Representation of the Solution of the Cauchy Problem via the Fundamental Solution

0.6. Representation of the Solution of the Cauchy Problem via the Fundamental Solution

0.6.1. Cauchy Problem for Parabolic Equations

0.6.1-1. General formula for the solution of the Cauchy problem. Let x = { ✜ 1 , ✰✱✰✱✰ , ✜

} and y = { 1 ,

, ✮ }, where x

and y

Consider a nonhomogeneous linear equation of the parabolic type with an arbitrary right-hand side,

(1) where the second-order linear differential operator ❚ x , ❯ is defined by relation (2) from Subsection

− ❚ x , ❯ [ ]= ❱ (x, ),

0.2.1. The solution of the Cauchy problem for equation (1) with an arbitrary initial condition,

= ❙ (x) at = 0,

can be represented as the sum of two integrals,

(x, ❙ )=

(y, ❲ ) ❳ ❳ (x, y, , ❲ ✧ ✧

) ✧ y ❲ + (y) ❳ ❳ (x, y, , 0) y, y= P 1 ✰✱✰✱✰ ✧ P .

Here, ❙

= ❳ ❳ (x, y, , ❲ ) is the fundamental solution of the Cauchy problem that satisfies, for > ❲ ≥ 0, the homogeneous linear equation

(2) with the nonhomogeneous initial condition of special form

− ❚ x , ❯ [ ❳ ❳ ]=0

= ❨ (x − y) at

(3) The quantities ❲ and y appear in problem (2)–(3) as free parameters, and ❨ (x) = ❨ ( ✜ 1 ) ✰✱✰✱✰❩❨ ( ✜ ✮ ) is

the ✯ ❬ -dimensional Dirac delta function. ❭❫❪ ❴ ❵❃❛ ❜ ❝

x , ❯ in (2) are independent of time ❙ , then the fundamental solution of the Cauchy problem depends on only three arguments, ❙ ❳ ❳ (x, y, , ❲ )=

If the coefficients of the differential operator ❙

(x, y, ❬ ❭❫❪ − ❴ ❵❃❛ ❲ ). ❞ ❝

If the differential operator ❚ x , ❯ has constant coefficients, then the fundamental ❙ ❙

solution of the Cauchy problem depends on only two arguments, ❳ ❳ (x, y, , ❲ )= ❳ ❳ (x − y, − ❲ ).

0.6.1-2. The fundamental solution allowing incomplete separation of variables. Consider the special case where the differential operator ❚ x , ❯ in equation (1) can be represented as

the sum

(4) where each term depends on a single space coordinate and time,

x , ❯ [ ]= ❚ 1 , ❯ [ ]+ ❏✱❏✱❏ + ❚ ✮ , ❯ [ ],

= 1, , . Equations of this form are often encountered in applications. The fundamental solution of the

Cauchy problem for the ✯ -dimensional equation (1) with operator (4) can be represented in the product form

where ❳ ❳ ❡ = ❳ ❳ ❡ ( ✜ ❡ , P ❡ , , ❲ ) are the fundamental solutions satisfying the one-dimensional equations

with the initial conditions

= ❨ ( ✜ ❡ − P ❡ ) at

In this case, the fundamental solution of the Cauchy problem (5) admits incomplete separation of ❙

variables; the fundamental solution is separated in the space variables ✜ 1 , ✰✱✰✱✰ , ✜ ✮ but not in time .

0.6.2. Cauchy Problem for Hyperbolic Equations

Consider a nonhomogeneous linear equation of the hyperbolic type with an arbitrary right-hand side,

(6) where the second-order linear differential operator ❚ x , ❯ is defined by relation (2) from Subsection

2 + ✐ (x, ) ◗ ❙ − ❚ x , ❯ [ ]= ❱ (x, ),

0.2.1, with x ❑ ▲ ✮ . The solution of the Cauchy problem for equation (6) with general initial conditions,

can be represented as the sum

(x, y, ✧ , 0) y, y= P 1 ✰✱✰✱✰ ✧ P .

Here, ❙

= ❳ ❳ (x, y, , ❲ ) is the fundamental solution of the Cauchy problem that satisfies, for > ❲ ≥ 0, the homogeneous linear equation

(7) with the semihomogeneous initial conditions of special form

(x, ) ◗ ❳ ❳ 2 ❙ − ❚ x , ❯ [ ❳ ❳ ]=0

at

= ❨ (x − y) at

The quantities ❬ ❭❫❪ ❴ ❵❃❛ ❜ ❝ ❲ and y appear in problem (7)–(8) as free parameters (y ❑ ▲ ✮ ). If the coefficients of the differential operator ❙

x , ❯ in (7) are independent of time ❙ , then the fundamental solution of the Cauchy problem depends on only three arguments, ❙ ❧ ❙ ❙ ❳ ❳ (x, y, , ❲ ❧ )=

(x, y, ❬ ❭❫❪ − ❴ ❵❃❛ ❲ ). Here, ❞ ❝ ♠ ❳ ❳ (x, y, , ❲ ♥ ♠ ) ♥ =0 =− ♠ ❯ ❳ ❳ ♠ (x, y, ). If the differential operator ❚ x , ❯ has constant coefficients, then the fundamental ❙ ❙

solution of the Cauchy problem depends on only two arguments, ❳ ❳ (x, y, , ❲ )= ❳ ❳ (x − y, − ❲ ▼❖◆ ).

References for Section 0.6: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), G. E. Shilov (1965), A. D. Polyanin (2000a, 2000b, 2000c, 2001a).