Equations with ✸ Space Variables

3.4. Equations with ✸ Space Variables

3.4.1. Equations of the Form ✹ ✺ = ✼ ✽ ✾ ✺ + ✿ ( ❀ 1 , ❁ ❁ ❁ , ❀ ✾ , ✻ )

This is an ✻

-dimensional nonhomogeneous heat equation. In the Cartesian system of coordinates, it ✡

is represented as

The solutions of various problems for this equation can be constructed on the basis of incomplete separation of variables (see Paragraphs 0.6.1-2 and 0.9.2-1) taking into account the results of Subsections 1.1.1 and 1.1.2. Some examples of solving such problems can be found below in Paragraphs 3.4.1-2 through 3.4.1-4.

3.4.1-1. Homogeneous equation ( ≡ 0).

1 ✡ ✖ . Particular solutions: ✡

where x = { ❂ 1 ,

, }; ❈ , ✮ ✜ , ❆ ✜ , and ❇ 0 are arbitrary constants.

2 ✖ . Fundamental solution: ● ●

3 P . Suppose ◗ = ◗ ( 1 , ✭✘✭✘✭ ,

, ❇ ) is a solution of the homogeneous equation. Then the functions

, and ❑ ❳ are arbitrary constants, are also solutions of the equation. The signs at ❙ in the formula for ◗ 1 can be taken independently of one another.

3.4.1-2. Domain: ❵ ❑ = {− ❛ < < ❛ ; ❜ = 1, ✭✘✭✘✭ , ❝ }. Cauchy problem. An initial condition is prescribed: ❖

= ❞ (x) at ❬ = 0.

Solution:

| x−y| 2 (x, ❬ )= ❍

1 | x−y|

4 ❱ ( ❬ − ❦ ) where y = { ❧ ✭✘✭✘✭ ❧

Reference : V. S. Vladimirov (1988).

3.4.1-3. Domain: q = {0 ≤ ≤ r ; ❜ = 1, ✭✘✭✘✭ , ❝ }. First boundary value problem. The following conditions are prescribed: ❖

= ▼ ❞ (x)

at ❬ ▼ =0

(initial condition),

= ▼ s (x, ❬

) ▼ at =0 (boundary conditions),

(boundary conditions). Solution: ❖

where the following notation is used: ▼

= {0 ≤ ❧ ⑩ ≤ r❶⑩ for ❷ = 1, ✭✘✭✘✭ , ❜ −1, ❜ +1, ✭✘✭✘✭ , ❝ }. The Green’s function can be represented in the product form ✇

where the ✈ ( , ❧ , ❬ ) are the Green’s functions of the respective one-dimensional boundary value problems (see Paragraph 1.1.2-5): ❖

3.4.1-4. Domain: q = {0 ≤ ❽ ❾ ≤ r❿❾ ; ➄ = 1, ✭✘✭✘✭ , ➅ }. Second boundary value problem. The following conditions are prescribed:

= ➆ (x)

at ➃ =0

(initial condition),

= (x, ) at

(boundary conditions),

(x, ) at ❽ ❾ = r❿❾

(boundary conditions).

The Green’s function can be represented as the product (1) of the corresponding one-dimensional Green’s functions of the form (see Paragraph 1.1.2-6)

3.4.2. Other Equations Containing Arbitrary Parameters ➦ ➦

This is a special case of equation 3.4.3.1. The transformation

leads to the ➎

-dimensional heat equation ➳ ➵ =

that is dealt with in Subsection 3.4.1.

The transformation ( ➩

is any number)

leads to the ➭

-dimensional heat equation ➹

that is dealt with in Subsection 3.4.1.

This is a special case of equation 3.4.3.2 with ❒ ( ➾ )= ➯ and ➒ ( ➾ )= ❮ .

The substitution ➧

leads to the ➎

-dimensional heat equation ➳ ➵

that is dealt with in Subsection 3.4.1.

This is a special case of equation 3.4.3.4.

This is the n-dimensional Schr ¨odinger equation, ➧

Fundamental solution:

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

3.4.3. Equations Containing Arbitrary Functions

The transformation é

leads to the ù

-dimensional heat equation ö ÷✃ñ = ó ø ö úüûýúüû ñ that is discussed in Subsection 3.4.1.

1 . Case æ > 0. The transformation

1 exp ➴ 2 ✞ ó í✠✟ , ✭✘✭✘✭ , ò = ê exp ✡ 2 ✞ ó í✠✟ , ✂ =

exp ✡ 4 ✞ ó í✠✟ + ☛ ,

where ☛ is an arbitrary constant, leads to an equation of the form 3.4.3.1: ☞ ë ☞

2 . Case ✁ < 0. The transformation

also leads to an equation of the form 3.4.3.1 (this equation is not specified here).

The solutions of various problems for this equation can be constructed on the basis of incomplete ✖ separation of variables (see Paragraphs 0.6.1-2 and 0.9.2-1) taking into account the results of ☞

Subsections 1.1.1 and 1.1.2. Some examples of solving such problems are given below. It is assumed that 0 < ó ( í )< ✢

1 . Domain: ✤ = {− ✢ < ø ê < ✢ ; ✣ é = 1, ✭✘✭✘✭ , ➷ }. Cauchy problem. An initial condition is prescribed:

x={ ê ☞ 1 , ✭✘✭✘✭ ☞ , ê ✮ }, y={ ✲ 1 , ✭✘✭✘✭ , ✲ ✮ }, ✭ y= ✭ ✲ 1 ✭ ✲ 2 ✭✘✭✘✭ ✭ ✲ ✮ .

2 ✷ . Domain: ✸ = {0 ≤ ✹ ≤ ✺ ; ✣ = 1, ✭✘✭✘✭ , ➷ }. First boundary value problem. The following conditions are prescribed:

= ☞ (x) at =0

(initial condition),

= ☞ (x, ✬ ) at ✹ =0 (boundary conditions),

(boundary conditions). Solution:

where the following notation is used: ☞ ☞ ☞

, ❁ ( ) = {0 ≤ ✲ ❅ ≤ ✺❆❅ for ❇ = 1, ✭✘✭✘✭ , ✣ −1, ✣ +1, ✭✘✭✘✭ , ❈ }. The Green’s function can be represented in the product form

where the ✫ ( ✹ , ✲ , ✬ , ✪ ) are the Green’s functions of the respective boundary value problems, ❊ ❊ ❊

3 ✷ . Domain: ✸ = {0 ≤ ✹ ≤ ✺ ; ❏ = 1, ✭✘✭✘✭ , ❈ }. Second boundary value problem.

The following conditions are prescribed:

= ❊ ▲ (x)

at ✬ ❊ =0

(initial condition),

= (x, ✬ ) at ✹ =0

(boundary conditions),

(boundary conditions). Solution:

The Green’s function can be represented as the product (1) of the corresponding one-dimensional ❊ ❊ ❊

Green’s functions ❊ ❊ ❊

=1 ✵ Reference : A. D. Polyanin (2000a, 2000b). ❬ ❬

Let us perform the transformation ❊

where the functions ▲ ( ✬ ), ◆ ( ✬ ), ✼ ( ✬ ), and ❝ ( ✬ ) are given by ( ❞ , ❡ , ❢ , and ❣ are arbitrary constants):

As a result, we arrive at an equation of the form 3.4.3.3 for the new dependent variable ② =

The substitution

( ❾ ① 1 , ❿➀❿➀❿ , ❾ , ➁ ) = exp ✐ ✈ ▲ ( ➁ ) ❾ 2 ♥ ② ( ❾ 1 , ❿➀❿➀❿ , ❾ , ➁ ),

where the functions ✈

= ▲ ( ➁ ) are solutions of the Riccati equation

=4 r ( ➁ ) ▲ 2 +2 ❧ ( ➁ ) ▲ + q ( ➁ )

( ➃ = 1, ❿➀❿➀❿ , ➄ ), leads to an equation of the form 3.4.3.4 for ② = ② ( ❾ 1 , ❿➀❿➀❿ , ❾ , ➁ ).

( ❾ , ➁ )< ✢ for all ➃ . We introduce the notation x = { ❾ 1 , ❿➀❿➀❿ , ❾ }, y = { ➈ 1 , ❿➀❿➀❿ , ➈ } and consider the domain ➉ ={ ➊

≤ ❾ ≤ ➋ , ➃ = 1, ❿➀❿➀❿ , ➄ }, which is an ➄ -dimensional parallelepiped. ✈ ✈

1 ➌ . First boundary value problem. The following conditions are prescribed: ❽

(x)

at ➁ =0

(initial condition),

= ① ➍ (x, ➁ ) at ❾ = ➊

(boundary conditions),

(boundary conditions). Solution: ❽

= 1, ❿➀❿➀❿ , ➃ −1, ➃ +1, ❿➀❿➀❿ , ➄ ✈ }. The Green’s function can be represented in the product form ① ① ①

Here, the ➒ = ➒ ( ❾ , ➈ , ➁ , ➑ ) are auxiliary Green’s functions that, for ➁ > ➑ ≥ 0, satisfy the one-dimensional linear homogeneous equations ① ① ① ③ ① ① ③ ① ① ③

( ➃ = 1, ❿➀❿➀❿ ❾ ➄ , ) (2) with nonhomogeneous initial conditions of a special form, ① ① ①

(3) and homogeneous boundary conditions of the first kind, ① ①

In determining the function ➒ , the quantities ➈ and ➑ play the role of parameters; ➥ ( ❾ ) is the Dirac delta function.

2 ➌ . The second and third boundary value problems. The following conditions are prescribed: ❽

= ▲ (x)

at ➁ =0

(initial condition),

− ➦ ① ③ ❽ = ➍ (x, ➁ ) at ❾ ①

(boundary conditions),

(boundary conditions). The second boundary value problem corresponds to ➝

(x, ) at

The Green’s function can be represented as the product (1) of the corresponding one-dimensional Green’s functions satisfying the linear equations (2) with the initial conditions (3) and the homogen- ① ① ① ① ①

eous boundary conditions

Reference : A. D. Polyanin (2000a, 2000b).

The problems considered below are assume to refer to a bounded domain ➉ with smooth surface ➜ . We introduce the brief notation x = { ❾ 1 , ❿➀❿➀❿ , ❾ } and assume that the condition

is satisfied; this condition imposes the requirement that the differential operator on the right-hand side of the equation is elliptic.

1 ➌ . First boundary value problem. The following conditions are prescribed: ❽

= ▲ (x)

at

=0 (initial condition),

= ➍ (x, ➁ ) for x ➻ ➜

(boundary condition).

Here, the Green’s function is given by

where the ✈ and (x) are the eigenvalues and corresponding eigenfunctions of the Sturm–Liouville problem for the following elliptic second-order equation with homogeneous boundary condition of ✈ ➚ ✈

the first kind:

(x)

− ➱ (x) + ➺

=0 ➷ ➬ for x ➻ ➜ .

The integration in solution (1) is carried out with respect to ➘ 1 , ➴➀➴➀➴ , ➘ ✃ ; ❐ ❐ ❒ ❮ is the differential operator defined as

where N = { Ð 1 , ➴➀➴➀➴ , Ð ✃ } is the unit outward normal to the surface ➷ ➷ Ñ . In the special case where

(x) = 1 and ➵ ➸ (x) = 0 for Ò ≠ Ó , the operator of (5) coincides with the usual operator of differentiation

along the direction of the outward normal to the surface Ï Ñ . General properties of the Sturm–Liouville problem (3)–(4):

1. There are countably many eigenvalues. All eigenvalues are real and can be ordered so that

as Ö Õ ✢ ; consequently, there can exist only finitely many negative eigenvalues.

1 ≤ ➺ 2 ≤ ➺ 3 ≤ Ô➀Ô➀Ô , with ➺ ✃ Õ

2. For ➱ (x) ≥ 0 all eigenvalues are positive: × ✃ > 0.

3. The eigenfunctions are defined up to a constant multiplier. Any two eigenfunctions ✃ (x) and

(x) corresponding to different eigenvalues × ✃ and × are orthogonal in the domain ➹ :

(x) (x) Ú ➹ =0 for Ö ≠ Û .

there generally correspond finitely many linearly indepen- dent eigenfunctions (1) ✃ , (2) ✃ , ➴➀➴➀➴ , ✃ ( ) . These functions can always be replaced by their linear combinations

To each eigenvalue Ø

= 1, 2, ➴➀➴➀➴ , Û , such that ¯ (1) ✃ ,¯ (2) ✃ ➚ , ➴➀➴➀➴ ,¯ ( ✃ ) ➚ are now pairwise orthogonal. Thus, without loss of generality, we ➚ ➚

assume that all eigenfunctions are orthogonal. ➚ ➚ ➚ Ø

2 ➌ . Second boundary value problem. The following conditions are prescribed:

= ç (x)

at

=0 (initial condition),

(x, ) for x ê Ñ

(boundary condition).

Here, the left-hand side of the boundary condition is determined with the help of (5), where ➶

, æ ➘ , y, and ➘ ã must be replaced by , , x, and ã , respectively. Solution: ❰

Here, the Green’s function is defined by (2), where the ➶

and ✃ (x) are the eigenvalues and corre- sponding eigenfunctions of the Sturm–Liouville problem for the elliptic second-order equation (3) ➚

with a homogeneous boundary condition of the second kind:

=0 for x ê Ñ .

For ➱ (x) > 0 the general properties of the eigenvalue problem (3), (7) are the same as for the ➷ è é

first boundary value problem (see Item 1 ➌ ). For ➱ (x) ≡ 0 the zero eigenvalue × 0 = 0 arises which

corresponds to the eigenfunction 0 = const.

It should be noted that the Green’s function of the second boundary value problem can be ➚

expressed in terms of the Green’s function of the third boundary value problem (see Item 3 ➌ ).

3 ➌ . Third boundary value problem. The following conditions are prescribed:

(initial condition),

(x) = ➍ (x, ) for x ê Ñ

(boundary condition).

The solution of the third boundary value problem is given by relations (6) and (2), where the ➶

and ✃

(x) are the eigenvalues and corresponding eigenfunctions of the Sturm–Liouville problem for the second-order elliptic equation (3) with a homogeneous boundary condition of the third kind: ➚

(x) ≥ 0 and å (x) > 0, the general properties of the eigenvalue problem (3), (8) are the same ➷ è é

as for the first boundary value problem (see Item 1 ➌ ). Let å (x) = å = const. Denote the Green’s functions of the second and third boundary value problems by 2 (x, y, ) and 3 (x, y, , å ), respectively. Then the following relations hold:

where ➹ 0 = ➏ Ú ➹ is the volume of the domain in question.

õ❱ö

References : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), A. D. Polyanin (2000a, 2000b).

Chapter 4

Hyperbolic Equations with One Space Variable