Higher-Order Linear Equations with Constant Coefficients

9.5. Higher-Order Linear Equations with Constant Coefficients

Throughout Section 9.5 the following notation is used: ↔ ↔

1 ➭ , ➶✳➶✳➶ , ➪ }, y={ ↔ ❍ x={ ➭ 1 , ➶✳➶✳➶ , ❍ }, ➹ ={ ↔ ➘ 1 ➶✳➶✳➶ ➘ }, ➴ ={ 1 ➶✳➶✳➶ ↔ , ↔ , , , },

9.5.1. Fundamental Solutions. Cauchy Problem ↔

9.5.1-1. Domain: ➮ = {− ➱ < ➪ ✃ < ➱ ; ❐ = 1, ➶✳➶✳➶ , ➟ }.

Let Ð be a constant coefficient linear differential operator such that

1 , ➶✳➶✳➶ , Ö are nonnegative integers, ➺ 1 , ÒÓÒÓÒ , are some constants, and × is the order of the operator. A generalized function (distribution) Ø Ø (x) = Ø Ø ( ➪ 1 , ➶✳➶✳➶ , ➪ ) that satisfies the equation

(x) = (x),

where ❮

(x) = Û ( ➪ 1 ) ➶✳➶✳➶ÜÛ ( ➪ ) is the Dirac delta function in the ➟ Ù -dimensional Euclidian space, is called the fundamental solution corresponding to the operator . Any constant coefficient linear differential operator has a fundamental solution Ø Ø Ô à (x). The fundamental solution is not unique — it is defined up to an additive term ➩ Ù Ý Þ Þ 0 (x) that is an arbitrary solution of the homogeneous equation

0 (x) = 0.

The solution of the nonhomogeneous equation

= á (x)

with an arbitrary right-hand side has the form

(x) â á (x) = ã ä Ø Ø (x − y) á (y) å y. Here, å y= å ❍ 1 ➶✳➶✳➶✱å ❍ and the convolution Ø Ø â ➽✦➾ á is assumed to be meaningful.

(x) = ↔ Ø Ø (x) â á (x),

References : G. E. Shilov (1965), S. G. Krein (1972), L. H ¨ormander (1983), V. S. Vladimirov (1988).

9.5.1-2. Domain: 0 ≤ æ < ➱ Ô à ,− ➱ < ➪ ✃ < ➱ ; ❐ = 1, ➶✳➶✳➶ , ç . Cauchy problem.

be a constant coefficient linear differential operator of order 1 é with respect to æ . Then a distribution Ø Ø ( æ , x) = Ø Ø ( æ , ➪ 1 , ➶✳➶✳➶ , ➪ ), which is a solution of the equation

( æ æ ➪ , ➶✳➶✳➶ , ❮ Ø Ø ➪ , x) = 0

and satisfies the initial conditions*

is called a fundamental solution of the Cauchy problem corresponding to the operators Ù . The solution of the Cauchy problem for the linear differential equation

with the special initial conditions ❮

is given by

( æ , x) = Ø Ø ( æ , x) â í (x),

Reference : S. G. Krein (1972).

9.5.1-3. Solution of the Cauchy problem for general initial conditions. If the general initial conditions ì

are set, the solution of equation (2) is sought in the form

Each term in (4) satisfies equation (2), and the functions ë

−1 , î −2 , ➶✳➶✳➶ , î 0 are determined successively from the linear system

0 (x) = î −1 (x),

This system of equations is obtained by successively differentiating relation (4) followed by substi- ë tuting æ ➽✦➾ = 0 and taking into account the initial conditions (1) and (3).

Reference : G. E. Shilov (1965). * The number of initial conditions can be less than ï

(see Paragraph 9.5.4-1).

9.5.2. Elliptic Equations

9.5.2-1. Homogeneous elliptic differential operator.

A constant coefficient linear homogeneous differential operator of order Ð ÐÕÔ ❐ has the form ↔

where Ö 1 , ➶✳➶✳➶ , Ö are nonnegative integers. From now on, we adopt the notation Ð ÐÕÔ Ð ÐÕÔ

A linear homogeneous differential operator of order ÷ possesses the property

≠ 0 is an arbitrary constant.

A linear homogeneous differential operator ü is called elliptic if, on replacing in the symbols

1 by variables

1 , ➶✳➶✳➶ , þ , one obtains a polynomial Ð ÐÕÔ ( þ 1 , ➶✳➶✳➶ , þ ) that does not vanish

if Ð ➹ ≠ 0, i.e., ÐÕÔ

A linear differential equation

is called elliptic if the linear homogeneous differential operator ú is elliptic.

9.5.2-2. Elliptic differential operator of general form. In general, a constant coefficient linear differential operator of order ÷ has the form

where ✁ is the leading part of the operator and

( ✂ = 0, 1 ➶✳➶✳➶ , ÷ ) is a linear homogeneous ù ú

differential operator of order ✂ . The operator is said to be elliptic if its leading part is elliptic.

A linear differential equation

is called elliptic if the linear differential operator ú

is elliptic.

A linear elliptic operator and a linear elliptic differential equation can only be of even order ÷ =2 é , where é is a positive integer.

9.5.2-3. Fundamental solution of a homogeneous elliptic equation. The fundamental solution of the homogeneous elliptic equation (1) with ↔ ÷ =2 é is given by

if is even and 2 ≥ ;

is even and 2 < ç . (2 )

| ⋅ x|

if ç

Here, the integration is performed over the surface of the ↔

-dimensional sphere ù ù ✑ of unit radius defined by the equation | ➹ | = 1; ➹ ⋅ x= þ 1 ➪ 1 + ✔✕✔✕✔ + þ ➪ and 2 ✏ ( ➹ )= 2 ✏ ( þ 1 , ➶✳➶✳➶ , þ ).

A fundamental solution is an ordinary function, analytic at any point x ≠ 0; this function is described, in a neighborhood of the origin of coordinates (as |x| û ↔ ↔ ✖ 0), by the relations

, ✏ | x|

if is odd or

is even and >2 ;

(x) = ✒

, ✏ | x|

ln |x| if ç is even and ç ≤2 ✒ .

Here, ✘ ,

and , ✏ are some nonzero constants. If 2 ✒ > ç , the fundamental solution has continuous derivatives up to order 2 ✒ − ç − 1 inclusive at the origin.

9.5.2-4. Fundamental solution of a general elliptic equation. The fundamental solution of the general elliptic equation (2) with ÷ =2 ✒ is determined from the

Here, the function

( , ) is a fundamental solution of the constant coefficient linear ordinary differential equation

If ç is odd, the fundamental solution (3) can be represented as ↔

References : I. M. Gel’fand, G. E. Shilov (1959), S. G. Krein (1972).

9.5.3. Hyperbolic Equations ✭

Let ü

be a constant coefficient linear homogeneous differential operator of ù

such that

order ✒ with respect to æ . The operator is called hyperbolic if for any numbers þ 1 , ➶✳➶✳➶ , þ ✰

= 1, the ✒ th-order algebraic equation

with respect to ✜ has ✒ different real roots. Fundamental solution of the Cauchy problem for ✒ ≥ ç − 1:

− ✰ −1 [sign( ✷ ⋅ x+ æ ✏ æ −1 ✷ æ )] å ✢

( , x) =

if ç is odd; 2(2 ) ( − ç − 1)! ✶

is even,

and å ✢ ✶ is the element of the surface ✸ = 0. Fundamental solution of the Cauchy problem for ✒ < ç − 1:

if ç is odd; (2 ✌ ) −1 ✿ ❀

if is even.

References : I. M. Gel’fand, G. E. Shilov (1959), S. G. Krein (1972).

9.5.4. Regular Equations. Number of Initial Conditions in the Cauchy Problem

9.5.4-1. Equations with two independent variables (0 ≤ ❈ < ❏ ,− ❏ < ❑ < ❏ ).

1 ▲ . Consider the constant coefficient linear differential equation

) is a polynomial of degree ÷ , 2 = −1. Let ❯ = ❯ ( ❆ ) be the number of roots (taking into account their multiplicities) of the characteristic equation P

whose real parts are nonpositive (or bounded above) for given a ❆ . If ❯ is the same (up to a set of measure zero) for all ❆ ❲ (− ❏ , ❏ ), the equation (1) will be called regular with regularity index ❯ .

Classical equations such as the heat, wave, and Laplace equations are regular.

2 ▲ . In the Cauchy problem for the regular equation (1), one should set ❯ initial conditions of the form

It should be emphasized that the regularity index ❯ can, in general, differ from the equation order ▼ ❴

with respect to ❈ . In particular, for the two-dimensional Laplace equation ❨❫❨ ◆

=− ❴ ◆ , we have

= 1 and ❊ = 2; here, ❍ is replaced by ❈ and the first boundary value problem in the upper half-plane ▼

≥ 0 is considered. For the heat equation ❨ ◆

= ▼ ◆ and the wave equation

= ❴ ◆ , we have

= ❊ = 1 and ❯ = ❊ = 2, respectively.

Example 1. Below are the regularity indices for some fourth-order equations:

. The special solution = ( ❈ , ❑ ) that satisfies the initial conditions

is called fundamental.

The fundamental solution can be found by applying the Fourier transform in the space variable ❇ ❇

to equation (1) (with ◆ = ) and the initial conditions (4).

Example 2. Consider the polyharmonic equation:

Taking into account the representation 2

, we rewrite the characteristic equation (2) in the form

It has only one solution whose real part is nonpositive, specifically, ✐ ♣ = −| q |. Considering the multiplicity of the root, we find that the regularity index ❡ ❇ is equal to ❇ r .

In equation (5) with ❛ = and the initial conditions (4) with ❡ = r , we perform the Fourier transform with respect to the space variable,

)= t ✉

As a result, we arrive at the ordinary differential equation ✉ ✈✡✇②①

(6) and the initial conditions s ④ ④⑥⑤

The bounded solution of problem (6), (7) is given by ✐

By applying the inverse Fourier transform, we obtain the fundamental solution of the polyharmonic equation in the form ✈ ①

( , q ) ③❶q =

−| | ③❶q

2 ⑩ ( r ✐ − 1)! −

2 ③❶q

③❶q

( r ✐ − 1)! 2 + 2

4 ▲ . For general initial conditions of the form (3), the solution of equation (1) is determined on the basis of the fundamental solution from the relation

−1 ( ❑ ). (8) Each term in (8) satisfies equation (1), and the functions ❸ ❪ −1 , ❸ ❪ −2 , ➶✳➶✳➶ , ❸ 0 are calculated succes-

sively by solving the linear system

−1 ( ❑ ), ÷ = 2, ➶✳➶✳➶ , ❯ − 1. This system of equations is obtained by successively differentiating relation (8) followed by substi-

tuting ❈ = 0 and taking into account the initial conditions (3) and (4). In the special case ❬ 0 ( ❑ )= ❬ 1 ( ❑ )= ❹✕❹✕❹ = ❬ ❪ −2 ( ❑ ) = 0, one should set ❸ 0 ( ❑ )= ❬ ❪ −1 ( ❑ ) and

1 ( ❑ )= ❹✕❹✕❹ = ❸ ❪ −1 ( ❍✳■ ❑ ) = 0 in (8).

Reference : G. E. Shilov (1965).

9.5.4-2. Equations with many independent variables (0 ≤ ❈ < ❏ ,x ❲ ❺ ❂ ). Solving the Cauchy problem for the constant coefficient linear differential equation

(9) with arbitrarily many space variables ❑ 1 , ➶✳➶✳➶ , ❑ ❂ can be reduced to solving the Cauchy problem for

an equation with one space variable ❻ . We take an auxiliary linear differential operator

that depends on two independent variables ❈ and ❻ so that the Cauchy problem for the equation

(10) is well posed. Then the fundamental solution of the Cauchy problem for the original equation (9) is

given by

, ➋ ) is the fundamental solution of the Cauchy problem for the auxiliary equation (10). If the number of space variables is odd, one can use the simpler formula ➇

The above relations hold for all equations for which the Cauchy problem is well posed. ➧✳➨

References : I. M. Gel’fand, G. E. Shilov (1959), S. G. Krein (1972).

9.5.4-3. Stationary homogeneous regular equations (x ❲ ➩ ➐ ).

A linear differential operator ➫ ➭ ↔ ➯ ➯ ➲ 1 , ➳✕➳✕➳ , ➯ ➯ ➲ ↕ is called regular if it is homogeneous and if the gradient of the function ➫ ➭ ( ➵ 1 , ➳✕➳✕➳ , ➵ ➐ ) on the set defined by the equation ➛ ➫ ➭ ( ➵ 1 , ➳✕➳✕➳ , ➵ ➐ ) = 0 is everywhere nonzero whenever | ➸ | ≠ 0. The fundamental solution of the linear regular differential equation ➫ ➭ ↔ ➯

1 , , ➯ ➲ ↕❶➺ =0 generated by the linear regular differential operator ➫ ➭ is expressed as

where the function ➻

( ➼ ) is defined by

if ➔ is even and ➽ ≥ ➔ ;

if ➔ is even and ➽ < ➔

The integral in (11) is understood in the sense of its regularized value, i.e., ➻

is the set of points on a sphere of unit radius for which | ➫ ➭ ( ➸ )| > ➹ .

References : I. M. Gel’fand, G. E. Shilov (1959), S. G. Krein (1972).

9.5.5. Some Special-Type Equations

The condition Re ❒

( ➼ )≤ Ð < Ñ is assumed to be met for all real ➼ .

1 Ò . Domain: − Ñ < Ó < Ñ . Cauchy problem. An initial condition is prescribed:

. The solution of the Cauchy problem with the initial condition (1) for the nonhomogeneous equation

is given by ❒

where the function ➇

( Ó , ➊ ) is defined in Item 1 Ò .

References : S. G. Krein (1972), V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, et al. (1974). ➇ ➇

1 ✡ . General solution (two representations): ☛

where the = ( ✔ ) are arbitrary functions.

2 ✡ . Fundamental solution:

1 ✡ . General solution (two representations): ☛

where the =

( ☞ , ✌ ) are arbitrary functions that satisfy the heat equations ✙ ✚

2 ✡ . Fundamental solution: ✕ ✕

3 ✡ . Domain: − ✤ < ☞ < ✤ . Cauchy problem. ☛ ✥

Initial conditions are prescribed: ✥

Reference : G. E. Shilov (1965).

1 ✡ . General solution (two representations): ☛ ✏ ✯✰✒

where the = ( ✳ ) and ✱ = ✱ ( ✔ ) are arbitrary functions.

2 ✡ . Fundamental solution:

Reference : G. E. Shilov (1965).

This is the polyharmonic equation of order ✖ with two independent variables.

1 ✡ . General solution (two representations): ☛ ✏ ✘

where the ( ☞ , ✳ ) are arbitrary harmonic functions ( ✺ = 0). In the second relation, ☞ can be replaced by ✳ .

2 ✡ . Domain: − ✤ < ☞ < ✤ ,− ✤ < ✳ < ✤ . Fundamental solution:

. Domain: − ✤ < ☞ < ✤ ,0≤ ☛ ✥ ✳ < ✤ . Boundary value problem. ☛ ✥

Boundary conditions are prescribed: ✥

− 1)! ✍ ☞ 2 + ✳ 2 . See also Example 2 in Paragraph 9.5.4-1. ✩

References : G. E. Shilov (1965), L. D. Faddeev (1998).

This is a nonhomogeneous polyharmonic equation of order ✖ with two independent variables.

Particular solution: ( ☞

The general solution is given by the sum of any particular solution of the nonhomogeneous equation ❀ and the general solution of the homogeneous equation (see equation 9.5.5.5, Item 1 ✡ ).

This is the polyharmonic equation of order ❆ with ✖ independent variables. For ❆ = 1, see Sections 7.1 and 8.1. For ❆ = 2, see Subsection 9.4.1. For ✖ = 2, see equations 9.5.5.5 and 9.5.5.6.

1 ✡ . Particular solutions:

where the (x) are arbitrary harmonic functions ( ✺

2 ✡ . Fundamental solution for ❆ ≥ 1 and ✖

if is odd or ✖ is even and ✖ >2 ❆ ; (x) = ■ ✍ ❇

| x|

2 ❇ − , ✍ | x| ln |x| if ✖ is even and ✖ ≤2 ❆ .

2. The expression of the coefficient ■ , can be obtained formally from the expression of , by removing the multiplier (2 ❆ ✬✮✭ ❍ 0 − ✖ ) equal to zero from the denominator. ✍ ❇

Reference ✍ ❇ : G. E. Shilov (1965).

Particular solutions: ✘

where the ✍ are solutions of the Helmholtz equations

= 0 and the ▲ are roots of the characteristic equation ▼ ✍

Reference : A. V. Bitsadze and D. F. Kalinichenko (1985).

Here, ❖ is any constant coefficient linear differential operator with arbitrarily many independent variables ☞ 1 , ✦✧✦✧✦ , ☞ .

Particular solutions: ☛

where the ✏ are solution of the equations ❈ ❖ [ ]− ▲ = 0 the ▲ are roots of the characteristic

equation ❇ ✓ ▲ = 0, and the

are arbitrary constants.