Nonhomogeneous Boundary Value Problems with

0.8. Nonhomogeneous Boundary Value Problems with

Many Space Variables. Representation of Solutions via the Green’s Function

0.8.1. Problems for Parabolic Equations

0.8.1-1. Statement of the problem. In general, a nonhomogeneous linear differential equation of the parabolic type in ✯ space variables

has the form

be some simply connected domain in ▲ ⑨ with a sufficiently smooth boundary ⑩ = ⑧ . We consider the nonstationary boundary value problem for equation (1) in the domain ⑧ with an arbitrary initial condition,

Let ◗

(3) and nonhomogeneous linear boundary conditions,

(x) at

(4) In the general case, ❶ x ,

x , ❯ [ ]= ♣ (x, ) for x ❑ ⑩ .

is a first-order linear differential operator in the space coordinates with ❙

coefficients dependent on x and .

0.8.1-2. Representation of the problem solution in terms of the Green’s function. The solution of the nonhomogeneous linear boundary value problem (1)–(4) can be represented as

where ❼ (x, y, ❽ , ❻ ) is the Green’s function; for ❽ > ❻ ≥ 0, it satisfies the homogeneous equation

(6) with the nonhomogeneous initial condition of special form ❸

− ➃ ❽ x , [ ❼ ]=0

(7) and the homogeneous boundary condition

= ❨ (x − y) at ❽ = ❻

(8) The vector y = { ❸

x , [ ❼ ]=0 for x ➄ ⑩ .

1 ⑥ , ➆✱➆✱➆ , ➅ ⑨ } appears in problem (6)–(8) as an ⑥ ➇ -dimensional free parameter (y ➄ ⑧ ), and ❨ (x − y) = ❨ ( 1 − ➅ 1 ) ➆✱➆✱➆❩❨ ( ⑨ − ➅ ⑨ ) is the ➇ -dimensional Dirac delta function. The Green’s

TABLE 8

The form of the function ➁ (x, y, ❽ , ❻ ) for the basic types of nonstationary boundary value problems Type of problem

Function ➁ (x, y, ❽ , ❻ ) 1st boundary value problem

Form of boundary condition (4)

(x, y, ❽ , ❻ )=− ♠ ➉ (x, y, ❽ ♠ ➊ ➋ , ❻ ) 2nd boundary value problem

= ♣ (x, ❽ ) for x ➄ ⑩

(x, y, ❽ , ❻ )= ❼ (x, y, ❽ , ❻ ) 3rd boundary value problem

function ❼ is independent of the functions ❺ , ➏ , and ♣ that characterize various nonhomogeneities of the boundary value problem. In (5), the integration is everywhere performed with respect to y,

with ❾ ⑧ ❿ = ❾ ➅ 1 ➆✱➆✱➆❃❾ ➅ ⑨ . The function ➁ (x, y, ❽ , ❻ ) involved in the integrand of the last term in solution (5) can be expressed via the Green’s function ❼ (x, y, ❽ , ❻ ). The corresponding formulas for ➁ (x, y, ❽ , ❻ ) are given in Table 8 for the three basic types of boundary value problems; in the third boundary value problem, the coefficient ➎ can depend on x and ❽ . The boundary conditions of the second and third kind, as well as the solution of the first boundary value problem, involve operators of differentiation along the conormal of operator (2); these operators act as follows:

where N = { ➣ 1 , ➆✱➆✱➆ , ➣

} is the unit outward normal to the surface ⑩ . In the special case where

(x, ❽ ) = 1 and ➓ ➔ (x, ❽ ) = 0 for ↕ ≠ ➙ , operator (9) coincides with the ordinary operator of differen- tiation along the outward normal to → → ➛ . If the coefficient of equation (6) and the boundary condition (8) are independent of ❽ , then the Green’s function depends on only three arguments, ❼ (x, y, ❽ , ❻ )= ❼ (x, y, ❽ − ❻ ).

( ↕ = 1, ➆✱➆✱➆ , ➡ ) be different portions of the surface ➛ such that ➛ =

and let

boundary conditions of various types be set on the ➛ ➓ ➓ ,

= 1, ➆✱➆✱➆ , ➡ . (10) Then formula (5) remains valid but the last term in (5) must be replaced by the sum ❸

0.8.2. Problems for Hyperbolic Equations

0.8.2-1. Statement of the problem. The general nonhomogeneous linear differential hyperbolic equation in ➇ space variables can be

written as

(12) where the operator ❸ ➃ x , [ ➈ ] is explicitly defined in (2). We consider the nonstationary boundary value problem for equation (12) in the domain ❸ ➨ with arbitrary initial conditions,

(14) and the nonhomogeneous linear boundary condition (4). ❸

= ➏ 1 (x) at ❽ = 0,

0.8.2-2. Representation of the problem solution in terms of the Green’s function. The solution of the nonhomogeneous linear boundary value problem (12)–(14), (4) can be represented

+ ❷ ❹ ➭ ➏ 1 (y) + ➏ 0 (y) ➧ (y, 0) ➯❫❼ (x, y, ❽ , 0) ❾ ➨ ❿ + ❷ ❸ ❷ ➀ ♣ (y, ❻ ) ➁ (x, y, ❽ , ❻ ) ❾ ➛ ❿ ❾ ❻ . (15)

Here, ❼ (x, y, ❽ , ❻ ) is the Green’s function; for ❽ > ❻ ≥ 0 it satisfies the homogeneous equation

(16) with the semihomogeneous initial conditions ❸

and the homogeneous boundary condition (8). ❸ If the coefficients of equation (16) and the boundary condition (8) are independent of time ❽ ,

then the Green’s function depends on only three arguments, ❧ ❼ (x, y, ❽ , ❻ )= ❼ (x, y, ❽ − ❻ ❧ ). In this case, one can set ♠ ❼ ♠ (x, y, ❽ , ❻ ) ♥ ♥ =0 =− ♠ ❼

(x, y, ❽ ) in solution (15).

The function (x, y, ❽ , ❻ ) involved in the integrand of the last term in solution (15) can be ❸

expressed via the Green’s function ❼ (x, y, ❽ , ❻ ). The corresponding formulas for ➁ are given in Table 8 for the three basic types of boundary value problems; in the third boundary value problem, the coefficient ➎ can depend on x and ❽ .

Let ➤

( ↕ = 1, ➆✱➆✱➆ , ➡ ) be different portions of the surface ➛ such that ➛ = ➓ ➢ ➛ ➓ and let

boundary conditions of various types (10) be set on the ➛ ➓ . Then formula (15) remains valid but the last term in (15) must be replaced by the sum (11).

0.8.3. Problems for Elliptic Equations

0.8.3-1. Statement of the problem. In general, a nonhomogeneous linear elliptic equation can be written as

(17) where

− ➃ x [ ➈ ]= ❺ (x),

x [ ➈ ]≡ ➲

(x) ➂ ↔ ➂ ↔

(x) ➓ + ➵ (x) ➈ . (18)

Two-dimensional problems correspond to ➇ = 2 and three-dimensional problems, to ➇ = 3. We consider equation (17)–(18) in a domain ➨ and assume that the equation is subject to the general linear boundary condition

(19) The solution of the stationary problem (17)–(19) can be obtained by passing in (5) to the limit as

x [ ➈ ]= ♣ (x) for x ➄ ➛ .

. To this end, one should start with equation (1) whose coefficients are independent of ❽ and take the homogeneous initial condition (3), with ➏ (x) = 0, and the stationary boundary condition (4).

TABLE 9 The form of the function ➁ (x, y) involved in the integrand of the last term in solution (20) for the basic types of stationary boundary value problems

Function ➁ (x, y) 1st boundary value problem

Type of problem

Form of boundary condition (19)

(x, y) = − ♠ ➉ ♠ ➊ ➋ (x, y) 2nd boundary value problem

= ♣ (x) for x ➄ ➛

(x, y) = (x, y) 3rd boundary value problem

0.8.3-2. Representation of the problem solution in terms of the Green’s function. The solution of the linear boundary value problem (17)–(19) can be represented as the sum

(20) Here, the Green’s function ❼ (x, y) satisfies the nonhomogeneous equation of special form

(x) = ❷ ❹ ❺ (y) ❼ (x, y) ❾ ➨ ❿ + ❷ ➀ ♣ (y) ➁ (x, y) ❾ ➛ ❿ .

(21) with the homogeneous boundary condition

− ➃ x [ ❼ ]= ➻ (x − y)

(22) The vector y = { ➅ 1 , ➆✱➆✱➆ , ➅ } appears in problem (21), (22) as an ➇ -dimensional free parameter (y ➄ ➨ ).

x [ ❼ ]=0 for x ➄ ➛ .

Note that ❼ is independent of the functions ➲ ❺ and ♣ characterizing various nonhomogeneities of the original boundary value problem.

The function ➁ (x, y) involved in the integrand of the second term in solution (20) can be expressed via the Green’s function ❼ (x, y). The corresponding formulas for ➁ are given in Table 9 for the three basic types of boundary value problems. The boundary conditions of the second and third kind, as well as the solution of the first boundary value problem, involve operators of differentiation along the conormal of operator (18); these operators are defined by (9); in this case, the coefficients ➓ ➼ ➽❫➜ ➝ ➞❃➟ ➠ ➔ depend on only x.

For the second boundary value problem with → ➵ (x) ≡ 0, the thus defined Green’s function must not necessarily exist; see Remark 2 in Paragraph 8.2.1-2.

0.8.4. Comparison of the Solution Structures for Boundary Value

Problems for Equations of Various Types

Table 10 lists brief formulations of boundary value problems for second-order equations of elliptic, ➥

parabolic, and hyperbolic types. The coefficients of the differential operators ↔ ↔ ➃ x and x in the space variables 1 , ➆✱➆✱➆ ,

are assumed to be independent of time ❽ ; these operators are the same for the problems under consideration. ➲

Below are the respective general formulas defining the solutions of these problems with zero

initial conditions ( ➏ = ➏ 0 = ➏ 1 = 0):

0 (x) =

(y) ❼ 0 (x, y) ❾ ➨ ❿

(y) ➾ ➭ ❼ 0 (x, y) ➯ ❾ ➛ ❿ ,

1 (x, ❽ )= ❷ ❸ ❷ ❹ ❺ (y, ❻ ) ❼ 1 (x, y, ❽ − ❻ ) ❾ ➨ ❿ ❾ ❻ + ❷ ❸ ❷ ➀ ♣ (y, ❻ ) ➾ ➭ ❼ 1 (x, y, ❽ − ❻ ) ➯ ❾ ➛ ❿ ❾ ❻ ,

2 (x, ❽ )= ❷ ❸ ❷ ❹ ❺ (y, ❻ ) ❼ 2 (x, y, ❽ − ❻ ) ❾ ➨ ❿ ❾ ❻ + ❷ ❸ ❷ ➀ ♣ (y, ❻ ) ➾ ➭ ❼ 2 (x, y, ❽ − ❻ ) ➯ ❾ ➛ ❿ ❾ ❻ ,

TABLE 10 Formulations of boundary value problems for equations of various types

Type of equation Form of equation

Boundary conditions Elliptic ➥ −

Initial conditions

where the ❼ are the Green’s functions, the subscripts 0, 1, and 2 refer to the elliptic, parabolic, and hyperbolic problem, respectively. All solutions involve the same operator ➲ ➾ [ ❼ ]; it is explicitly defined in Subsections 0.8.1–0.8.3 (see also Section 0.7) for different boundary conditions.

It is apparent that the solutions of the parabolic and hyperbolic problems with zero initial conditions have the same structure. The structure of the solution to the problem for a parabolic equation differs from that for an elliptic equation by the additional integration with respect to ❽ ➪❖➶ .

References for Section 0.8: P. M. Morse and H. Feshbach (1953), V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), E. Butkov (1968), A. G. Butkovskiy (1979, 1982), E. Zauderer (1989), A. N. Tikhonov and A. A. Samarskii (1990), A. D. Polyanin (2000a, 2000c, 2001a).