Representation of Solutions to Boundary Value Problems via the Green’s Functions 14.9.2-1.. The solution of the third boundary value problem for equation 14.9.1.1 in terms of the Green’s
Trang 1Example Consider a boundary value problem for the Laplace equation
∂2w
∂x2 +∂
2w
∂y2 = 0
in a strip 0 ≤x≤l, –∞ < y < ∞ with mixed boundary conditions
w = f1(y) at x= 0 , ∂w
∂x = f2(y) at x = l.
This equation is a special case of equation (14.9.1.1) with a(x) =1and b(x) = c(x) = Φ(x, t) = 0 The corresponding Sturm–Liouville problem (14.9.1.5)–(14.9.1.7) is written as
u xx + λy =0 , u= 0 at x= 0 , u x= 0 at x = l.
The eigenfunctions and eigenvalues are found as
u n (x) = sin
π(2n– 1)x l
, λ n= π
2 ( 2n– 1 )2
l , n= 1 , 2, Using formulas (14.9.1.3) and (14.9.1.4) and taking into account the identities ρ(ξ) =1 andy n 2 = l/2
(n =1 , 2, ) and the expression forΨnfrom the first row in Table 14.8, we obtain the Green’s function in the form
G(x, y, ξ, η) = 1
l
∞
n=1
1
σ n sin(σ n x) sin(σ n ξ)e–σ n|y–η| , σ n=√
λ n= π(2n– 1 )
l .
14.9.2 Representation of Solutions to Boundary Value Problems via
the Green’s Functions
14.9.2-1 First boundary value problem
The solution of the first boundary value problem for equation (14.9.1.1) with the boundary conditions
w = f1(y) at x = x1, w = f2(y) at x = x2,
w = f3(x) at y =0, w = f4(x) at y = h
is expressed in terms of the Green’s function as
w (x, y) = a(x1)
h
0 f1(η)
∂
∂ξ G (x, y, ξ, η)
ξ=x1
dη – a(x2)
h
0 f2(η)
∂
∂ξ G (x, y, ξ, η)
ξ=x2
dη
+
x2
x1
f3(ξ)
∂
∂η G (x, y, ξ, η)
η=0
dξ–
x2
x1
f4(ξ)
∂
∂η G (x, y, ξ, η)
η=h
dξ
+
x2
x1
h
0 Φ(ξ, η)G(x, y, ξ, η) dη dξ.
14.9.2-2 Second boundary value problem
The solution of the second boundary value problem for equation (14.9.1.1) with boundary conditions
∂ x w = f1(y) at x = x1, ∂ x w = f2(y) at x = x2,
∂ y w = f3(x) at y =0, ∂ y w = f4(x) at y = h
Trang 2is expressed in terms of the Green’s function as
w (x, y) = – a(x1)
h
0 f1(η)G(x, y, x1, η) dη + a(x2)
h
0 f2(η)G(x, y, x2, η) dη –
x2
x1 f3(ξ)G(x, y, ξ,0) dξ +
x2
x1 f4(ξ)G(x, y, ξ, h) dξ
+
x2
x1
h
0 Φ(ξ, η)G(x, y, ξ, η) dη dξ.
14.9.2-3 Third boundary value problem
The solution of the third boundary value problem for equation (14.9.1.1) in terms of the Green’s function is represented in the same way as the solution of the second boundary value problem (the Green’s function is now different)
Solutions of various boundary value problems for elliptic equations can be found in
Section T8.3
14.10 Boundary Value Problems with Many Space
Variables Representation of Solutions via the Green’s Function
14.10.1 Problems for Parabolic Equations
14.10.1-1 Statement of the problem
In general, a nonhomogeneous linear differential equation of the parabolic type in n space
variables has the form
∂w
∂t – Lx,t [w] = Φ(x, t), (14.10.1.1) where
Lx,t [w]≡ n
i,j=1
a ij (x, t) ∂
2w
∂x i ∂x j +
n
i=1
b i (x, t) ∂x ∂w
i + c(x, t)w,
x ={x1, , x n},
n
i,j=1
a ij (x, t)ξ i ξ j ≥σ
n
i=1
ξ2
i σ>0
(14.10.1.2)
Let V be some simply connected domain inRnwith a sufficiently smooth boundary
S = ∂V We consider the nonstationary boundary value problem for equation (14.10.1.1)
in the domain V with an arbitrary initial condition,
and nonhomogeneous linear boundary conditions,
Γx,t [w] = g(x, t) for xS (14.10.1.4)
In the general case,Γx,tis a first-order linear differential operator in the space coordinates
with coefficients dependent on x and t.
Trang 314.10.1-2 Representation of the problem solution in terms of the Green’s function The solution of the nonhomogeneous linear boundary value problem defined by (14.10.1.1)– (14.10.1.4) can be represented as the sum
w (x, t) =
t
0
V Φ(y, τ)G(x, y, t, τ) dV y dτ +
V f (y)G(x, y, t,0) dV y
+
t
0
S g (y, τ )H(x, y, t, τ ) dS y dτ, (14.10.1.5)
where G(x, y, t, τ ) is the Green’s function; for t > τ ≥ 0, it satisfies the homogeneous equation
∂G
with the nonhomogeneous initial condition of special form
and the homogeneous boundary condition
Γx,t [G] =0 for xS (14.10.1.8)
The vector y ={y1, , y n}appears in problem (14.10.1.6)–(14.10.1.8) as an n-dimensional
free parameter (yV ), and δ(x – y) = δ(x1– y1) δ(x n – y n ) is the n-dimensional Dirac delta function The Green’s function G is independent of the functions Φ, f, and g that
characterize various nonhomogeneities of the boundary value problem In (14.10.1.5), the
integration is performed everywhere with respect to y, with dV y = dy1 dy n.
The function H(x, y, t, τ ) involved in the integrand of the last term in solution (14.10.1.5) can be expressed via the Green’s function G(x, y, t, τ ) The corresponding formulas for
H (x, y, t, τ ) are given in Table 14.9 for the three basic types of boundary value problems;
in the third boundary value problem, the coefficient k can depend on x and t The boundary
conditions of the second and third kind, as well as the solution of the first boundary value problem, involve operators of differentiation along the conormal of operator (14.10.1.2); these operators act as follows:
∂G
∂M x ≡n
i,j=1
a ij (x, t)N j ∂G ∂x
i,
∂G
∂M y ≡ n
i,j=1
a ij (y, τ )N j ∂G ∂y
i, (14.10.1.9)
where N ={N1, , N n}is the unit outward normal to the surface S In the special case where a ii (x, t) =1and a ij (x, t) =0for i≠j, operator (14.10.1.9) coincides with the ordinary
operator of differentiation along the outward normal to S.
TABLE 14.9
The form of the function H(x, y, t, τ ) for the basic types of nonstationary boundary value problems
Type of problem Form of boundary condition (14.10.1.4) Function H(x, y, t, τ )
First boundary value problem w = g(x, t) for xS H (x, y, t, τ ) = – ∂G
∂M y (x, y, t, τ )
Second boundary value problem ∂w
∂M x = g(x, t) for xS H (x, y, t, τ ) = G(x, y, t, τ )
Third boundary value problem ∂w
∂M x + kw = g(x, t) for xS H (x, y, t, τ ) = G(x, y, t, τ )
Trang 4If the coefficient of equation (14.10.1.6) and the boundary condition (14.10.1.8) are
independent of t, then the Green’s function depends on only three arguments, G(x, y, t, τ ) =
G (x, y, t – τ ).
Remark. Let S i (i =1, , p) be different portions of the surface S such that S =p
i=1S iand let boundary
conditions of various types be set on the S i,
Γ (i)
x,t [w] = g i (x, t) for xS i, i= 1, , p. (14 10 1 10 ) Then formula (14.10.1.5) remains valid but the last term in (14.10.1.5) must be replaced by the sum
p
i=1
t 0
S i
g i (y, τ )H i (x, y, t, τ ) dS y dτ (14 10 1 11 )
14.10.2 Problems for Hyperbolic Equations
14.10.2-1 Statement of the problem
The general nonhomogeneous linear differential hyperbolic equation in n space variables
can be written as
∂2w
∂t2 + ϕ(x, t)
∂w
∂t – Lx,t [w] = Φ(x, t), (14.10.2.1)
where the operator Lx,t [w] is explicitly defined in (14.10.1.2).
We consider the nonstationary boundary value problem for equation (14.10.2.1) in the
domain V with arbitrary initial conditions,
w = f0(x) at t=0,
∂ t w = f1(x) at t=0,
(14.10.2.2) (14.10.2.3) and the nonhomogeneous linear boundary condition (14.10.1.4)
14.10.2-2 Representation of the problem solution in terms of the Green’s function The solution of the nonhomogeneous linear boundary value problem defined by (14.10.2.1)– (14.10.2.3), (14.10.1.4) can be represented as the sum
w (x, t) =
t
0
V Φ(y, τ)G(x, y, t, τ) dV y dτ–
V f0(y)
∂
∂τ G (x, y, t, τ )
τ=0dV y +
V
f1(y) + f0(y)ϕ(y,0)
G (x, y, t,0) dV y
+
t
0
S g (y, τ )H(x, y, t, τ ) dS y dτ. (14.10.2.4)
Here, G(x, y, t, τ ) is the Green’s function; for t > τ≥ 0it satisfies the homogeneous equation
∂2G
∂t2 + ϕ(x, t)
∂G
∂t – Lx,t [G] =0 (14.10.2.5) with the semihomogeneous initial conditions
∂ t G = δ(x – y) at t = τ ,
and the homogeneous boundary condition (14.10.1.8)
Trang 5If the coefficients of equation (14.10.2.5) and the boundary condition (14.10.1.8)
are independent of time t, then the Green’s function depends on only three arguments,
G (x, y, t, τ ) = G(x, y, t – τ ) In this case, one can set ∂τ ∂ G (x, y, t, τ )
τ=0 = –∂t ∂ G (x, y, t) in
solution (14.10.2.4)
The function H(x, y, t, τ ) involved in the integrand of the last term in solution (14.10.2.4) can be expressed via the Green’s function G(x, y, t, τ ) The corresponding formulas for H
are given in Table 14.9 for the three basic types of boundary value problems; in the third
boundary value problem, the coefficient k can depend on x and t.
Remark. Let S i (i =1, , p) be different portions of the surface S such that S =p
i=1S iand let boundary
conditions of various types (14.10.1.10) be set on the S i Then formula (14.10.2.4) remains valid but the last term in (14.10.2.4) must be replaced by the sum (14.10.1.11).
14.10.3 Problems for Elliptic Equations
14.10.3-1 Statement of the problem
In general, a nonhomogeneous linear elliptic equation can be written as
where
Lx[w]≡ n
i,j=1
a ij(x) ∂
2w
∂x i ∂x j +
n
i=1
b i(x)∂x ∂w
i + c(x)w. (14.10.3.2)
Two-dimensional problems correspond to n =2and three-dimensional problems to n =3
We consider equation (14.10.3.1)–(14.10.3.2) in a domain V and assume that the
equa-tion is subject to the general linear boundary condiequa-tion
Γx[w] = g(x) for xS (14.10.3.3) The solution of the stationary problem (14.10.3.1)–(14.10.3.3) can be obtained by
passing in (14.10.1.5) to the limit as t → ∞ To this end, one should start with
equa-tion (14.10.1.1), whose coefficients are independent of t, and take the homogeneous initial
condition (14.10.1.3), with f (x) =0, and the stationary boundary condition (14.10.1.4)
14.10.3-2 Representation of the problem solution in terms of the Green’s function The solution of the linear boundary value problem (14.10.3.1)–(14.10.3.3) can be repre-sented as the sum
w(x) =
V Φ(y)G(x, y) dV y +
S g (y)H(x, y) dS y. (14.10.3.4)
Here, the Green’s function G(x, y) satisfies the nonhomogeneous equation of special form
with the homogeneous boundary condition
Trang 6The vector y ={y1, , y n}appears in problem (14.10.3.5), (14.10.3.6) as an n-dimensional
free parameter (yV ) Note that G is independent of the functions Φ and g characterizing
various nonhomogeneities of the original boundary value problem
The function H(x, y) involved in the integrand of the second term in solution (14.10.3.4) can be expressed via the Green’s function G(x, y) The corresponding formulas for H are
given in Table 14.10 for the three basic types of boundary value problems The boundary conditions of the second and third kind, as well as the solution of the first boundary value problem, involve operators of differentiation along the conormal of operator (14.10.3.2);
these operators are defined by (14.10.1.9); in this case, the coefficients a ij depend on x
only
TABLE 14.10
The form of the function H(x, y) involved in the integrand of the last term in
solution (14.10.3.4) for the basic types of stationary boundary value problems
Type of problem Form of boundary condition (14.10.3.3) Function H(x, y)
First boundary value problem w = g(x) for xS H(x, y) = – ∂G
∂M y(x, y)
Second boundary value problem ∂w
∂M x = g(x) for xS H(x, y) = G(x, y)
Third boundary value problem ∂w
∂M x + kw = g(x) for xS H(x, y) = G(x, y)
Remark. For the second boundary value problem with c(x)≡ 0 , the thus defined Green’s function must not necessarily exist; see Polyanin (2002).
14.10.4 Comparison of the Solution Structures for Boundary Value
Problems for Equations of Various Types
Table 14.11 lists brief formulations of boundary value problems for second-order equations
of elliptic, parabolic, and hyperbolic types The coefficients of the differential operators
LxandΓxin the space variables x1, , x n are assumed to be independent of time t; these
operators are the same for the problems under consideration
TABLE 14.11 Formulations of boundary value problems for equations of various types
Type of equation Form of equation Initial conditions Boundary conditions Elliptic –Lx[w] =Φ(x) not set Γx[w] = g(x) for xS
Parabolic ∂ t w – Lx[w] = Φ(x, t) w = f (x) at t =0 Γx[w] = g(x, t) for xS
Hyperbolic ∂ tt w – Lx[w] = Φ(x, t) w = f0(x) at t =0,
∂ t w = f1(x) at t =0 Γx[w] = g(x, t) for xS
Below are the respective general formulas defining the solutions of these problems with
Trang 7zero initial conditions (f = f0= f1 =0):
w0(x) =
V Φ(y)G0(x, y) dV y +
S g(y)HG0(x, y)
dS y,
w1(x, t) =
t
0
V Φ(y, τ)G1(x, y, t – τ ) dV y dτ+
t
0
S g (y, τ ) HG1(x, y, t – τ )
dS y dτ,
w2(x, t) =
t
0
V Φ(y, τ)G2(x, y, t – τ ) dV y dτ+
t
0
S g (y, τ ) HG2(x, y, t – τ )
dS y dτ,
where the G nare the Green’s functions, and the subscripts0,1, and2refer to the elliptic, parabolic, and hyperbolic problem, respectively All solutions involve the same opera-torH[G]; it is explicitly defined in Subsections 14.10.1–14.10.3 (see also Section 14.7) for
different boundary conditions
It is apparent that the solutions of the parabolic and hyperbolic problems with zero initial conditions have the same structure The structure of the solution to the problem for
a parabolic equation differs from that for an elliptic equation by the additional integration
with respect to t.
14.11 Construction of the Green’s Functions General
Formulas and Relations
14.11.1 Green’s Functions of Boundary Value Problems for
Equations of Various Types in Bounded Domains
14.11.1-1 Expressions of the Green’s function in terms of infinite series
Table 14.12 lists the Green’s functions of boundary value problems for second-order
equa-tions of various types in a bounded domain V It is assumed that Lx is a second-order linear self-adjoint differential operator (e.g., see Zwillinger, 1997) in the space variables
x1, , x n, and Γx is a zeroth- or first-order linear boundary operator that can define a
boundary condition of the first, second, or third kind; the coefficients of the operators Lx
andΓxcan depend on the space variables but are independent of time t The coefficients λ k and the functions u k(x) are determined by solving the homogeneous eigenvalue problem
It is apparent from Table 14.12 that, given the Green’s function in the problem for a parabolic (or hyperbolic) equation, one can easily construct the Green’s functions of the corresponding problems for elliptic and hyperbolic (or parabolic) equations In particular, the Green’s function of the problem for an elliptic equation can be expressed via the Green’s function of the problem for a parabolic equation as follows:
G0(x, y) =
∞
0 G1(x, y, t) dt. (14.11.1.3)
Here, the fact that all λ kare positive is taken into account; for the second boundary value
problem, it is assumed that λ =0is not an eigenvalue of problem (14.11.1.1)–(14.11.1.2)