Nonhomogeneous Boundary Value Problems with One Space Variable. Representation of Solutions via the Green’s Function

0.7. Nonhomogeneous Boundary Value Problems with One Space Variable. Representation of Solutions via the Green’s Function

0.7.1. Problems for Parabolic Equations

0.7.1-1. Statement of the problem ( ❙ ≥ 0,

In general, a nonhomogeneous linear differential equation of the parabolic type with variable coef- ficients in one dimension can be written as

(1) where

( ✜ ✜ ✜ , ) > 0. (2) Consider the nonstationary boundary value problem for equation (1) with an initial condition of

general form,

(3) and arbitrary nonhomogeneous linear boundary conditions,

1 = ❣ 1 ( ), ❣ 2 = ❣ 2 ( ) in (4) and (5), we obtain the first, second, third, and mixed boundary value problems for equation (1).

By appropriately choosing the coefficients ❙ 1 , 2 and the functions

0.7.1-2. Representation of the problem solution in terms of the Green’s function. The solution of the nonhomogeneous linear boundary value problem (1)–(5) can be represented as

( ✜ , P , , ❲ ) is the Green’s function that satisfies, for > ❲ ≥ 0, the homogeneous equation

− ❚ ✦ , ❯ [ q ]=0

TABLE 7 ❙

Expressions of the functions ❙

1 ( ✜ , , ❲ ) and r 2 ( ✜ , , ❲ )

involved in the integrands of the last two terms in solution (6) Type of problem ❙ Form of boundary conditions Functions

rs

1 ( ) at ✜ = ✜ 1 r 1 ( ✜ , , ❲ )= q ( ✜ , First boundary value problem P , , ❲ ) ♥

( 1 = 2 = 0, ❣ 1 = ❣ 2 = 1) = ♣ 2 ( ) at ✜ = ✜ 2 r 2 ( ✜ , , ❲ )=− q ( ✜ , P , , ❲ ) ♥ ♥ = ✦ 2

Second boundary value problem ❙

Third boundary value problem ❙

Mixed boundary value problem t =

Mixed boundary value problem ❙

with the nonhomogeneous initial condition of special form

(8) and the homogeneous boundary conditions

(10) The quantities P and ❲ appear in problem (7)–(10) as free parameters ( ✜ 1 ≤ P ≤ ✜ 2 ), and ❨ ( ✜ ) is the

Dirac delta function. The initial condition (8) implies the limit relation

) = lim ✧

for any continuous function ✖

The functions ❙

1 ( ✜ , , ❲ ) and r 2 ( ✜ , , ❲ ) involved in the integrands of the last two terms in ❙

solution (6) can be expressed in terms of the Green’s function ❙ q ( ✜ , P , , ❲ ). The corresponding formulas for r s ( ✜ , , ❲ ) are given in Table 7 for the basic types of boundary value problems.

It is significant that the Green’s function ✖ q and the functions r 1 , r 2 are independent of the functions ❱ , , ♣ 1 , and ♣ 2 that characterize various nonhomogeneities of the boundary value problem. If the coefficients of equation (1)–(2) and the coefficients ❙ ❣ 1 , ❣ 2 in the boundary conditions (4) and (5) are independent of time , i.e., the conditions

= ✫ ( ✜ ), ✬ = ✬ ( ✜ ), ❢ = ❢ ( ✜ ), ❣ 1 = const, ❣ 2 = const (11) hold, then the Green’s function depends on only three arguments,

( ✜ , P , , ❲ )= q ( ✜ , P , − ❲ ).

In this case, the functions ❙

depend on only two arguments, r s = r s ( ✜ , − ❲ ), ✇ = 1, 2. Formula (6) also remains valid for the problem with boundary conditions of the third kind if ❙ ❙

1 = ❣ 1 ( ) and ❣ 2 = ❣ 2 ( ). Here, the relation between r s ( ✇ = 1, 2) and the Green’s function q is the same as that in the case of constants ❣ 1 and ❣ 2 ; the Green’s function itself is now different. ❘ ① ①

The condition that the solution must vanish at infinity,

0 as ✜

, is often set for the first, second, and third boundary value problems that are considered on the interval ✜ 1 ≤ ✜ < ✩ . In this case, the solution is calculated by formula (6) with r 2 = 0 and r 1 specified in Table 7 .

0.7.2. Problems for Hyperbolic Equations

0.7.1-2. Statement of the problem ( ❙ ≥ 0,

In general, a one-dimensional nonhomogeneous linear differential equation of hyperbolic type with variable coefficients is written as ◗ ❘

(12) where the operator ❘

, ❯ [ ] is defined by (2). Consider the nonstationary boundary value problem for equation (12) with the initial conditions ❘ ❙

and arbitrary nonhomogeneous linear boundary conditions (4)–(5). 0.7.2-2. Representation of the problem solution in terms of the Green’s function.

The solution of problem (12), (13), (4), (5) can be represented as the sum

+ ✧ ♣ 1 ( ❲ ✧ ) ✫ ( ✜ 1 , ❲ ) r 1 ( ✜ , , ❲ ) ❲ + ♣ 2 ( ❲ ) ✫ ( ✜ 2 , ❲ ) r 2 ( ✜ , , ❲ ) ❲ . (14)

Here, the Green’s function q ( ✜ , P , , ❲ ◗ ) is determined by solving the homogeneous equation

(15) with the semihomogeneous initial conditions

(17) and the homogeneous boundary conditions (9) and (10). The quantities P and ❲ appear in problem

= ❨ ( ✜ − P ) at

(15)–(17), (9), (10) as free parameters ( ❙ ✜ 1 ❙ ≤ P ≤ ✜ 2 ), and ❨ ( ✜ ) is the Dirac delta function. The functions r 1 ( ✜ , , ❲ ) and r 2 ( ✜ , , ❲ ) involved in the integrands of the last two terms in ❙

solution (14) can be expressed via the Green’s function ❙ q ( ✜ , P , , ❲ ). The corresponding formulas for r s ( ✜ , , ❲ ) are given in Table 7 for the basic types of boundary value problems.

It is significant that the Green’s function q and r 1 , r 2 are independent of the functions ❱ , ② 0 ,

1 , ♣ 1 , and ♣ 2 that characterize various nonhomogeneities of the boundary value problem. If the coefficients of equation (12) and the coefficients ❙ ❣ 1 , ❣ 2 in the boundary conditions (4) and (5) are independent of time ❙ ❙ , then the Green’s function depends on only three arguments, ❧ ❙ ❧ ❙

( ✜ , P , , ❲ )= q ( ✜ , P , − ❲ ). In this case, one can set ♠ q ( ✜ , P , , ❲ ) ♥ =0 =− ♠ ❯ q ♠ ♥ ♠ ( ✜ , P , ) in solu- tion (14). ▼❖◆

References for Section 0.7: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), E. Butkov (1968), A. G. Butkovskiy (1982), E. Zauderer (1989), A. D. Polyanin (2000a, 2000b, 2000c, 2001a).