Stegun (1964) and Carslaw and Jaeger (1984).
+ ♦ s ♣ q –1 ❦
The substitution ❦
leads to a constant coefficient equation of the form 4.4.1.2: