Representation of the problem solution in terms of the Green’s function... The corresponding formulas forΛm x, t, τ are given in Table 14.6 for the basic types of boundary value problem
Trang 1L x,t [w]≡a (x, t) ∂
2w
∂x2 + b(x, t)
∂w
∂x + c(x, t)w, a (x, t) >0 (14.7.1.2) Consider the nonstationary boundary value problem for equation (14.7.1.1) with an initial condition of general form
and arbitrary nonhomogeneous linear boundary conditions
α1∂w
∂x + β1w = g1(t) at x = x1, (14.7.1.4)
α2∂w ∂x + β2w = g2(t) at x = x2 (14.7.1.5)
By appropriately choosing the coefficients α1, α2, β1, and β2in (14.7.1.4) and (14.7.1.5), we obtain the first, second, third, and mixed boundary value problems for equation (14.7.1.1)
14.7.1-2 Representation of the problem solution in terms of the Green’s function The solution of the nonhomogeneous linear boundary value problem (14.7.1.1)–(14.7.1.5) can be represented as
w (x, t) =
t
0
x2
x1
Φ(y, τ)G(x, y, t, τ) dy dτ +
x2
x1
f (y)G(x, y, t,0) dy +
t
0 g1(τ )a(x1, τ )Λ1(x, t, τ ) dτ +
t
0 g2(τ )a(x2, τ )Λ2(x, t, τ ) dτ (14.7.1.6)
Here, G(x, y, t, τ ) is the Green’s function that satisfies, for t > τ ≥ 0, the homogeneous equation
∂G
with the nonhomogeneous initial condition of special form
and the homogeneous boundary conditions
α1∂G
∂x + β1G=0 at x = x1, (14.7.1.9)
α2∂G ∂x + β2G=0 at x = x2 (14.7.1.10)
The quantities y and τ appear in problem (14.7.1.7)–(14.7.1.10) as free parameters, with
x1≤y≤x2, and δ(x) is the Dirac delta function.
The initial condition (14.7.1.8) implies the limit relation
f (x) = lim
t→τ
x2
x1
f (y)G(x, y, t, τ ) dy for any continuous function f = f (x).
Trang 2620 LINEARPARTIALDIFFERENTIALEQUATIONS
TABLE 14.6 Expressions of the functions Λ 1(x, t, τ ) andΛ 2(x, t, τ ) involved
in the integrands of the last two terms in solution (14.7.1.6) Type of problem Form of boundary conditions Functions Λm (x, t, τ )
First boundary value problem
(α1= α2 = 0, β1 = β2 = 1 )
w = g1(t) at x = x1
w = g2(t) at x = x2
Λ 1(x, t, τ ) = ∂ y G (x, y, t, τ )
y=x1
Λ 2(x, t, τ ) = –∂ y G (x, y, t, τ )
y=x2
Second boundary value problem
(α1= α2= 1, β1 = β2= 0 )
∂ x w = g1(t) at x = x1
∂ x w = g2(t) at x = x2
Λ 1(x, t, τ ) = –G(x, x1, t, τ )
Λ 2(x, t, τ ) = G(x, x2, t, τ )
Third boundary value problem
(α1= α2= 1, β1< 0, β2> 0 )
∂ x w + β1w = g1(t) at x = x1
∂ x w + β2w = g2(t) at x = x2
Λ 1(x, t, τ ) = –G(x, x1, t, τ )
Λ 2(x, t, τ ) = G(x, x2, t, τ )
Mixed boundary value problem
(α1= β2 = 0, α2 = β1 = 1 )
w = g1(t) at x = x1
∂ x w = g2(t) at x = x2
Λ 1(x, t, τ ) = ∂ y G (x, y, t, τ )
y=x1
Λ 2(x, t, τ ) = G(x, x2, t, τ )
Mixed boundary value problem
(α1= β2= 1, α2 = β1= 0 )
∂ x w = g1(t) at x = x1
w = g2(t) at x = x2
Λ 1(x, t, τ ) = –G(x, x1, t, τ )
Λ 2(x, t, τ ) = –∂ y G (x, y, t, τ )
y=x2
The functions Λ1(x, t, τ ) and Λ2(x, t, τ ) involved in the integrands of the last two terms in solution (14.7.1.6) can be expressed in terms of the Green’s function G(x, y, t, τ ).
The corresponding formulas forΛm (x, t, τ ) are given in Table 14.6 for the basic types of
boundary value problems
It is significant that the Green’s function G and the functionsΛ1,Λ2are independent of the functionsΦ, f, g1 , and g2that characterize various nonhomogeneities of the boundary value problem
If the coefficients of equation (14.7.1.1)–(14.7.1.2) are independent of time t, i.e., the
conditions
a = a(x), b = b(x), c = c(x) (14.7.1.11) hold, then the Green’s function depends on only three arguments,
G (x, y, t, τ ) = G(x, y, t – τ ).
In this case, the functionsΛmdepend on only two arguments,Λm=Λm (x, t – τ ), m =1,2 Formula (14.7.1.6) also remains valid for the problem with boundary conditions of the
third kind if β1 = β1(t) and β2 = β2(t) Here, the relation between Λm (m = 1, 2) and
the Green’s function G is the same as that in the case of constants β1and β2; the Green’s function itself is now different
The condition that the solution must vanish at infinity, w → 0as x → ∞, is often set
for the first, second, and third boundary value problems that are considered on the interval
x1 ≤x <∞ In this case, the solution is calculated by formula (14.7.1.6) with Λ2=0and
Λ1specified in Table 14.6
14.7.2 Problems for Equation
s(x) ∂w
∂t =
∂
∂x
*
p(x) ∂w
∂x
+
–q(x)w+Φ(x, t)
14.7.2-1 General formulas for solving nonhomogeneous boundary value problems Consider linear equations of the special form
s (x) ∂w
∂t = ∂
∂x
p (x) ∂w
∂x
– q(x)w + Φ(x, t). (14.7.2.1)
Trang 3TABLE 14.7 Expressions of the functions Λ 1(x, t) andΛ 2(x, t) involved in the integrands of the
last two terms in solutions (14.7.2.2) and (14.8.2.2); the modified Green’s function
G(x, ξ, t) for parabolic equations of the form (14.7.2.1) are found by formula (14.7.2.3),
and that for hyperbolic equations of the form (14.8.2.1), by formula (14.8.2.3)
Type of problem Form of boundary conditions Functions Λm (x, t)
First boundary value problem
(α1= α2= 0, β1 = β2= 1 )
w = g1(t) at x = x1
w = g2(t) at x = x2
Λ 1(x, t) = ∂ ξ G(x, ξ, t)
ξ=x1
Λ 2(x, t) = –∂ ξ G(x, ξ, t)
ξ=x2
Second boundary value problem
(α1= α2= 1, β1 = β2= 0 )
∂ x w = g1(t) at x = x1
∂ x w = g2(t) at x = x2
Λ 1(x, t) = – G(x, x1, t)
Λ 2(x, t) = G(x, x2, t)
Third boundary value problem
(α1= α2= 1, β1< 0, β2> 0 )
∂ x w + β1w = g1(t) at x = x1
∂ x w + β2w = g2(t) at x = x2
Λ 1(x, t) = – G(x, x1, t)
Λ 2(x, t) = G(x, x2, t)
Mixed boundary value problem
(α1= β2 = 0, α2 = β1 = 1 )
w = g1(t) at x = x1
∂ x w = g2(t) at x = x2
Λ 1(x, t) = ∂ ξ G(x, ξ, t)
ξ=x1
Λ 2(x, t) = G(x, x2, t)
Mixed boundary value problem
(α1= β2= 1, α2 = β1= 0 )
∂ x w = g1(t) at x = x1
w = g2(t) at x = x2
Λ 1(x, t) = – G(x, x1, t)
Λ 2(x, t) = –∂ ξ G(x, ξ, t)
ξ=x2
They are often encountered in heat and mass transfer theory and chemical engineering
sciences Throughout this subsection, we assume that the functions s, p, p x , and q are continuous and s >0, p >0, and x1≤x≤x2.
The solution of equation (14.7.2.1) under the initial condition (14.7.1.3) and the arbitrary linear nonhomogeneous boundary conditions (14.7.1.4)–(14.7.1.5) can be represented as the sum
w (x, t) =
t
0
x2
x1
Φ(ξ, τ)G(x, ξ, t – τ) dξ dτ +
x2
x1
s (ξ)f (ξ) G(x, ξ, t) dξ
+ p(x1)
t
0 g1(τ )Λ1(x, t – τ ) dτ + p(x2)
t
0 g2(τ )Λ2(x, t – τ ) dτ
(14.7.2.2)
Here, the modified Green’s function is given by
G(x, ξ, t) =∞
n=1
y n (x)y n (ξ)
y n 2 exp(–λ n t), y n 2 =
x2
x1
s (x)y2n (x) dx, (14.7.2.3)
where the λ n and y n (x) are the eigenvalues and corresponding eigenfunctions of the
fol-lowing Sturm–Liouville problem for a second-order linear ordinary differential equation:
[p(x)y x ] x + [λs(x) – q(x)]y =0,
α1y
x + β1y=0 at x = x1,
α2y
x + β2y=0 at x = x2
(14.7.2.4)
The functions Λ1(x, t) and Λ2(x, t) that occur in the integrands of the last two terms in
solution (14.7.2.2) are expressed via the Green’s function (14.7.2.3) The corresponding formulas forΛm (x, t) are given in Table 14.7 for the basic types of boundary value problems.
Trang 4622 LINEARPARTIALDIFFERENTIALEQUATIONS
14.7.2-2 Properties of Sturm–Liouville problem (14.7.2.4) Heat equation example
1◦ There are infinitely many eigenvalues All eigenvalues are real and different and can
be ordered so that λ1< λ2< λ3<· · · , with λ n → ∞ as n → ∞ (therefore, there can exist
only finitely many negative eigenvalues) Each eigenvalue is of multiplicity 1
2◦ The different eigenfunctions y
n (x) and y m (x) are orthogonal with weight s(x) on the
interval x1≤x≤x2:
x2
x1
s (x)y n (x)y m (x) dx =0 for n≠m
3◦ If the conditions
q (x)≥ 0, α1β1≤ 0, α2β2 ≥ 0 (14.7.2.5)
are satisfied, there are no negative eigenvalues If q≡ 0and β1= β2=0, then λ1 =0is the
least eigenvalue, to which there corresponds the eigenfunction ϕ1= const Otherwise, all eigenvalues are positive, provided that conditions (14.7.2.5) are satisfied
Other general and special properties of the Sturm–Liouville problem (14.7.2.4) are given in Subsection 12.2.5; various asymptotic and approximate formulas for eigenvalues and eigenfunctions can also be found there
Example Consider the first boundary value problem in the domain0 ≤x≤lfor the heat equation with a source
∂w
∂t = a ∂
2w
∂x2 – bw
under the initial condition (14.7.1.3) and boundary conditions
w = g1(t) at x= 0 ,
w = g2(t) at x = l. (14.7 2 6 )
The above equation is a special case of equation (14.7.2.1) with s(x) =1, p(x) = a, q(x) = b, and Φ(x, t) =0 The corresponding Sturm–Liouville problem (14.7.2.4) has the form
ay xx + (λ – b)y =0 , y= 0 at x =0 , y= 0 at x = l.
The eigenfunctions and eigenvalues are found to be
y (x) = sin
πnx l
, λ n = b + aπ
2n2
l , n= 1 , 2,
Using formula (14.7.2.3) and taking into account thaty n 2= l/2 , we obtain the Green’s function
G(x, ξ, t) = 2
l e
–bt ∞
n=1
sin
πnx l
sin
πnξ l
exp
–aπ
2n2
l t
Substituting this expression into (14.7.2.2) with p(x1) = p(x2) = s(ξ) =1, x1 = 0, and x2 = l and taking into
account the formulas
Λ 1(x, t) = ∂ ξ G(x, ξ, t)
ξ=x1 , Λ 2(x, t) = –∂ ξ G(x, ξ, t)
ξ=x2
(see the first row in Table 14.7), one obtains the solution to the problem in question.
Solutions to various boundary value problems for parabolic equations can be found in
Section T8.1
Trang 514.8 Boundary Value Problems for Hyperbolic
Equations with One Space Variable Green’s
Function Goursat Problem
14.8.1 Representation of Solutions via the Green’s Function
14.8.1-1 Statement of the problem (t≥ 0, x1 ≤x≤x2)
In general, a one-dimensional nonhomogeneous linear differential equation of hyperbolic type with variable coefficients is written as
∂2w
∂t2 + ϕ(x, t)
∂w
∂t – L x,t [w] = Φ(x, t), (14.8.1.1)
where the operator L x,t [w] is defined by (14.7.1.2).
Consider the nonstationary boundary value problem for equation (14.8.1.1) with the initial conditions
w = f0(x) at t=0,
∂ t w = f1(x) at t=0 (14.8.1.2) and arbitrary nonhomogeneous linear boundary conditions
α1∂w
∂x + β1w = g1(t) at x = x1, (14.8.1.3)
α2∂w ∂x + β2w = g2(t) at x = x2 (14.8.1.4)
14.8.1-2 Representation of the problem solution in terms of the Green’s function The solution of problem (14.8.1.1), (14.8.1.2), (14.8.1.3), (14.8.1.4) can be represented as the sum
w (x, t) =
t
0
x2
x1
Φ(y, τ)G(x, y, t, τ) dy dτ
–
x2
x1
f0(y)
∂
∂τ G (x, y, t, τ )
τ=0dy+
x2
x1
f1(y) + f0(y)ϕ(y,0)G (x, y, t,0) dy +
t
0 g1(τ )a(x1, τ )Λ1(x, t, τ ) dτ +
t
0 g2(τ )a(x2, τ )Λ2(x, t, τ ) dτ (14.8.1.5)
Here, the Green’s function G(x, y, t, τ ) is determined by solving the homogeneous equation
∂2G
∂t2 + ϕ(x, t)
∂G
∂t – L x,t [G] =0 (14.8.1.6) with the semihomogeneous initial conditions
G=0 at t = τ ,
∂ t G = δ(x – y) at t = τ ,
(14.8.1.7) (14.8.1.8)
Trang 6624 LINEARPARTIALDIFFERENTIALEQUATIONS
and the homogeneous boundary conditions
α1∂G
∂x + β1G=0 at x = x1, (14.8.1.9)
α2∂G ∂x + β2G=0 at x = x2 (14.8.1.10)
The quantities y and τ appear in problem (14.8.1.6)–(14.8.1.8), (14.8.1.9), (14.8.1.10) as free parameters (x1≤y≤x2), and δ(x) is the Dirac delta function.
The functions Λ1(x, t, τ ) and Λ2(x, t, τ ) involved in the integrands of the last two terms in solution (14.8.1.5) can be expressed via the Green’s function G(x, y, t, τ ) The
corresponding formulas for Λm (x, t, τ ) are given in Table 14.6 for the basic types of
boundary value problems
It is significant that the Green’s function G andΛ1,Λ2are independent of the functions
Φ, f0 , f1, g1, and g2 that characterize various nonhomogeneities of the boundary value problem
If the coefficients of equation (14.8.1.1) 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 (14.8.1.5).
14.8.2 Problems for Equation
s(x) ∂2w
∂t2 = ∂
∂x
*
p(x) ∂w
∂x
+
– q(x)w+Φ(x, t)
14.8.2-1 General relations for solving nonhomogeneous boundary value problems Consider linear equations of the special form
s (x) ∂
2w
∂t2 =
∂
∂x
p (x) ∂w
∂x
– q(x)w + Φ(x, t). (14.8.2.1)
It is assumed that the functions s, p, p x , and q are continuous and the inequalities s >0,
p>0hold for x1 ≤x≤x2.
The solution of equation (14.8.2.1) under the general initial conditions (14.8.1.2) and the arbitrary linear nonhomogeneous boundary conditions (14.8.1.3)–(14.8.1.4) can be represented as the sum
w (x, t) =
t 0
x2
x1
Φ(ξ, τ)G(x, ξ, t – τ) dξ dτ
+ ∂
∂t
x2
x1
s (ξ)f0(ξ) G(x, ξ, t) dξ +
x2
x1
s (ξ)f1(ξ) G(x, ξ, t) dξ
+ p(x1)
t
0 g1(τ )Λ1(x, t – τ ) dτ + p(x2)
t
0 g2(τ )Λ2(x, t – τ ) dτ (14.8.2.2) Here, the modified Green’s function is determined by
G(x, ξ, t) =∞
n=1
y n (x)y n (ξ) sin t √
λ n
y n 2√
λ n , y n 2 =
x2
x1
s (x)y2n (x) dx, (14.8.2.3)
Trang 7where the λ n and y n (x) are the eigenvalues and corresponding eigenfunctions of the Sturm–
Liouville problem for the second-order linear ordinary differential equation
[p(x)y x ] x + [λs(x) – q(x)]y =0,
α1y
x + β1y=0 at x = x1,
α2y
x + β2y=0 at x = x2
(14.8.2.4)
The functions Λ1(x, t) and Λ2(x, t) that occur in the integrands of the last two terms
in solution (14.8.2.2) are expressed in terms of the Green’s function of (14.8.2.3) The corresponding formulas forΛm (x, t) are given in Table 14.7 for the basic types of boundary
value problems
14.8.2-2 Properties of the Sturm–Liouville problem The Klein–Gordon equation The general and special properties of the Sturm–Liouville problem (14.8.2.4) are given
in Subsection 12.2.5; various asymptotic and approximate formulas for eigenvalues and eigenfunctions can also be found there
Example Consider the second boundary value problem in the domain0 ≤x≤lfor the Klein–Gordon equation
∂2w
∂t2 = a2∂
2w
∂x2 – bw,
under the initial conditions (14.8.1.2) and boundary conditions
∂ x w = g1(t) at x= 0 ,
∂ x w = g2(t) at x = l.
The Klein–Gordon equation is a special case of equation (14.8.2.1) with s(x) =1, p(x) = a2, q(x) = b, and Φ(x, t) =0 The corresponding Sturm–Liouville problem (14.8.2.4) has the form
a2y xx + (λ – b)y =0 , y x= 0 at x= 0 , y x = 0 at x = l.
The eigenfunctions and eigenvalues are found to be
y n+1 (x) = cos
πnx l
, λ n+1 = b + aπ
2n2
l , n= 0 , 1,
Using formula (14.8.2.4) and taking into account that y12= l and y n 2= l/2(n =1 , 2, ), we find the
Green’s function:
G(x, ξ, t) = 1
l √
bsin t √
b + 2
l
∞
n=1
cos
πnx l
cos
πnξ l
sin t
(aπn/l)2+ b
(aπn/l)2+ b . Substituting this expression into (14.8.2.3) with p(x1) = p(x2) = s(ξ) =1, x1 = 0, and x2 = l and taking into
account the formulas
Λ 1(x, t) = – G(x, x1, t), Λ 2(x, t) = G(x, x2, t)
(see the second row in Table 14.7), one obtains the solution to the problem in question.
Solutions to various boundary value problems for hyperbolic equations can be found in
Section T8.2
... x1 = 0, and x2 = l and taking intoaccount the formulas
Λ 1(x,... solution to the problem in question.
Solutions to various boundary value problems for hyperbolic equations can be found in
Section T8.2