Equations with Ñ Space Variables

6.6. Equations with Ñ Space Variables

Throughout this section the following notation is used: →

2 , x={ 1 , ➘✄➘✄➘ , }, y={ Õ 1 , ➘✄➘✄➘ , Õ }, | x| = Ö

6.6.1. Wave Equation ×

6.6.1-1. Fundamental solution: →

if ≥ 2 is even;

if ê ≥ 3 is odd;

î❺ï where ( í ) is the Heaviside unit step function and ë ( í ) is the Dirac delta function.

Reference è : V. S. Vladimirov (1988).

6.6.1-2. Properties of solutions. Suppose ð ( ñ 1 , ò✄ò✄ò , ñ , Þ ) is a solution of the wave equation. Then the functions

are also solutions of this equation everywhere they are defined; ó , ö 1 , ò✄ò✄ò , ö +1 , ÷ , and õ are arbitrary constants. The signs at õ in the expression of ð 1 can be taken independently of one é

another. 6.6.1-3. Domain: − ✂ <

< ; = 1, ò✄ò✄ò , ê . Cauchy problem.

Initial conditions are prescribed:

Here, ✡ ☛ [ ✄ (x)] is the average of ✄ over the surface of the sphere of radius ë ý with center at x:

|x−y|=

where ✌ ý −1

is the area of the surface of an ê -dimensional sphere of radius ý , ☞ ✍ ✎ is the area

element of this surface, and |x − y| 2 =( ñ 1 − ✓ 1 ) 2 + ✔✕✔✕✔ +( ñ − ✓ ) 2 .

For odd ê , the solution can be alternatively represented as é é

(x, )=

(x)]

[ (x)] .

For even ê , the solution can be alternatively represented as

References : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), R. Courant and D. Hilbert (1989), D. Zwillinger ë

6.6.1-4. Domain: 0 ≤ ✂

≤ ✘✙✁ ; = 1, ò✄ò✄ò , ê . Boundary value problems.

For solutions of the first, second, third, and mixed boundary value problems with nonhomoge- neous conditions of general form, see Paragraphs 6.6.2-2, 6.6.2-3, 6.6.2-4, and 6.6.2-5 for ✚ ≡ 0, respectively.

6.6.2. Nonhomogeneous Wave Equation ✛

6.6.2-1. Domain: − ✂ <

< ; = 1, ò✄ò✄ò , ê . Cauchy problem.

Initial conditions are prescribed:

[ ë ✄ (x)] is the average of ✄ over the spherical surface of radius ý with center at x:

|x−y|= ☛

where ✌ ý −1 is the area of the surface of an ê -dimensional sphere of radius ý and ☞ ✍ ✎ is the area element of this surface. é é î❺ï

References : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), R. Courant and D. Hilbert (1989).

= {0 ≤ ñ ✁ ≤ ✘✙✁ ; = 1, ò✄ò✄ò , ê }. First boundary value problem. The following conditions are prescribed:

6.6.2-2. Domain: ✂

= ✄ 0 (x)

at û =0

(initial condition),

= ✄ 1 (x)

at û =0

(initial condition),

= ✆ ✁ (x, û ) at ñ ✁ =0 (boundary conditions),

= ✭ ✁ (x, û ) at ñ ✁ = ✘✙✁

(boundary conditions).

where x={ ✁

1 , ✹✕✹✕✹ , ✸ }, y = { ✓ 1 , ✹✕✹✕✹ , ✓ }, ☞ y= ☞ ✓ 1 ☞ ✓ 2 ✹✕✹✕✹✺☞ ✓ , ☞ ✍ ( ✎ ) = ☞ ✓ 1 ✹✕✹✕✹✺☞ ✓ ✁ −1 ☞ ✓ ✁ +1 ✹✕✹✕✹✻☞ ✓ ,

= {0 ≤ ✓ ✼ ≤ ✘✽✼ for ✾ = 1, ✹✕✹✕✹ , − 1, + 1, ✹✕✹✕✹ , ✿ }. Green’s function:

sin( ❅ 1 ✸ 1 ) sin( ❅ 2 ✸ 2 ) ✹✕✹✕✹

× sin( ❅ 1 ✓ 1 ) sin( ❅ 2 ✓ 2 ✹✕✹✕✹

6.6.2-3. Domain: ✬ = {0 ≤ ✸ ❋ ≤ ✘✙❋ ; ● = 1, ✹✕✹✕✹ , ✿ }. Second boundary value problem. The following conditions are prescribed:

= ■ 0 (x)

at ❀ =0

(initial condition),

= ■ 1 (x)

at ❀ =0

(initial condition),

= ▲ ✴ ❋ ❑ (x, ❀ ) at ✸ ❋ =0 (boundary conditions),

= ✭ ❋ (x, ❀ ) at ✸ ✴ ❋ ❑ = ✘✙❋ (boundary conditions). Solution:

1 ) cos( ❅ 2 ✸ 2 ) ✹✕✹✕✹ cos( ❅

) cos( ❅ 1 ✓ 1 ) cos( ❅ 2 ✓ 2 ) ✹✕✹✕✹ cos( ❅ ✓ ),

The summation is performed over the indices satisfying the condition ❨

1 + ✔✕✔✕✔ + ❴❵❝ > 0; the term corresponding to ❴ 1 = ✔✕✔✕✔ = ❴❵❝ = 0 is singled out.

= {0 ≤ ✸ ❞ ≤ ❞ ; ❡ = 1, ✹✕✹✕✹ , ❜ }. Third boundary value problem. The following conditions are prescribed:

6.6.2-4. Domain: ❳

= ❢ 0 (x)

at ❣ =0

(initial condition),

= ❢ (x)

at ❣ =0

(initial condition),

− ❧❉❞ ❍ = ♠ ❞ (x, ❣ ) at ✸ ❞ =0 (boundary conditions),

(boundary conditions). The solution ❍ (x, ❣ ) is determined by the formula in Paragraph 6.6.2-3 where

sin q ❩❇s ❣❉t ✉ q❄r 2 + ✉ q 1 2 2 + ✔✕✔✕✔ q + ✉ 2 P q❄r

× sin( q

1 + ✇ q 1 ✉ ) sin( q 2 ✸

2 q❄r ✇ 2 ) sin( ✉ ✸ ❝ + q❄r ✹✕✹✕✹ + ✇ ) × sin( ✉ 1 ① 1 + ✇ 1 ) sin( ✉ 2 ① 2 + ✇ 2 ) ✹✕✹✕✹ sin( ✉ ① ❝ + ✇

are positive roots of the transcendental equations

6.6.2-5. Domain: ✬ = {0 ≤ ✸ ❞ ≤ ⑤✙❞ ; ❡ = 1, ✹✕✹✕✹ , ❜ }. Mixed boundary value problem. The following conditions are prescribed:

= ❢ 0 (x)

at ❣ =0

(initial condition),

= ❢ 1 (x)

at ❣ =0

(initial condition),

= ♠ ❞ (x, ❤ ❣ ❥ ❦ ) at ✸ ❞ =0 (boundary conditions),

(boundary conditions). Solution:

= ⑦ ❞ (x, ❣ ) at ✸ ❞ = ⑤✙❞

(x, ❣ )= ⑧

(y, ✩ ) ❶ (x, y, ❣ − ✩ ) ❷ y ❷ ✩

+ ❤ ⑧ ⑨ ❢ ❣ 0 (y) ❶ (x, y, ❣ ) ❷ y+ ⑧ ⑨ ❢ 1 (y) ❶ (x, y, ❣ ) ❷ y

2 + ❻ ⑧ ⑧ ❸ ❦ ♠ ❞ (y, ✩ ) ❤ ❶ ❣ ✩ ❺ ❻ ❞

(x, y, − ) ❦ ❷ ❯ ( ) ❷ ✩

(y, ✩ ) ❶ (x, y, ❣ − ✩ ) ❺ ❻ ❦ ❦ ❷ ❯ ( ❞ ) ❷ ✩ ,

q❄r

(x, y, ♦ )= ✹✕✹✕✹ sin( ✉

1 ) sin( ✉ 2 ✸ 2 ) ✹✕✹✕✹ sin( ✉ ✸ ⑤ ❝

sin ❽ s ❣❉t q ✉ 2 ✔✕✔✕✔ q❄r ✉ 2

× sin( ✉ 1 ① 1 ) sin( ✉

6.6.3. Equations of the Form ➁ = – + ➇ ( ➈ 1 , ➉ ➉ ➉ ➅ , ➈ ,

6.6.3-1. Domain: − ➊ < ✸ ➋ < ➊ ; ➌ = 1, ✹✕✹✕✹ , ➍ . Cauchy problem. Initial conditions are prescribed:

= ➏ (x) at ➐ = 0,

= ➓ (x) at ➐ = 0,

where x = { ✸ 1 , ✹✕✹✕✹ , ✸ ➔ }.

1 → . Let ➣ =− ↔ 2 < 0 and ⑩ ≡ 0. The solution is sought by the descent method in the form ➎

where ↕ is the solution of the Cauchy problem for the auxiliary ( ➍ ➑ + 1)-dimensional wave equation

(2) with the initial conditions

= exp( ↔✫✸ ➔ +1 ) ➏ (x) at ➐ = 0,

= exp( ↔✫✸ ➔ +1 ) ➓ (x) at ➐ = 0.

For the solution of problem (2), (3), see Paragraph 6.6.1-3.

2 → . Let ➣ = ↔ 2 > 0 and ⑩ ≡ 0. In this case the function exp( ↔✫✸ ➔ +1 ) in (1) and (3) must be replaced ➛➝➜ by cos( ↔✫✸ ➔ +1 ).

Reference : R. Courant and D. Hilbert (1989).

6.6.3-2. Domain: ➞ = {0 ≤ ✸ ➋ ≤ ⑤✙➋ ; ➌ = 1, ✹✕✹✕✹ , ➍ }. First boundary value problem. The following conditions are prescribed: ➎

= ➏ 0 (x) at ➑ ➐ ➒ =0

(initial condition),

= ➏ 1 (x)

at ➐ =0

(initial condition),

= ➓ ➋ (x, ➐ ) at ✸ ➋ =0 (boundary conditions),

(boundary conditions). Solution:

= ⑦ ➋ (x, ➐ ) at ✸ ➋ = ⑤✙➋

(x, ➐ )= ⑧

(y, ➟ ) ❶ (x, y, ➐ − ➟ ) ❷ y ❷ ➟

+ ➑ ⑧ ⑨ ➏ ➐ 0 (y) ❶ (x, y, ➐ ) ❷ y+ ⑧ ⑨ ➏ 1 (y) ❶ (x, y, ➐ ) ❷ y

(y, ➟ ) ➑ ➤ ❶ (x, y, ➐ − ➟ ) ❺ ❻

(y, )

(x, y, − )

( ✡ x={ ✠ 1 , , }, y = { 1 , , }, y= 1 2 , ) = ✝ ✆ 1 ✂✄✂✄✂✞✝ ✆ ✠ −1 ✝ ✆ ✠ +1 ✂✄✂✄✂☛✝ ✆ ☎ ,

Green’s function:

(x, y, ✕

sin( ✙ ✖ 1 ✁ 1 ) sin( ✙ ✖ 2 ✁ 2 ) ✂✄✂✄✂ sin( ✙ ✖✘✗ ✁ ✌ ☎

× sin( 1 1 ) sin( 2 2 )

6.6.3-3. Domain: ✩ = {0 ≤ ✁ ✠ ≤ ✌ ✠ ; ✏ = 1, ✂✄✂✄✂ , ✑ }. Second boundary value problem. The following conditions are prescribed:

0 (x)

at ✓ =0

(initial condition),

at ✓ ✬ =0 ✮ ✯

= ✫ 1 (x)

(initial condition),

= ✰ ✠ (x, ✓ ) at ✁ ✠ =0

(boundary conditions),

(boundary conditions). Solution:

cos( ✙ ✖ ✁ 1 ) cos( ✙ ✖ 2 ✌ ✁ ✌ ✂✄✂✄✂✞✌✍☎ 1 2 ) ✂✄✂✄✂ cos( ✙ ✖✘✗ ✁ ☎ )

× cos( ✙ ✖ 1 ✆ 1 ) cos( ✙ ✖ 2 ✆ 2 ) ✂✄✂✄✂ cos( ✙ ✖✘✗ ✆ ☎

where

1 for ☞ = 0,

1 ✑ 2 2 for

6.6.3-4. Domain: ✩ = {0 ≤ ✁ ✠ ≤ ✌ ✠ ; ✏ = 1, ✂✄✂✄✂ , ✑ }. Third boundary value problem. The following conditions are prescribed:

0 (x)

at ✓ =0

(initial condition),

(initial condition),

− ✥ ✪ ✠ = ✰ ✠ (x, ✓ ) at ✁ ✠ =0 (boundary conditions),

(boundary conditions). The solution ✪ (x, ✓ ) is determined by the formula in Paragraph 6.6.3-3 where

× sin( ✙ ✖ 1 ✁ 1 + ✿ ✖ 1 ) sin( ✙ ✖ 2 ✁ 2 + ✿ ✖ 2 ) ✂✄✂✄✂ sin( ✙ ✖✘✗ ✁ ☎ + ✿ ✖✘✗ ) × sin( ✙ ✖ 1 ✆ 1 + ✿ ✖ 1 ) sin( ✙ ✖ 2 ✆ 2 + ✿ ✖ 2 ) ✂✄✂✄✂ sin( ✙ ✖✘✗ ✆ ☎ + ✿ ✖✘✗ ). Here,

the ✙ ✖✘✻ are positive roots of the transcendental equations

6.6.3-5. Domain: ✩ = {0 ≤ ✁ ✠ ≤ ✌ ✠ ; ✏ = 1, ✂✄✂✄✂ , ✑ }. Mixed boundary value problem. The following conditions are prescribed:

= ✫ (x)

at ✓ =0

(initial condition),

= ✫ 1 (x)

at ✓ =0

(initial condition),

= ✰ ✠ (x, ✓ ) at ✁ ✠ =0

(boundary conditions),

(boundary conditions). Solution:

sin( ✙ ✖ 1 ✁ 1 ) sin( ✙ ✖ 2 ✁ 2 ) ✂✄✂✄✂ sin( ✙ ✖✘✗ ✁ ✌ ☎

× sin( ✙ ✖ 1 ✆ 1 ) sin( ✙ ✖

2 ) ✂✄✂✄✂ sin( ✙ ✖✘✗ ✆ ☎

where

6.6.4. Equations Containing the First Time Derivative

Nonhomogeneous telegraph equation with ❉ ✑ space variables.

1 ✪ . The substitution = exp ✚ − 1

leads to the equation

− ✚ ✥ − 1 2 4 1 ◆ ✦✢❖ + exp ✚ 2 ◆ ✓ ✦ ✴ ( ✁ 1 , ✂✄✂✄✂ , ✁ ☎ , ✓ ), which is considered in Subsection 6.6.3.

2 ▼ . Domain: ✩ = {0 ≤ ✁ ✠ ≤ ✌ ✠ ; ✏ = 1, ✂✄✂✄✂ , ✑ }. First boundary value problem. The following conditions are prescribed:

= ✫ 0 (x)

at ✓ =0

(initial condition),

= ✫ 1 (x)

at ✓ =0

(initial condition),

= ✰ ✠ (x, ✓ ) at ✁ ✠ =0 (boundary conditions),

(boundary conditions). Solution:

where x={ ✁ , ✂✄✂✄✂ , ✁ ☎ }, y = { ✆ , ✂✄✂✄✂ , ✆ ☎ }, ✝ y= ✝ ✆ ✝ ✆ ✂✄✂✄✂✞✝ ✆ ☎ , ✝ ✟ ✡ ( ✠ ) 1 1 1 2 = ✝ ✆ 1 ✂✄✂✄✂✞✝ ✆ ✠ −1 ✝ ✆ ✠ +1 ✂✄✂✄✂☛✝ ✆ ☎ ,

( ✠ ) = {0 ≤ ✆ ☞ ≤ ✌✍☞ for ✎ = 1, ✂✄✂✄✂ , ✏ − 1, ✏ + 1, ✂✄✂✄✂ , ✑ }.

Green’s function: ✭✛❙

(x, y, ✕ )=

sin( ✙ ✖ 1 ✁ 1 ) sin( ✙ ✖ 2 ✁ 2 ) ✂✄✂✄✂ sin( ✙ ✖✘✗ ✁ ✌ ☎ ✌

× sin( 1 1 ) sin( 2 2 )

3 ▼ . Domain: ✩ = {0 ≤ ✁ ✠ ≤ ✌ ✠ ; ✏ = 1, ✂✄✂✄✂ , ✑ }. Second boundary value problem. The following conditions are prescribed:

= ✫ 0 (x)

at ✓ =0

(initial condition),

= ✫ (x)

at ✓ =0

(initial condition),

= ✰ ✠ (x, ✓ ) at ✁ ✠ =0

(boundary conditions),

= ✱ ✠ (x, ✓ ) at ✁ ✠ = ✌ ✠

(boundary conditions).

1 1 ) cos( 2 ) cos(

1 ) cos( ✙ ✖ 2 ✆ 2 ) ✂✄✂✄✂ cos( ✙ ✖✘✗

4 ▼ . Domain: ✩ = {0 ≤ ✁ ✠ ≤ ✌ ✠ ; ✏ = 1, ✂✄✂✄✂ , ✑ }. Third boundary value problem. The following conditions are prescribed:

= ✫ 0 (x)

at ✓ =0

(initial condition),

= ✫ (x)

at ✓ =0

(initial condition),

− ✥ ✪ ✠ = ✰ ✠ (x, ✓ ) at ✁ ✠ =0 (boundary conditions),

+ ✽ ✪ ✠ = ✱ ✠ (x, ✓ ) at ✁ ✠ = ✌ ✠ (boundary conditions). The solution ✪ (x, ✓ ) is given by the formula in Item 3 ▼ with

sin ✚ ✓ ✜ ✣ 2 ( ✙ ✖ 2 2 2 2 ❘ ❚

(x, y, ✓ )=2 − 2 ✂✄✂✄✂

( ✙ 2 ✖ =1 ✖✘✗ =1 1 + ✙ ✖ 2 + ✤✄✤✄✤ + ✙ ✖✘✗ )+ ✥ − ◆ 4 × sin( ✙ ✖ 1 ✁ 1 + ✿ ✖ 1 ) sin( ✙ ✖ 2 ✁ 2 + ✿ ✖ 2 ) ✂✄✂✄✂ sin( ✙ ✖✘✗ ✁ ☎ + ✿ ✖✘✗ )

× sin( ✙ ✖ 1 ✆ 1 + ✿ ✖ 1 ) sin( ✙ ✖ 2 ✆ 2 + ✿ ✖ 2 ) ✂✄✂✄✂ sin( ✙ ✖✘✗ ✆ ☎ + ✿ ✖✘✗ ). Here,

the ✙ ✖✘✻ are positive roots of the transcendental equation

= cot( ✌❄☞ ✙

5 ▼ . Domain: ✩ = {0 ≤ ✁ ✠ ≤ ✌ ✠ ; ✏ = 1, ✂✄✂✄✂ , ✑ }. Mixed boundary value problem. The following conditions are prescribed:

= ✫ 0 (x)

at ✓ =0

(initial condition),

= ✫ 1 (x)

at ✓ =0

(initial condition),

= ✰ ✮ ✠ ✯ (x, ✓ ) at ✁ ✠ =0 (boundary conditions),

(boundary conditions). Solution:

sin( ✙ ✖ 1 ✁ 1 ) sin( ✙ ✖ 2 ✁ 2 ) ✂✄✂✄✂ sin( ✙ ✖✘✗ ✁ ☎ )

× sin( 1 1 ) sin( ✙ ✖ 2 ✆ 2 ) ✂✄✂✄✂ sin( ✙ ✖✘✗ ✆ ☎ )

The transformation ❇

leads to the equation

which is considered in Subsection 6.6.3. ❩❭❬

Reference : R. Courant and D. Hilbert (1989). 2 ❈

Darboux equation. ❉ Cauchy problem. Initial conditions are prescribed:

= ✫ (x) at ✓ = 0,

at ✓ = 0.

| x−y|=

where ❡ −1 is the area of the surface of an ❞ -dimensional sphere of radius , and ❵ ❛ ❜ is the area element of this surface (i.e., the solution ✪ is the average of the function ✫ over the sphere for radius ❡

with center at x).

❩❭❬

Reference : R. Courant and D. Hilbert (1989).

Chapter 7

Elliptic Equations with Two Space Variables