Two-Dimensional Nonstationary Fourth-Order Equations

9.3. Two-Dimensional Nonstationary Fourth-Order Equations

9.3.1. Equations of the Form ➃ ➄ + ➆ 2 ➃ ➄

9.3.1-1. Domain: 0 ≤ ✈ ≤ ➌ 1 ,0≤ ➁ ≤ ➌ 2 . Solution in terms of the Green’s function. We consider boundary value problems in a rectangular domain 0 ≤ ✈ ≤ ➌ 1 ,0≤ ➁ ≤ ➌ 2 with the general

initial condition

= ❥ ( ✈ , ➁ ) at r =0

and various homogeneous boundary conditions. The solution can be represented in terms of the Green’s function as

( ✈ , ➁ , r )= ❜ ➍ ❜ ➍ ❥ ( ➎ , ➏ ) ➐ ( ✈ , ➁ , ➎ , ➏ , r ) ❡ ➏ ❡ ➎ + ❜

Below are the Green’s functions for various types of boundary conditions.

9.3.1-2. The function and its first derivatives are prescribed at the sides of a rectangle:

=0 at ✈ = 0,

=0 at ✈ = ❦ ➌ ➑ ❦ ➑ 1 ,

=0 at ➁ = ➌ 2 . Green’s function:

( ➓ )= sinh( 1 ) − sin( ➌ 1 ) ➀ ❾ cosh( ✈ ) − cos( ✈ ➓ ) ➀

− ❾ cosh( ➣ ➌

1 ) − cos( → ➌ 1 ) ➀ ❾ sinh( ➣ ✈ ) − sin( ➣ → → ✈ ) ➀ ,

( ➁ )= ❾ sinh( ➔ ➣ → ➌ 2 ) − sin( ➔ ➣

2 ➣ ) ➀ ❾ cosh( ➔

) − cos( → ➔ ➁ ) ➀

− ❾ cosh( ➔ ➌ 2 ) − cos( ➔ ➌ 2 ) ➀ ❾ sinh( ➔ ➁ ) − sin( ➔ ➁ ) ➀ , where the

and ➔ are positive roots of the transcendental equations ➣ ➣ ➣

cosh( ➌ 1 ) cos( ➌ 1 ) = 1, cosh( ➌ 2 ) cos( ➌ 2 )=1

9.3.1-3. The function and its second derivatives are prescribed at the sides of a rectangle:

=0 at ✈ = 0,

=0 at ✈ = ➌ ,

=0 at ➁ = ➌ 2 . Green’s function:

sin( ✈ ) sin( ➔ ➁ ) sin( ➎ ) sin( ➔ ➏ ) exp ❾ −( 4 + ➔ 4 ) ③ 2 r ➀ ,

9.3.1-4. The first and third derivatives are prescribed at the sides of a rectangle:

=0 at ➁ = ➌ 2 . Green’s function:

9.3.1-5. The second and third derivatives are prescribed at the sides of a rectangle: ➢ ➢

= ➥ =0 at = 0, = ➝ =0 at

= ➥ ➝ =0 at ➁ = 0, = ➝ =0 at ➁ = ➌ 2 . Green’s function:

1 ➓ ) − sin( ➌ 1 ) ➀ ❾ ➓ ➢ sinh( ➢ cosh( ) + cos( ) ➀

− ❾ cosh( → ➣ ➌ 1 ) − cos( → ➣ ➌ 1 ) ➀ ❾ sinh( → ➣

) + sin( → ➣ ) ➀ ,

( ➣ ➁ )= ❾ sinh( ➔

2 ) − sin( ➔ → ➌ 2 ) ➀ ❾ cosh( ➔ → ➁ ) + cos( → ➔ ➁ ) ➀

− ❾ cosh( ➔ ➌ 2 ) − cos( ➔ ➌ 2 ) ➀ ❾ sinh( ➔ ➁ ) + sin( ➔ ➁ ) ➀ , where the

and ➔ are positive roots of the transcendental equations ➣ ➣

cosh( ➌ 1 ) cos( ➌ 1 ) = 1, cosh( ➌ 2 ) cos( ➌ 2 ) = 1.

9.3.1-6. Mixed boundary conditions are prescribed at the sides of a rectangle: ➢ ➢

= ➥ =0 at = ➌ ,

=0 ⑨ at = 0, =

= ➥ =0 at = 0, = =0 at ➁ = ➌ 2 . Green’s function: ➢

) sin( ➔ ➁ ) sin( ➎ ) sin( ➔ ➏

9.3.2. Two-Dimensional Equations of the Form ➃ ➄ 2 + ➆ 2

This equation governs two-dimensional free transverse vibration of a thin elastic plate; the un- ➅ known ➥ is the deflection (transverse displacement) of the plate’s midplane points relative to the

original plane position. Here, ➧ ➧ = ➧ 2 and ➧ is the Laplace operator that is defined as

in the Cartesian coordinate system,

+ 1 2 2 2 in the polar coordinate system.

9.3.2-1. Particular solutions: ➢ ➢

( ➢ , , )= 1 sin( 1 )+ ➺ 1 cos( ➸ 1 ) ➻ ➳✓➵ 2 sin( ➸ 2 ➲ )+ ➺ 2 cos( ➸ 2 ➲ ) ➻ sin ➳ ( ➸ 2

( ➢ , ➲ , ➠ )= ➳✓➵ 1 sin( ➸ 1 )+ ➢ ➺ 1 cos( ➸

1 ➢ ) ➻ ➳✓➵ sin( ➸

2 2 ➲ )+ ➺ 2 cos( ➸ 2 ➲ ) ➻ cos ➳ ( ➸ 1 + ➸ 2 ) ③ ➠✔➻ ,

( ➢ , ➠ )= ➳✓➵ 1 sinh( ➸ 1 )+ ➺ 1 cosh( ➸ 1 ) ➻ ➳✓➵ 2 sinh( ➸ 2 ➲ )+ ➺ 2 cosh( ➸ 2 ➲ ) ➻ sin ➳ ( ➸ 2

( , ➲ , ➠ )= ➳✓➵ 1 sinh( ➸ 1 )+ ➺ 1 cosh( ➸

1 ) ➻ ➳✓➵ 2 sinh( ➸ 2 ➲ )+ ➺ 2 cosh( ➸ 2 ➲ ) ➻ cos ➳ ( ➸ 1 + ➸ 2 ) ③ ➠✔➻ ,

( ➼ , ➽ , ➠ )= ➳✓➵ 1 ➾ ➚ ( ➸ ➼ )+ ➵ 2 ➪ ➚ ( ➸ ➼ )+ ➵ 3 ➶❱➚ ( ➸ ➼ )+ ➵ 4 ➹ ➚ ( ➸ ➼ ) ➻ cos( ➘ ➽ ) sin( ➸ 2 ➴ ➷ ),

1 ➷ ➾ ➚ ( ➸ ➼ )+ ➵ 2 ➪ ➚ ( ➸ ➼ )+ ➵ 3 ➶❱➚ ( ➸ ➼ )+ ➵ 4 ➹ ➚ ( ➸ ➼ ) ➻ sin( ➘ ➽ ) cos( ➸ 2 ), where ➵ 1 , ➵ 2 , ➵ 3 , ➵ 4 , ➺ 1 , ➺ 2 , ➸ , ➸ 1 , ➸ 2 are arbitrary constants, the ➾ ➚ ( ➮ ) and ➪ ➚ ( ➮ ) are the Bessel

functions of the first and second kind, the ➶ ➚ ( ➮ ) and ➹ ➚ ( ➮ ) are the modified Bessel functions of the

first and second kind, ➼ = ➱ ✃ 2 + ➲ 2 , and ➘ = 0, 1, 2, ❐✖❐✖❐

9.3.2-2. Domain: − ❒ < ✃ < ❒ ,− ❒ < ➲ < ❒ . Cauchy problem. Initial conditions are prescribed:

= ➷ ( , ) at = 0, = Ð ( ✃ , ➲ ) at = 0. Poisson solution:

Green’s function: Û

References : A. N. Krylov (1949), I. Sneddon (1951), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

9.3.2-3. Domain: 0 ≤ ✃ ≤ à 1 ,0≤ ➲ ≤ à 2 . Solution in terms of the Green’s function. We consider boundary value problems in a rectangular domain 0 ≤ ✃ ≤ à 1 ,0≤ ➲ ≤ à 2 with the general

initial conditions

= ➷ ( , ) at =0 and various homogeneous boundary conditions. The solution can be represented in terms of the

= ➷ ❮ ( ✃ , ➲ ) at = 0,

Green’s function as

Paragraphs 9.3.2-4 through 9.3.2-6 present the Green’s functions for three types of boundary conditions.

9.3.2-4. Domain: 0 ≤ ✃ ≤ à 1 ,0≤ ➲ ≤ à 2 . All sides of the plate are hinged. Boundary conditions are prescribed:

= ➬ =0 at = 0, = ❰ =0

at ✃ = à 1 ,

=0 ➬ at ➲ = 0, = ❰ ➫ ➫ ➬ =0 at ➲ = à 2 . Green’s function: Û

sin( ä ➚ ✃ ) sin( å ã ➲ ) sin( ä ➚ ➮ ) sin( å ã × )

9.3.2-5. Domain: 0 ≤ ✃ ≤ à 1 ,0≤ ➲ ≤ à 2 . The 1st and 3rd derivatives are prescribed at the sides:

= ➬ ❰ ➬ =0 at ✃ = 0, ❰ = ❰ =0 at ✃ = à 1 ,

= ➬ ❰ ➫ è❱è❱è ➫ ➫ =0 at ➲ = 0, ❰ ➫ è = ❰ ➫ è❱è❱è ➫ ➫ ➬ =0 at ➲ = à 2 . Green’s function: Û

1 sin( ) ( ✃ , ➲ , ➮ , × , )= ➴

cos( ä ➚ ✃ ) cos( å ã ➲ ) cos( ä ➚ à ➮ à ) cos( å ã × )

= ➷ = 0, the ratio sin( æ ➚ ã ) ë æ ➚ ã must be replaced by .

9.3.2-6. Domain: 0 ≤ ✃ ≤ à 1 ,0≤ ➲ ≤ à 2 . Mixed boundary conditions are set at the sides:

= ➬ =0 at = 0, = =0 at ✃ = à 1 ,

= ➬ ➫ ❰ ➫ è❱è❱è =0 at ➲ = 0, ❰ è = ➫ ➫ ❰ è❱è❱è ➬ =0 at ➲ = à 2 . Green’s function: Û

sin( ➚ ) ( , , ➮ , × , )= ➴

sin( ä ➚ ✃ ) cos( å ã à ➲ à ) sin( ä ➚ ➮ ) cos( å ã × ) ,

= ê 1 for = 0,

2 for ≠ 0.

9.3.2-7. Domain: 0 ≤ Ñ ➼ < ❒ ,0≤ ➽ ≤2 . Cauchy problem. Initial conditions for the symmetric case in the polar coordinate system:

= ➷ ❮ ( ➼ ) at = 0, ❰ Ï =0 at = 0. Solution:

, )= ➴ ➷ Ò Ó ➮ ❮ ( ➮ ) ➾ 0 ì ➴ ➷ í sin ì

Þ✓ß where ➾ 0 ( ) is the zeroth Bessel function.

References : I. Sneddon (1951), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

9.3.2-8. Domain: 0 ≤ Ñ ➼ ≤ ï ,0≤ ➽ ≤2 . Transverse vibration of a circular plate. Initial and boundary conditions for symmetric transverse vibrations of a circular plate of radius ï

with clamped contour in the polar coordinate system:

( ➷ ➼ ) at = 0, ❰ Ï = Ð ( ) at = 0;

at ➼ = ï . Solution: ➭

where the ➸ ➚ are positive roots of the transcendental equation (the prime denotes the derivative)

and the coefficients ➵ ➚ and ➺ ➚ are given by

Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

9.3.3. Three- and õ ô -Dimensional Equations of the Form

9.3.3-1. Three-dimensional case. Cauchy problem. Domain: − ❒ < ✃ < ❒ ,− ❒ < ú < ❒ ,− ❒ < î < ❒ . Initial conditions are prescribed:

= ➷ ( , , ) at = 0, =0 at = 0. Solution:

Reference Ó : V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, et al. (1974).

9.3.3-2. Three-dimensional case. Boundary value problem. Domain: 0 ≤ ✃ ≤ à 1 ,0≤ ú ≤ à 2 ,0≤ î ≤ à 3 (rectangular parallelepiped).

Initial conditions:

= ➷ ❮ ( ✃ , ú , î ) at = 0, ❰ Ï = Ð ( ✃ , ú , î ) at = 0. Boundary conditions:

= ➬ ❰ ý✡ý =0 at î = 0, = ❰ ý✡ý =0 at î = à 3 . Solution:

sin( ä ✃ ) sin( å ã ú

) sin( û ) sin( ÿ æ ã ),

9.3.3-3. ÿ ✄ -dimensional case. Cauchy problem.

Domain: ✂ = {− < < ; = 1, ❐✖❐✖❐ , ✄ }. Initial conditions are prescribed:

= 0, ✆ ❰ Ï =0 at ✝ = 0, where x = { ✃ 1 , ❐✖❐✖❐ , ✃ }.

4 where y = { Ù ú

, ú } and y= ú 1 ☛☞☛☞☛ ú .

Reference : V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, et al. (1974).

9.3.3-4. ÿ ✄ -dimensional case. Boundary value problem. ÿ Domain: ✌

= {0 ≤ ✂ ✍ ≤ à ; = 1, 2, ☛☞☛☞☛ ✄ }( ✄ -dimensional rectangular parallelepiped). Initial conditions:

= ✑ (x) at ✝ = 0. Boundary conditions:

= ✡ (x) at

= ✎ ü ✒☞ü ✒ ✆ =0 at ✍

= 0, ✆ = ✎ ü ✒☞ü ✒ ✆ =0 at ✍ = à .

Solution: Û

(x, ✡

(y) (x, y, ✝ Ù

) Ù y+ ✑ (y) (x, y, ✝ ) y, ) Ù y+ ✑ (y) (x, y, ✝ ) y,

(x, y, ✘ )=

sin( ✛ 1 ✜ 1 ) sin( ✛ 2 ✜ 2 ) ☛☞☛☞☛ sin( ✛ ✠ ✜ ✢ )

× sin( ✛ ✘ 1 ) sin( ✛ ✘ 2 ) ☛☞☛☞☛ sin( ✛ ✘ ✠

9.3.4. Equations of the Form ✭ ✬ + ø 2

9.3.4-1. Domain: 0 ≤ ✕

≤ 1 ,0≤ ✣ ≤ 2 . Solution in terms of the Green’s function.

We consider boundary value problems in a rectangular domain 0 ≤ ✕

≤ 1 ,0≤ ✣ ≤ 2 with the general initial conditions

= ✶ ( ✜ , ✣ ) at ✦ =0 and various homogeneous boundary conditions. The solution can be represented in terms of the

= ( ✜ , ✣ ) at ✦ = 0,

Green’s function as

Paragraphs 9.3.4-2 through 9.3.4-4 present the Green’s functions for three types of boundary conditions.

9.3.4-2. The function and its second derivatives are prescribed at the sides of a rectangle: ✳ ✳ ✳ ✳

= ✴ ❀ ❀ ✳ =0 at ✜ = 0,

=0 at ✜ = 1 , = ✴ ❁❂❁

=0 at ✣ = 2 . Green’s function:

sin( ✛ ✢ ✜ ) sin( ❄ ❃ ✣ ) sin( ✛ ✢ ✸ ) sin( ❄ ❃ ✹ )

9.3.4-3. The first and third derivatives are prescribed at the sides of a rectangle: ✳ ✳ ✳ ✳

=0 at ✜ = 0,

=0 at ✜ = 1 ,

=0 at ✣ = 2 . Green’s function:

cos( ✛ ✢ ✜ ) cos( ❄ ❃ ✣ ) cos( ✛ ✢ ✸ ) cos( ❄ ❃ ✹ )

1 for = 0,

2 for ❆ ≠ 0.

9.3.4-4. Mixed boundary conditions are prescribed at the sides of a rectangle: ✳ ✳ ✳ ✳

=0 at ✣ = ❋ 2 . Green’s function:

) sin( ✢ ) cos(

9.3.5. Equations of the Form ✬

9.3.5-1. Domain: 0 ≤ ✜ ≤ ❋ 1 ,0≤ ✣ ≤ ❋ 2 . Solution in terms of the Green’s function. We consider boundary value problems in a rectangular domain 0 ≤ ✜ ≤ ❋ 1 ,0≤ ✣ ≤ ❋ 2 with the general

initial conditions

= ✶ ( ✜ , ✣ ) at ✦ =0 and various homogeneous boundary conditions. The solution can be represented in terms of the

, ✣ ) at ✦ = 0,

Green’s function as

Paragraphs 9.3.5-2 through 9.3.5-4 present the Green’s functions for three types of boundary conditions.

9.3.5-2. The function and its first derivatives are prescribed at the sides of a rectangle: ✳ ✳ ✳ ✳

= ✴ ❀ ✳ =0 at ✜ = 0,

= ✴ ❀ ✳ =0 at ✜ = ❋ 1 , = ✴ ❁

=0 at ✣ = ❋ 2 . Green’s function:

( ✜ )= sinh( ✛ ✢ ❋ 1 ) − sin( ✛ ✢ ❋

1 ) ◗ cosh( ✛ ❑ ✢ ✜ ) − cos( ✛ ✢ ✜ ) ◗

− cosh( ✛ ✢ ❋ 1 ) − cos( ✛ ✢ ❋ 1 ) ◗ sinh( ✛ ✢ ✜ ) − sin( ✛ ✢ ✜ ) ◗ ,

( ✣ )= sinh( ❄ ❃ ❋ 2 ) − sin( ❄ ❃ ❋ 2 ) ◗ cosh( ❄ ❃ ✣ ) − cos( ❄ ❑ ❃ ❑ ✣ ) ◗

− cosh( ❄ ❃ ❋ 2 ) − cos( ❄ ❃ ❋ 2 ) ◗ sinh( ❄ ❃ ✣ ) − sin( ❄ ❃ ✣ ) ◗ , where the ✛ ✢ and ❄ ❃ are positive roots of the transcendental equations

cosh( ✛ ❋ 1 ) cos( ✛ ❋ 1 ) = 1, cosh( ❄ ❋ 2 ) cos( ❄ ❋ 2 ) = 1.

9.3.5-3. The function and its second derivatives are prescribed at the sides of a rectangle:

=0 at ✞ = ☎ 2 . Green’s function:

) sin( ✑ ✎ ✞ ) sin( ✏ ✍

9.3.5-4. The first and third derivatives are prescribed at the sides of a rectangle:

=0 at ✞ = ☎ 2 . Green’s function:

cos( ✏ ✍ ✄ ) cos( ✑

) cos( ✏ ✍ ✠ ) cos( ✑ ✎ ✡ ) ,