Higher-Order Linear Equations with Variable Coefficients

9.6. Higher-Order Linear Equations with Variable Coefficients

9.6.1. Equations Containing the First Time Derivative

9.6.1-1. Statement of the problem for an equation with two independent variables. ☛

Consider the linear nonhomogeneous partial differential equation ☛

(1) where ☛

, ✚ is a general linear differential operator of order ☛ ❀ ✖ ✏ with respect to the space variable ☞ ✏ ,

whose coefficients ✓ = ✓ ( ☞ , ✌ ) are sufficiently smooth functions of both arguments for ✌ ≥ 0 and

1 ≤ ☞ ≤ ☞ 2 . The subscripts ☞ and ✌ indicate that the operator ❖ ✛ , ✚ is dependent on the variables ☞

and ✌

We set the initial condition

(3) and the general nonhomogeneous boundary conditions ☛ ✏ ☛

where ◗ ≥ 1 and ✖ ≥ ◗ + 1. We assume that both sets of the boundary forms (1) [ ]( ❆ = 1, ✦✧✦✧✦ , ◗ ) and

= ◗ + 1, ✦✧✦✧✦ , ✖ ) are linearly independent, which means that for any nonzero ❈ ❇ ❏ ❘ = ❘ ( ✌ ) the following relations hold: ❇

In what follows, we deal with the nonstationary boundary value problem (1), (3), (4). ❇

9.6.1-2. The case of general homogeneous boundary conditions. The Green’s function. The solution of equation (1) with the initial condition (3) and the homogeneous boundary conditions ☛

can be written as ❇ ✒

( ☞ , ✳ , ✌ , ❚ ) is the Green’s function that satisfies, for ✌ > ❚ ≥ 0, the homogeneous equation

(8) and the homogeneous boundary conditions

[ ✾ ]=0 at ☞ = ☞ 2 ( ❆ = ◗ + 1, ✦✧✦✧✦ , ✖ ). The quantities ✳

and ❇ ❚ appear in problem (7)–(9) as free parameters ( ☞ 1 ≤ ✳ ≤ ☞ 2 ), and

( ☞ ) is the ✒

Dirac delta function. It should be emphasized that the Green’s function ✏ ✏ ✏ ✾ is independent of the functions ( ☞ , ✌ ), ( ☞ ),

(1) ( ✌ ), and ✱ (2) ( ✌ ) that characterize various nonhomogeneities of the boundary value problem. If the coefficients (1)

, ❍ , and determining the differential operator (2) and boundary conditions (4) are independent of time ❇ ✌ ❍ , then the Green’s function depends only on three arguments, ❇ ✾ ( ☞ , ✳ , ✌ , ❚ )=

Reference : Mathematical Encyclopedia (1977, Vol. 1).

9.6.1-3. The case of nonhomogeneous boundary conditions. Preliminary transformations. To solve the problem with nonhomogeneous boundary conditions (1), (3), (4), we choose a suffi-

ciently smooth “test function” ❯ = ❯ ( ☞ , ✌ ) that satisfies the same boundary conditions as the unknown function; thus,

( ✌ ) at ☞ = ☞ 2 ( ❆ = ◗ + 1, ✦✧✦✧✦ , ✖ ). Otherwise the choice of the “test function” ❇ ❇ ❯ is arbitrary and is not linked to the solution of the

equation in question; there are infinitely many such functions.

Let us pass from ❱ = ❱ ( ❲ , ❳ ) to the new unknown ❨ = ❨ ( ❲ , ❳ ) by the relation

(11) Substituting (11) into (1), (3), and (4), we arrive at the problem for an equation with a modified right-hand side,

( ❲ , ❳ )= ❴ ( ❲ , ❳ )− ❬ ❳ + ❭ ❪ , ❫ [ ❩ ], (12) subject to the nonhomogeneous initial condition

(13) and the homogeneous boundary conditions ❛

The solution of problem (12)–(14) can be found using the Green’s function by formula (6) in which one should replace

by ❨ , ❴ ( ❲ , ❳ ) by ❴ ( ❲ , ❳ ), and ❵ ( ❲ ) by ❵ ( ❲ )= ❵ ( ❲ )− ❩ ( ❲ , 0). Taking into account relation (11), for ❱ we obtain the representation

Changing the order of integration and integrating by parts with respect to ♥ , we find, with reference to the initial condition (8) for the Green’s function,

= ❩ ( ❥ , ❳ ) q ( ❲ − ❥ )− ❩ ( ❥ , 0) ❦ ( ❲ , ❥ , ❳ , 0) − ✐

We transform the inner integral of the last term in (15) using the Lagrange–Green formula [see Kamke (1977)] to obtain

where ⑩ ❶ r , ❷ [ ❸ ] is the differential form adjoint with ⑩ ❶ , ❷ [ ❸ ] of (2); ⑥ = ⑥ ( ❥ , ♥ ); and ❹ and ❺ are nonnegative integers.

Using relations (16) and (17), we rewrite solution (15) in the form

( ❼ , ❽ )= ❾ ❿ ( ➁ ) ➂ ( ❼ , ➁ , ❽ , 0) ➃ ➁ + ❾ ➄ ❾ ❿ ( ➁ , ➆ ) ➂ ( ❼ , ➁ , ❽ , ➆ ) ➃ ➁ ➃ ➆ + ❾ ➄ s s [ ➇ , ➂ ➈ = 2 ] = ➃ ➆ . (18)

This formula was derived taking into account the fact that the Green’s function with respect to ❿

and ➆ satisfies the adjoint equation*

For subsequent analysis, it is convenient to represent the bilinear differential form ➉

Note that in the special case where operator (2) is binomial, ➊

= const,

the differential forms in (19) are written as ➎

9.6.1-4. The case of special nonhomogeneous boundary conditions. Consider the following nonhomogeneous boundary conditions of special form that are often encoun- ➐➙↕

tered in applications: ➊

( ❽ ) at ❼ = ❼ 2 ( ❆ = ➜ + 1, ✦✧✦✧✦ , ➝ ). Without loss of generality, we assume that the following inequalities hold: ❈ ❈ ❈

The Green’s function satisfies the corresponding homogeneous boundary conditions that can be ➎ obtained from (20) by replacing ❻ by ➂ and setting ➛ (1) ( ❽ )= ➛ (2) ( ❽ ) = 0.

* This equation can be derived by considering the case of homogeneous initial and boundary conditions and using arbitrariness in the choice of the test function ⑥ = ⑥ ( ➠ , ➡ ); it should be taken into account that the solution itself must be independent of the specific form of ⑥ , because ⑥ does not occur in the original statement of the problem. By appropriately selecting the test function, one can also derive the boundary conditions (21).

The adjoint homogeneous boundary conditions, with respect to (20), which must be met by the Green’s function with respect to ➑ ➐➤➢ ➁ and ➆ have the form

[ ➂ ]=0 at ❼ = ❼ 1 ( ➞ ➥ ≠ ➞ , ➦ = ➜ + 1, ✦✧✦✧✦ , ➝ ; ➧ = 1, ✦✧✦✧✦ , ➜ ), (21) [ ➂ ]=0 at ❼ = ❼ 2 ( ➞ ➥ ≠ ➞ ❇ ➨ , ➑ ➦ ➐ = 1, ✦✧✦✧✦ , ➜ ; ➧ = ➜ + 1, ✦✧✦✧✦ , ➝ ).

These conditions involve the linear differential forms [ ➂ ] defined in (19). For each endpoint of the interval in question, the set { ➞ ➥ } of the indices in the boundary operators (21) together with the set { ➞ ➨ } of the orders of derivatives in the boundary conditions (20) make up a complete set of nonnegative integers from 0 to ➝ − 1.

Taking into account the fact that the test function ➇ must satisfy the boundary conditions (20) and the Green’s function ➂ to conditions (21), we rewrite solution (18) to obtain

where the ➩ [ ] are differential operators with respect to ➁ , which are defined in (19). If the Green’s function is known, formula (22) can be used to immediately obtain the solution ➐➙↕

of the nonhomogeneous boundary value problem (1), (3), (20) for arbitrary (1) ( ❼ , ❽ ), ( ❼ ), ➛ ( ❽ ) ( ➀ ➧ = 1, ✦✧✦✧✦ , ➜ ), and ➛ (2) ➅ ( ❽ )( ➧ = ➜ + 1, ✦✧✦✧✦ , ➝ ).

9.6.1-5. The case of general nonhomogeneous boundary conditions. On solving (4) for the highest derivatives, we reduce the boundary conditions (4) to the canonical ➐➙↕ ➐➙↕ ➫ ➊ ➊

= 2 ( = + 1, , ), where the leading terms in different boundary conditions are different,

The sums in (23) do not contain the derivatives of orders ➎

1 , ✦✧✦✧✦ , ➞ (for ❼ = ❼ 1 ) and ➞ +1 , ✦✧✦✧✦ , ➞

(for ❼ = ❼ 2 ); thus,

It can be shown that the solution of problem (1), (3), (23) is given by ➯

where the ➩ [ ➂ ] are differential operators with respect to ➁ , which are defined in (19). Relation (24)

is similar to (22) but contains the Green’s function satisfying the more complicated boundary ➐➙↕ conditions that can be obtained from (23) by substituting (2) ➂ ❻ for and setting ➲ (1) ( ❽ )= ➲ ( ❽ ) = 0.

9.6.2. Equations Containing the Second Time Derivative

9.6.2-1. The case of homogeneous initial and boundary conditions. Consider the linear nonhomogeneous differential equation ➐ ➊ ➊ ➊

We set the homogeneous initial conditions

and the homogeneous boundary conditions ➄

where the boundary operators ❻ ➨ ❻ ➺ [ ] and (2) ➨ [ ] are defined in Paragraph 9.6.1-1. The solution of problem (1)–(3) can be represented in the form*

= ➂ ( ❼ , ➁ , ❽ , ➆ ) is the Green’s function; for ➊ ➊ ❽ > ➆ ≥ 0, it satisfies the homogeneous equation ➊ ➐

with the special semihomogeneous initial conditions

and the corresponding homogeneous boundary conditions ➄

[ ➂ ]=0 at ❼ = ❼ 2 ( ➧ = ➜ + 1, ✦✧✦✧✦ , ➝ ). The quantities ➁ and ➆ appear in problem (5)–(7) as free parameters ( ❼ 1 ≤ ➁ ≤ ❼ 2 ), and ➻ ( ❼ ) is the

Dirac delta function. One can verify by direct substitution into the equation and the initial and boundary conditions (1)–(3) that formula (4) is correct, taking into account the properties (5)–(7) of the Green’s function.

9.6.2-2. The case of nonhomogeneous initial and boundary conditions. Consider the linear nonhomogeneous differential equation (1) with the general nonhomogeneous

initial conditions

* Problem (1)–(3) is assumed to be well posed.

and the nonhomogeneous boundary conditions, reduced to the canonical form (see Paragraph 9.6.1-5):

Introducing a test function ➇ = ➇ ( ❼ , ❽ ) that satisfies the nonhomogeneous initial and boundary conditions (8), (9) and using the same line of reasoning as in Paragraph 9.6.1-3 for a simpler equation, we arrive at the solution of problem (1), (8), (9) in the form

where the ➩ [

] are differential operators with respect to ➁ , which are defined in relations (19), Paragraph 9.6.1-3. ➾ ➚➙➪ ➶ ➹➤➘ ➴

If the coefficients of equation (1) and those of the boundary conditions (9) are time independent, i.e.,

= (1) ➸ ( ❼ ), = ( ❼ ), ➨ = const,

(2) = const,

then in solution (10) one should set ➯

9.6.3. Nonstationary Problems with Many Space Variables

9.6.3-1. Equations with the first-order partial derivative with respect to ❽ .

Consider the following linear differential operator with respect to variables ➐➙❐ ❼ 1 , ✦✧✦✧✦

The coefficients ➎

of the operator are assumed to be sufficiently smooth functions of

1 , ✦✧✦✧✦ , ❼ and ❽ (and also bounded if necessary). The coefficients of the highest derivatives are assumed to be everywhere nonzero. ➎

1 ❮ . Cauchy problem ( ❽ ≥ 0, x ❰ Ï ). The solution of the Cauchy problem for the linear nonhomo- geneous parabolic differential equation with variable coefficients ➎ ➊

− x, [ ]= (x, ❽ )

under the initial conditions

= (x) at ❽ =0

= Ñ Ñ (x, y, ❽ , ➆ ) is the fundamental solution of the Cauchy problem, which satisfies for ➊ ❽ > ➆ ≥0 the equation

(5) and the special initial condition ➄

− x, [ Ñ Ñ ]=0

(6) The quantities y and ➆ appear in problem (5), (6) as free parameters (y ➐ ➐➙❐ ➄ ❰ Ï ), and ➻ (x) is the

= ➍ = (x − y).

-dimensional Dirac delta function. ➮

of operator (1) are independent of time ❽ , then the fundamental solution depends on only three arguments, Ñ Ñ (x, y, ❽ , ➆ )= Ñ Ñ (x, y, ❽ − ➆ ). If the coefficients of operator (1) are constants, then Ñ Ñ (x, y, ❽ , ➆ )= Ñ Ñ (x − y, ❽ − ➆ ).

If the coefficients

2 ❮ . Boundary value problems ( ➊ ❽ ≥ 0, x ❰ Ò ). The solutions of linear boundary value problems in a spatial domain Ò for equation (2) with initial condition (3) and homogeneous boundary conditions for x ❰ Ò (these conditions are not written out here) are given by formula (4) in which the domain of integration Ï

should be replaced by ➊ Ò . Here, by Ñ Ñ we mean the Green’s function that must satisfy, apart from equation (5) and the boundary condition (6), the same homogeneous boundary ➎

conditions for x ❰

as the original equation (2). For boundary value problems, the parameter y belongs to the same domain as x, i.e., y ❰ Ò Ó✮Ô .

Reference : Mathematical Encyclopedia (1977, Vol. 1).

9.6.3-2. Equations with the second-order partial derivative with respect to Õ .

1 ❮ . Cauchy problem ( Õ ≥ 0, x ❰ Ï ). The solution of the Cauchy problem for the linear nonhomo- geneous differential equation with variable coefficients ➎ ➊

x, Ö [ ]= (x, Õ )

under the initial conditions ➅

is given by

Here, Ý Ý = Ý Ý (x, y, â , Ü ) is the fundamental solution of the Cauchy problem

where y and Ü play the role of parameters.

of operator (1) are independent of time â , then the fundamen- tal solution depends on only three arguments, ì ä ì Ý Ý (x, y, â , Ü ä )= Ý Ý (x, y, â − Ü ), and the relation

If the coefficients é ê

(x, y, â , Ü ) ç =0 ì =− Ý Ý

(x, y, â ) holds. If the coefficients of operator (1) are constants, then

(x, y, â , Ü )= Ý Ý (x − y, â − Ø Ü ).

2 í . The solution of the Cauchy problem for the more complicated linear nonhomogeneous differ- ential equation with variable coefficients

2 + ï (x, â ) á â − æ

x, î [ ]= Û (x, â )

with initial conditions (8) is expressed as Ø

Here, Ý Ý (x, y, â , Ü ) is the corresponding fundamental solution of the Cauchy problem,

. Boundary value problems ( â ≥ 0, x ò ó ). The solutions of linear boundary value problems in a spatial domain ó for equation (7) with initial condition (8) and homogeneous boundary conditions for x ò ó á (these conditions are not written out here) are given by formula (9) in which the domain of integration ô õ should be replaced by ó . Here, by Ý Ý we mean the Green’s function that must satisfy, apart from equation (7) and the initial conditions (8), the same homogeneous boundary conditions as the original equation (7).

9.6.4. Some Special-Type Equations

The transformation û

, â )=( ✁ , Ü ) exp à × ( â ) Þ â ã , ✁ = ÿ ✄ ( â )+ × å ( â ) ✄ ( â ) Þ â , Ü = × ☎ ( â ) ✄ õ ( â ) Þ â ,

where ✄ ( â ) = exp à × ß ( â ) Þ â ã , leads to the simpler constant coefficient equation

The transformation ú

=1 ✞ ÿ , = î ÿ 1− õ leads to the constant coefficient equation

The change of variable û

= ln | ÿ | leads to a constant coefficient equation.

The transformation ú

leads to a constant coefficient equation.

is a linear differential operator of any order with respect to the space variable ÿ whose coefficients can depend on ÿ .

1 í . General solution:

where the ê = ê ( ÿ , â ) are arbitrary functions that satisfy the original equation with ✗ = 1: (

. Fundamental solution:

where Ý Ý 1 ( ÿ , â ) is the fundamental solution of the equation with ✗ = 1.

The linear differential operator ✕ ✖ can involve arbitrarily many space variables.

is a linear differential operator of any order with respect to the space variable ÿ whose coefficients can depend on ÿ .

1 í . General solution:

where the ê = ê ( ÿ , â ) are arbitrary functions that satisfy the original equation with ✗ = 1: (

. Suppose that the Cauchy problem for the special case of the equation with ✗ = 1 is well posed if only one initial condition is set at â = 0; this means that the constant coefficient differential operator ✕ ✖ is such that the equation with ✗ = 1 is regular with regularity index ✣ = 1. Then the fundamental solution of the original equation can be found by the formula

where Ý Ý 1 ( ÿ , â ) is the fundamental solution for ✗ = 1.

The linear differential operator ✕ ✖ can involve arbitrarily many space variables.

Here, ✕ is any linear differential operator with arbitrarily many independent variables ÿ 1 , ✦✧✦✧✦ , ÿ . Particular solutions:

where the are solutions of the equations ✥ ✕ [ ]− ★ = 0, the ★ are roots of the characteristic equation ✩

= 0, and the

are arbitrary constants.

=0

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