First, second, third, and mixed boundary value problems.1◦.. Second boundary value problem.. Generalized Cauchy Problem with Initial Conditions Set Along a Curve 14.8.4-1.. Domain of inf
Trang 114.8.3 Problems for Equation
∂2w
∂t2 +a(t) ∂w
∂t = b(t)
∂x
*
p(x) ∂w
∂x
+
–q(x)w
4
+ Φ(x, t)
14.8.3-1 General relations to solve nonhomogeneous boundary value problems Consider the generalized telegraph equation of the form
∂2w
∂t2 + a(t)
∂w
∂t = b(t) ∂
∂x
*
p (x) ∂w
∂x
+
– q(x)w4
+Φ(x, t). (14.8.3.1)
It is assumed that the functions p, p x , and q are continuous and p >0for x1≤x≤x2.
The solution of equation (14.8.3.1) 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
–
x2
x1
f0(ξ)
∂
∂τ G (x, ξ, t, τ )
τ=0dξ+
x2
x1
f1(ξ) + a(0)f0(ξ)
G (x, ξ, t,0) dξ + p(x1)
t
0 g1(τ )b(τ )Λ1(x, t, τ ) dτ + p(x2)
t
0 g2(τ )b(τ )Λ2(x, t, τ ) dτ (14.8.3.2) Here, the modified Green’s function is determined by
G (x, ξ, t, τ ) =
∞
n=1
y n (x)y n (ξ)
y n 2 U n (t, τ ), y n 2=
x2
x1
y2
n (x) dx, (14.8.3.3)
where the λ n and y n (x) are the eigenvalues and corresponding eigenfunctions of the Sturm–
Liouville problem for the following second-order linear ordinary differential equation with homogeneous boundary conditions:
[p(x)y x] x + [λ – q(x)]y =0,
α1y
x + β1y=0 at x = x1,
α2y
x + β2y=0 at x = x2
(14.8.3.4)
The functions U n = U n (t, τ ) are determined by solving the Cauchy problem for the linear
ordinary differential equation
U
n + a(t)U n + λ n b (t)U n=0,
U n
t=τ =0, U
n
The prime denotes the derivative with respect to t, and τ is a free parameter occurring in
the initial conditions
The functions Λ1(x, t) and Λ2(x, t) that occur in the integrands of the last two terms
in solution (14.8.3.2) are expressed in terms of the Green’s function of (14.8.3.3) The corresponding formulas will be specified below when studying specific boundary value problems
The general and special properties of the Sturm–Liouville problem (14.8.3.4) are de-tailed in Subsection 12.2.5 Asymptotic and approximate formulas for eigenvalues and eigenfunctions are also presented there
Trang 214.8.3-2 First, second, third, and mixed boundary value problems.
1◦ First boundary value problem The solution of equation (14.8.3.1) with the initial
conditions (14.8.1.2) and boundary conditions (14.8.1.3)–(14.8.1.4) for α1 = α2 = 0 and
β1= β2 =1is given by relations (14.8.3.2) and (14.8.3.3), where
Λ1(x, t, τ ) = ∂
∂ξ G (x, ξ, t, τ )
ξ=x1
, Λ2(x, t, τ ) = – ∂
∂ξ G (x, ξ, t, τ )
ξ=x2
2◦ Second boundary value problem The solution of equation (14.8.3.1) with the initial
conditions (14.8.1.2) and boundary conditions (14.8.1.3)–(14.8.1.4) for α1 = α2 = 1 and
β1= β2 =0is given by relations (14.8.3.2) and (14.8.3.3) with
Λ1(x, t, τ ) = –G(x, x1, t, τ ), Λ2(x, t, τ ) = G(x, x2, t, τ ).
3◦ Third boundary value problem The solution of equation (14.8.3.1) with the initial
conditions (14.8.1.2) and boundary conditions (14.8.1.3)–(14.8.1.4) for α1 = α2 = 1 and
β1β2≠ 0is given by relations (14.8.3.2) and (14.8.3.3) in which
Λ1(x, t, τ ) = –G(x, x1, t, τ ), Λ2(x, t, τ ) = G(x, x2, t, τ ).
4◦ Mixed boundary value problem The solution of equation (14.8.3.1) with the initial
conditions (14.8.1.2) and boundary conditions (14.8.1.3)–(14.8.1.4) for α1 = β2 = 0 and
α2 = β1 =1is given by relations (14.8.3.2) and (14.8.3.3) with
Λ1(x, t, τ ) = ∂ξ ∂ G (x, ξ, t, τ )
ξ=x1 , Λ2(x, t, τ ) = G(x, x2, t, τ ).
5◦ Mixed boundary value problem The solution of equation (14.8.3.1) with the initial
conditions (14.8.1.2) and boundary conditions (14.8.1.3)–(14.8.1.4) for α1 = β2 = 1 and
α2 = β1 =0is given by relations (14.8.3.2) and (14.8.3.3) with
Λ1(x, t, τ ) = –G(x, x1, t, τ ), Λ2(x, t, τ ) = – ∂ξ ∂ G (x, ξ, t, τ )
ξ=x2
14.8.4 Generalized Cauchy Problem with Initial Conditions Set
Along a Curve
14.8.4-1 Statement of the generalized Cauchy problem Basic property of a solution Consider the general linear hyperbolic equation in two independent variables which is reduced to the first canonical form (see Paragraph 14.1.1-4):
∂2w
∂x∂y + a(x, y) ∂w
∂x + b(x, y) ∂w
∂y + c(x, y)w = f (x, y), (14.8.4.1)
where a(x, y), b(x, y), c(x, y), and f (x, y) are continuous functions.
Let a segment of a curve in the xy-plane be defined by
where ϕ(x) is continuously differentiable, with ϕ (x)≠ 0and ϕ (x)≠∞.
Trang 3The generalized Cauchy problem for equation (14.8.4.1) with initial conditions defined
along a curve (14.8.4.2) is stated as follows: find a solution to equation (14.8.4.1) that satisfies the conditions
w (x, y) | y=ϕ(x) = g(x), ∂w
∂x
y=ϕ(x) = h1(x),
∂w
∂y
y=ϕ(x) = h2(x), (14.8.4.3)
where g(x), h1(x), and h2(x) are given continuous functions, related by the compatibility
condition
g
x (x) = h1(x) + h2(x)ϕ x (x). (14.8.4.4)
Basic property of the generalized Cauchy problem: the value of the solution at any point
M (x0, y0) depends only on the values of the functions g(x), h1(x), and h2(x) on the arc
AB , cut off on the given curve (14.8.4.2) by the characteristics x = x0 and y = y0, and
on the values of a(x, y), b(x, y), c(x, y), and f (x, y) in the curvilinear triangle AM B; see Fig 14.3 The domain of influence on the solution at M (x0, y0) is shaded for clarity
characteristics
B
A
y = φ( ) x y
x x
y
0
0
O
M x y( ,0 0)
Figure 14.3 Domain of influence of the solution to the generalized Cauchy problem at a point M
Remark 1 Rather than setting two derivatives in the boundary conditions (14.8.4.3), it suffices to set either of them, with the other being uniquely determined from the compatibility condition (14.8.4.4).
Remark 2 Instead of the last two boundary conditions in (14.8.4.3), the value of the derivative along the normal to the curve (14.8.4.2) can be used:
∂w
∂n
1+ [ϕ x (x)]2
∂w
∂y – ϕ x (x) ∂w
∂x
y=ϕ(x)
= h3(x). (14 8 4 5 )
Denoting w x | y=ϕ(x) = h1(x) and w y | y=ϕ(x) = h2(x), we have
h2(x) – ϕ x (x)h1(x) = h3(x)
1+ [ϕ x (x)]2 (14 8 4 6 )
The functions h1(x) and h2(x) can be found from (14.8.4.4) and (14.8.4.6) Further substituting their
expres-sions into (14.8.4.3), one arrives at the standard formulation of the generalized Cauchy problem, where the compatibility condition for initial data (14.8.4.4) will be satisfied automatically.
14.8.4-2 Riemann function
A Riemann function, R = R(x, y; x0, y0), corresponding to equation (14.8.4.1) is defined
as a solution to the equation
∂2R
∂x∂y – ∂
∂x
*
a (x, y) R+– ∂
∂y
*
b (x, y) R++ c(x, y) R =0 (14.8.4.7)
Trang 4that satisfies the conditions
R = exp
y
y0
a (x0, ξ) dξ
at x = x0, R = exp
x
x0
b (ξ, y0) dξ
at y = y0
(14.8.4.8)
at the characteristics x = x0and y = y0 Here, (x0, y0) is an arbitrary point from the domain
of equation (14.8.4.1) The x0and y0appear in problem (14.8.4.7)–(14.8.4.8) as parameters
in the boundary conditions only
THEOREM If the functions a, b, c and the partial derivatives a x , b y are all continuous, then the Riemann function R(x, y; x0, y0) exists Moreover, the function R(x0, y0, x, y),
obtained by swapping the parameters and the arguments, is a solution to the homogeneous
equation (14.8.4.1), with f =0
Remark It is significant that the Riemann function depends on neither the shape of the curve (14.8.4.2) nor the initial data set on it (14.8.4.3).
Example 1 The Riemann function for the equation w xy= 0 is justR≡ 1
Example 2 The Riemann function for the equation
is expressed via the Bessel function J0(z) as
R = J0 4c (x0 – x)(y0 – y)
.
Remark Any linear constant-coefficient partial differential equation of the parabolic type in two inde-pendent variables can be reduced to an equation of the form (14.8.4.9); see Paragraph 14.1.1-6.
14.8.4-3 Solution of the generalized Cauchy problem via the Riemann function Given a Riemann function, the solution to the generalized Cauchy problem (14.8.4.1)–
(14.8.4.3) at any point (x0, y0) can written as
w (x0, y0) = 1
2(w R) A+
1
2(w R) B+
1 2
AB
R ∂w
∂x – w ∂ R
∂x +2bw R
dx
– 1 2
AB
R ∂w
∂y – w ∂R
∂y +2awR
dy+
ΔAMB fR dx dy.
The first two terms on the right-hand side are evaluated as the points A and B The third and fourth terms are curvilinear integrals over the arc AB; the arc is defined by equation
(14.8.4.2), and the integrands involve quantities defined by the initial conditions (14.8.4.3)
The last integral is taken over the curvilinear triangular domain AM B.
14.8.5 Goursat Problem (a Problem with Initial Data of
Characteristics)
14.8.5-1 Statement of the Goursat problem Basic property of the solution
The Goursat problem for equation (14.8.4.1) is stated as follows: find a solution to equation
(14.8.4.1) that satisfies the conditions at characteristics
w (x, y) | x=x1 = g(y), w (x, y) | y=y1 = h(x), (14.8.5.1)
Trang 5where g(y) and h(x) are given continuous functions that match each other at the point of
intersection of the characteristics, so that
g (y1) = h(x1)
Basic properties of the Goursat problem: the value of the solution at any point M (x0, y0)
depends only on the values of g(y) at the segment AN (which is part of the characteristic
x = x1), the values of h(x) at the segment BN (which is part of the characteristic y = y1),
and the values of the functions a(x, y), b(x, y), c(x, y), and f (x, y) in the rectangle N AM B; see Fig 14.4 The domain of influence on the solution at the point M (x0, y0) is shaded for clarity
N x y( , )
characteristics
B A
y
x x x
y
y = y
x = x
y
0 1
1
M x y( ,0 0)
1
0
1 1
1
O
Figure 14.4 Domain of influence of the solution to the Goursat problem at a point M
14.8.5-2 Solution representation for the Goursat problem via the Riemann function Given a Riemann function (see Paragraph 14.8.4-2), the solution to the Goursat problem
(14.8.4.1), (14.8.5.1) at any point (x0, y0) can be written as
w (x0, y0) = (w R) N+
A
N R g
y + bg
dy+
B
N R h
x + ah
dx+
NAMB f R dx dy.
The first term on the right-hand side is evaluated at the point of intersection of the
charac-teristics (x1, y1) The second and third terms are integrals along the characteristics y = y1 (x1≤x≤x0) and x = x1(y1≤y≤y0); these involve the initial data of (14.8.5.1) The last
in-tegral is taken over the rectangular domain N AM B defined by the inequalities x1≤x≤x0,
y1≤y≤y0.
The Goursat problem for hyperbolic equations reduced to the second canonical form (see Paragraph 14.1.1-4) is treated similarly
Example Consider the Goursat problem for the wave equation
∂2w
∂t2 – a2∂
2w
∂x2 = 0 with the boundary conditions prescribed on its characteristics
w = f (x) for x – at =0 ( 0 ≤x≤b),
w = g(x) for x + at =0 ( 0 ≤x≤c), (14.8 5 2 )
where f (0) = g(0 ).
Substituting the values set on the characteristics (14.8.5.2) into the general solution of the wave equation,
w = ϕ(x – at) + ψ(x + at), we arrive to a system of linear algebraic equations for ϕ(x) and ψ(x) As a result,
the solution to the Goursat problem is obtained in the form
w (x, t) = f
x + at
2
+ g
x – at
2
– f (0 ).
The solution propagation domain is the parallelogram bounded by the four lines
x – at =0 , x + at =0 , x – at =2c, x + at =2b.
Trang 614.9 Boundary Value Problems for Elliptic Equations
with Two Space Variables
14.9.1 Problems and the Green’s Functions for Equation
a(x) ∂2w
∂x2 +
∂2w
∂y2 + b(x)
∂w
∂x +c(x)w = –Φ(x, y)
14.9.1-1 Statements of boundary value problems
Consider two-dimensional boundary value problems for the equation
a (x) ∂
2w
∂x2 +
∂2w
∂y2 + b(x)
∂w
∂x + c(x)w = – Φ(x, y) (14.9.1.1)
with general boundary conditions in x,
α1∂w ∂x – β1w = f1(y) at x = x1,
α2∂w ∂x + β2w = f2(y) at x = x2,
(14.9.1.2)
and different boundary conditions in y It is assumed that the coefficients of equation
(14.9.1.1) and the boundary conditions (14.9.1.2) meet the requirements
a (x), b(x), c(x) are continuous (x1≤x≤x2); a>0, |α1|+|β1|>0, |α2|+|β2|>0
14.9.1-2 Relations for the Green’s function
In the general case, the Green’s function can be represented as
G (x, y, ξ, η) = ρ(ξ)
∞
n=1
u n (x)u n (ξ)
u n 2 Ψn (y, η; λ n) (14.9.1.3) Here,
ρ (x) = 1
a (x)exp
b (x)
a (x) dx
, u n 2 =
x2
x1
ρ (x)u2n (x) dx, (14.9.1.4)
and the λ n and u n (x) are the eigenvalues and eigenfunctions of the homogeneous boundary
value problem for the ordinary differential equation
a (x)u xx + b(x)u x + [λ + c(x)]u =0, (14.9.1.5)
α1u
x – β1u=0 at x = x1, (14.9.1.6)
α2u
x + β2u=0 at x = x2 (14.9.1.7) The functionsΨn for various boundary conditions in y are specified in Table 14.8.
Equation (14.9.1.5) can be rewritten in self-adjoint form as
[p(x)u x] x + [λρ(x) – q(x)]u =0, (14.9.1.8)
Trang 7TABLE 14.8 The functions Ψn in (14.9.1.3) for various boundary conditions.* Notation: σ n=√
λ n
Domain Boundary conditions Function Ψn (y, η; λ n)
–∞ < y < ∞ |w| <∞ for y → ∞ 21
σ n e
–σ n|y–η|
σ n
e–σ n y sinh(σ n ) for y > η,
e–σ n η sinh(σ n ) for η > y
σ n
e–σ n y cosh(σ n ) for y > η,
e–σ n η cosh(σ n ) for η > y
0 ≤y<∞ ∂ y w – β3w= 0 for y =0 1
σ n (σ n + β3)
e–σ n y [σ n cosh(σ n ) + β3 sinh(σ n )] for y > η,
e–σ n η [σ n cosh(σ n ) + β3sinh(σ n )] for η > y
0 ≤y≤h w= 0 at y= 0 ,
w= 0 at y = h
1
σ n sinh(σ n h)
sinh(σ n ) sinh[σ n (h – y)] for y > η, sinh(σ n ) sinh[σ n (h – η)] for η > y
0 ≤y≤h ∂ y w= 0 at y= 0 ,
∂ y w= 0 at y = h
1
σ n sinh(σ n h)
cosh(σ n ) cosh[σ n (h – y)] for y > η, cosh(σ n ) cosh[σ n (h – η)] for η > y
0 ≤y≤h w= 0 at y= 0 ,
∂ y w= 0 at y = h
1
σ n cosh(σ n h)
sinh(σ n ) cosh[σ n (h – y)] for y > η, sinh(σ n ) cosh[σ n (h – η)] for η > y where the functions p(x) and q(x) are given by
p (x) = exp
b (x)
a (x) dx
, q(x) = – c (x)
a (x)exp
b (x)
a (x) dx
,
and ρ(x) is defined in (14.9.1.4).
The eigenvalue problem (14.9.1.8), (14.9.1.6), (14.9.1.7) possesses the following prop-erties:
1◦ All eigenvalues λ1, λ2, are real and λ n → ∞ as n → ∞.
2◦ The system of eigenfunctions{u1(x), u2(x), }is orthogonal on the interval x1≤x≤x2
with weight ρ(x), that is,
x2
x1
ρ (x)u n (x)u m (x) dx =0 for n≠m
3◦ If the conditions
q (x)≥ 0, α1β1≥ 0, α2β2 ≥ 0 (14.9.1.9)
are satisfied, there are no negative eigenvalues If q ≡ 0and β1 = β2 = 0, then the least
eigenvalue is λ0 = 0and the corresponding eigenfunction is u0 = const; in this case, the
summation in (14.9.1.3) must start with n =0 In the other cases, if conditions (14.9.1.9) are satisfied, all eigenvalues are positive; for example, the first inequality in (14.9.1.9) holds
if c(x)≤ 0
Subsection 12.2.5 presents some relations for estimating the eigenvalues λ nand
eigen-functions u n (x).
* For unbounded domains, the condition of boundedness of the solution as y → ∞ is set; in Table 14.8,
this condition is omitted.
... x1), the values of h(x) at the segment BN (which is part of the characteristic y = y1),and the values of the functions a(x, y), b(x, y), c(x, y), and f (x, y) in the... properties of the Goursat problem: the value of the solution at any point M (x0, y0)
depends only on the values of g(y) at the segment AN (which is part of. .. right-hand side are evaluated as the points A and B The third and fourth terms are curvilinear integrals over the arc AB; the arc is defined by equation
(14.8.4.2), and the integrands