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.