Construction of the Green’s Functions. General Formulas and Relations

0.9. Construction of the Green’s Functions. General Formulas and Relations

0.9.1. Green’s Functions of Boundary Value Problems for Equations of Various Types in Bounded Domains

0.9.1-1. Expressions of the Green’s function in terms of infinite series. Table 11 lists the Green’s functions of boundary value problems for second-order equations of

various types in a bounded domain ➨ . It is assumed that ➃ x is a second-order linear self-adjoint ↔ ↔ ➥

differential operator (e.g., see Zwillinger, 1998) in the space variables 1 , ➆✱➆✱➆ , , and x is a zeroth- or first-order linear boundary operator that can define a boundary condition of the first, second, or ➥ ➲

third kind; the coefficients of the operators ➃ x and x can depend on the space variables but are independent of time ❽ . The coefficients ➹ ➘ and the functions ➴ ➘ (x) are determined by solving the homogeneous eigenvalue problem

x [ ➴ ]+ ➹ ➴ = 0,

(2) It is apparent from Table 11 that, given the Green’s function in the problem for a parabolic (or

x [ ➴ ]=0 for x ➄ ➛ .

hyperbolic) equation, one can easily construct the Green’s functions of the corresponding problems for elliptic and hyperbolic (or parabolic) equations. In particular, the Green’s function of the problem for an elliptic equation can be expressed via the Green’s function of the problem for a parabolic equation as follows:

0 (x, y) = ❷ ➷ ❼ 1 (x, y, ❽ ) ❾ ❽ .

Here, the fact that all ➹ ➘ are positive is taken into account; for the second boundary value problem, it is assumed that ➹ = 0 is not an eigenvalue of problem (1)–(2).

TABLE 11 The Green’s functions of boundary value problems for equations of various types in bounded ➥ ↔ ↔

domains. In all problems, the operators ➃ x and x are the same; x = { 1 , ➆✱➆✱➆ , } Initial and ➲

Equation Green’s function

boundary conditions

Elliptic equation ➥ x [

2 ➹ − ➘ ➃ x [ ➈ ]= ❺ (x) (no initial condition required)

(x) ➴ ➂ ➘ ➬ ➬ (y)

Parabolic equation ➒ =0

Hyperbolic equation ➒

0.9.1-2. Some remarks and generalizations.

Formula (3) can also be used if the domain ➨ is infinite. In this case, one should make sure that the integral on the right-hand side is convergent. ➼ ➽❫➜ ➝ ➞❃➟ ❮ ➠

Suppose the equations given in the first column of Table 11 contain − ❰ x [ Ï ]− Ð Ï

instead of − ❰ x [ Ï ], with Ð being a free parameter. Then the ➹ ➘ in the expressions of the Green’s function in the third column of Table 11 must be replaced by ➹ ➘ − Ð ; just as previously, the ➹ ➘ and

(x) were determined by solving the eigenvalue problem (1)–(2). ➼ ➽❫➜ ➝ ➞❃➟ Ñ ➠

The formulas for the Green’s functions presented in Table 11 will also hold for boundary value problems described by equations of the fourth or higher order in the space variables; provided that the eigenvalue problem for equation (1) subject to appropriate boundary conditions is self-adjoint.

0.9.2. Green’s Functions Admitting Incomplete Separation of Variables

0.9.2-1. Boundary value problems for rectangular domains.

1 Ò . Consider the parabolic equation Ó

(4) where each term ↔

= ❰ 1 , Õ [ Ï ]+ Ö✱Ö✱Ö + ❰ , Õ [ Ï ]+ × (x, ),

[ Ï ] depends on only one space variable, Ó Ó Ø , and time :

= 1, , . For equation (4) we set the initial condition of general form ➳

(5) Consider the domain ➨ ={ ß Ø ≤ Ú Ø ≤ Ð Ø , Û = 1, Ü✱Ü✱Ü , Ý } which is an Ý -dimensional paral-

= Þ (x) at

lelepiped. We set the following boundary conditions at the faces of the parallelepiped: Ó Ô Ô

+ à Ø Ú (1) ( ) Ï = ♣ (1) Ø Ø (x, ) at Ú Ø = ß Ø ,

(x, ) at

By appropriately choosing the coefficients ♦ (1) Ø , (2) Ø and functions à (1) Ø = à (1) Ø ( ), à (2) Ø = à (2) Ø ( ), we can obtain the boundary conditions of the first, second, or third kind. For infinite domains, the boundary

conditions corresponding to ß Ø =− ➺ or Ð Ø = ➺ are omitted.

2 Ò . The Green’s function of the nonstationary Ý -dimensional boundary value problem (4)–(6) can

be represented in the product form á

where the Green’s functions Ø Ó = Ø ( Ú Ø á , ä Ø , , â ) satisfy the one-dimensional equations

with the initial conditions

= å ( Ú Ø − ä Ø ) at

and the homogeneous boundary conditions á

Here, ä Ø and â are free parameters ( ß Ø ≤ ä Ø ≤ Ð Ø and ≥ â ≥ 0), and å ( Ú ) is the Dirac delta function.

It can be seen that the Green’s function (7) admits incomplete separation of variables; it separates Ô

in the space variables Ú 1 , Ü✱Ü✱Ü , Ú æ but not in time .

0.9.2-2. Boundary value problems for a cylindrical domain with arbitrary cross-section.

1 Ò . Consider the parabolic equation Ó

= ❰ x , Õ [ Ï ]+ ç è , Õ [ Ï ]+ × (x, é , ),

(8) where ❰ x , Õ is an arbitrary second-order linear differential operator in Ô Ú 1 , Ü✱Ü✱Ü , Ú æ with coefficients

dependent on x and , and ç è , Õ Ô is an arbitrary second-order linear differential operator in é with coefficients dependent on é and .

For equation (8) we set the general initial condition (5), where Þ (x) must be replaced by Þ (x, é ). We assume that the space variables belong to a cylindrical domain ê ={ x ë ì , é 1 ≤ é ≤ é 2 } with arbitrary cross-section ì . We set the boundary conditions*

where the linear boundary operators î (

= 1, 2, 3) can define boundary conditions of the first, í î

second, or third kind; in the last case, the coefficients of the differential operators Ô can be dependent on .

* If ï 1 =− ð or ï 2 = ð , the corresponding boundary condition is to be omitted.

. The Green’s function of problem (8)–(9), (5) can be represented in the product form

where = (x, y, , â ) and

( é , ñ , , â ) are auxiliary Green’s functions; these can be determined from the following two simpler problems with fewer independent variables: á ò á ó

Problem on the cross-section ó

Problem on the interval é 1 ≤ é ≤ é 2 :

Here, y, ñ , and â are free parameters (y ë ì , é 1 ≤ ñ ≤ é 2 , ≥ â ≥ 0).

It can be seen that the Green’s function (10) admits incomplete separation of variables; it Ô

separates in the space variables x and é but not in time .

0.9.3. Construction of Green’s Functions via Fundamental Solutions

0.9.3-1. Elliptic equations. Fundamental solution. Consider the elliptic equation

(11) where x = { Ú

x [ Ï ]+ é 2 = × (x, é ),

, 1 , } , , and ❰ x [ Ï ] is a linear differential operator that depends on

1 , Ü✱Ü✱Ü , Ú æ but is independent of é . For subsequent analysis it is significant that the homogeneous equation (with × ≡ 0) does not change under the replacement of é by − é and é by é + const. Let ù ù = ù ù (x, y, é − ñ ) be a fundamental solution of equation (11), which means that

x [ ù ù ]+ é 2 = å (x − y) å ( é − ñ ).

Here, y = { æ ä 1 , 1 Ü✱Ü✱Ü , ä æ } ë ø and ñ ë ø are free parameters. The fundamental solution of equation (11) is an even function in the last argument, i.e.,

(x, y, é )= ù ù (x, y, − é ).

Below, Paragraphs 0.9.3-2 and 0.9.3-3 present relations that permit one to express the Green’s functions of some boundary value problems for equation (11) via its fundamental solution.

0.9.3-2. Domain: x ë ø æ ,0≤ é < ú . Boundary value problems for elliptic equations.

1 Ò . First boundary value problem. The boundary condition:

= Þ (x) at é = 0.

Green’s function:

(x, y, é , ñ )= ù ù (x, y, é − ñ )− ù ù (x, y, é + ñ ).

Domain of the free parameters: y ë ø æ and 0 ≤ ñ < ú .

2 Ò . Second boundary value problem. The boundary condition: Ó

= Þ (x) at é = 0.

Green’s function:

(x, y, é , ñ )= ù ù (x, y, é − ñ )+ ù ù (x, y, é + ñ ).

3 Ò . Third boundary value problem. The boundary condition: Ó

− à Ï = Þ (x) at é = 0.

Green’s function: á (x, y, î❫þ

0.9.3-3. Domain: x ë ø æ ,0≤ é ≤ ☎ . Boundary value problems for elliptic equations.

1 ✆ . First boundary value problem. Boundary conditions:

= Þ 2 (x) at é = ☎ . Green’s function:

(x, y, é − ñ +2 ✠ ☎ )− ù ù (x, y, é + ñ +2 ✠ ☎ ) ✡ . (12)

Domain of the free parameters: y ü ë ø æ and 0 ≤ ñ ≤ ☎ .

2 ✆ . Second boundary value problem. Boundary conditions:

= Þ 2 (x) at é = ☎ . Green’s function:

(x, y, é − ñ +2 ✠ ☎ )+ ù ù (x, y, é + ñ +2 ✠ ☎ ) ✡ . (13)

3 ✆ . Mixed boundary value problem. The unknown function and its derivative are prescribed at the ü

left and right end, respectively:

= Þ 2 (x) at é = ☎ . Green’s function:

, ñ )= ü (−1) ù ù (x, y, é − ñ +2 ✠ ☎ )− ù ù (x, y, é + ñ +2 ✠ ☎ ) ✡ . (14)

4 ü . Mixed boundary value problem. The derivative and the unknown function itself are prescribed at the left and right end, respectively:

= Þ 2 (x) at é = ☎ . Green’s function:

(x, y, é − ñ +2 ✠ ☎ )+ ù ù (x, y, é + ñ +2 ✠ ☎ ) ✡ . (15)

One should make sure that series (12)–(15) are convergent; in particular, for the ü three-dimensional Laplace equation, series (12), (14), and (15) are convergent and series (13) is

divergent.

TABLE 12 Representation of the Green’s functions of some nonstationary boundary value problems in terms of the fundamental solution of the Cauchy problem

Boundary value Boundary conditions

Green’s functions problems

First problem

(x, y, é , ñ , é ✕ ø , ë ✖ ø ë )= ù ù (x, y, é − ñ , ✕ , ✖ )− ù ù (x, y, é + ñ , ✕ , ✖ )

1 = 0 at é =0

, Second problem

(x, y, é , ñ , ✕ , ✖ )= ù ù (x, y, é − ñ , ✕ , ✖ )+ ù ù (x, y, é + ë ñ , ✕ ø é ë ø , ✖ )

= 0 at é =0

(x, y, , , , )= (x, y, − , , )+ (x, y, + , , ) Third problem

First problem ✟ = 0 at

Second problem ✟

Mixed problem ✟ = 0 at

(x, y, , , , )= ü (−1) ù ù

Mixed problem ✟

= 0 at é = 0,

(x, y, é , ñ , ✕ , ✖ )= ü (−1) ù ù (x, y, é − ñ +2 ✠ ☎ , ✕ , ✖ )

0.9.3-4. Boundary value problems for parabolic equations. Let x ë ø æ , é ë ø 1 , and ✕ ≥ 0. Consider the parabolic equation

= ❰ x , ✘ [ Ï ]+ ✕ ☛ é 2 + ✙ (x, é , ✕ ),

(16) where ❰ x , ✘ [ Ï ] is a linear differential operator that depends on ✚ 1 , ✛✜✛✜✛ , ✚ ✢ and ✕ but is independent

of ✣ . Let ù ù = ù ù (x, y, ✣ − ñ , ✕ , ✖ ) be a fundamental solution of the Cauchy problem for equation (16), i.e.,

Here, y 1 ✥ ø ✢ , ñ ✥ ø , and ✖ ≥ 0 are free parameters. The fundamental solution of the Cauchy problem possesses the property

(x, y, ✣ , ✕ , ✖ )= ù ù (x, y, − ✣ , ✕ , ✖ ).

Table 12 presents formulas that permit one to express the Green’s functions of some nonstationary boundary value problems for equation (16) via the fundamental solution of the Cauchy problem. ✦★✧