First, second, third, and mixed boundary value problems x1≤x≤x2.. Nonhomogeneous boundary conditions can be reduced to homogeneous ones by the change of variable z = A2x2+A 1x +A0+y the
Trang 1where f = f (x), k = tan
π
2m
3◦ Let m be an odd integer Then,
y1=
⎧
⎪
⎪
|f (x)|– 1 4cos*
–1
ε
x
0
|f (x)|dx+ π
4
+
if x <0,
1
2k–1[f (x)]–1 4exp
*1
ε
x
0
f (x) dx
+
if x >0,
y2=
⎧
⎪
⎪
|f (x)|– 1 4cos*
–1
ε
x
0
|f (x)|dx– π
4
+
if x <0,
k [f (x)]–1 4exp*
–1
ε
x
0
f (x) dx+
if x >0,
where f = f (x), k = sin
π
2m
12.2.3-4 Equations not containing y x Equation coefficients are dependent on ε.
Consider an equation of the form
ε2y
xx – f (x, ε)y =0 (12.2.3.4)
on a closed interval a≤x≤b under the condition that f ≠ 0 Assume that the following asymptotic relation holds:
f (x, ε) =
∞
k=0
f k (x)ε k, ε →0
Then the leading terms of the asymptotic expansions of the fundamental system of solutions
of equation (12.2.3.4) are given by the formulas
y1= f0–1 4(x) exp
*
–1
ε
f0(x) dx + 1
2
f
1(x)
√
f0(x) dx
+
1+ O(ε)
,
y2= f0–1 4(x) exp
*1
ε
f0(x) dx + 1
2
f1(x)
√
f0(x) dx
+
1+ O(ε)
12.2.3-5 Equations containing y x
1◦ Consider an equation of the form
εy
xx + g(x)y x + f (x)y =0
on a closed interval 0 ≤ x≤ 1 With g(x) >0, the asymptotic solution of this equation,
satisfying the boundary conditions y(0) = C1 and y(1) = C2, can be represented in the form
y = (C1– kC2) exp
–ε–1g(0)x
+ C2exp* 1
x
f (x)
g (x) dx
+
+ O(ε), where k = exp
* 1
0
f (x)
g (x) dx
+
Trang 2
2◦ Now let us take a look at an equation of the form
ε2y
xx + εg(x)y x + f (x)y =0 (12.2.3.5)
on a closed interval a≤x≤b Assume
D (x)≡[g(x)]2–4f (x)≠ 0 Then the leading terms of the asymptotic expansions of the fundamental system of solutions
of equation (12.2.3.5), as ε →0, are expressed by
y1=|D (x)|– 1 4exp*
– 1
2ε
D (x) dx – 1
2
g
x (x)
√
D (x) dx
+
1+ O(ε)
,
y2=|D (x)|– 1 4exp* 1
2ε
D (x) dx – 1
2
g
x (x)
√
D (x) dx
+
1+ O(ε)
12.2.3-6 Equations of the general form
The more general equation
ε2y
xx + εg(x, ε)y x + f (x, ε)y =0
is reducible, with the aid of the substitution y = w exp
– 1
2ε
g dx
, to an equation of the form (12.2.3.4),
ε2w
xx + (f – 14g2– 12εg x )w =0,
to which the asymptotic formulas given above in Paragraph 12.2.3-4 are applicable
12.2.4 Boundary Value Problems
12.2.4-1 First, second, third, and mixed boundary value problems (x1≤x≤x2).
We consider the second-order nonhomogeneous linear differential equation
y
xx + f (x)y x + g(x)y = h(x). (12.2.4.1)
1◦ The first boundary value problem: Find a solution of equation (12.2.4.1) satisfying the
boundary conditions
y = a1 at x = x1, y = a2 at x = x2 (12.2.4.2)
(The values of the unknown are prescribed at two distinct points x1 and x2.)
2◦ The second boundary value problem: Find a solution of equation (12.2.4.1) satisfying
the boundary conditions
y
x = a1 at x = x1, y
x = a2 at x = x2. (12.2.4.3)
(The values of the derivative of the unknown are prescribed at two distinct points x1 and x2.)
Trang 33◦ The third boundary value problem: Find a solution of equation (12.2.4.1) satisfying the
boundary conditions
y
x – k1y = a1 at x = x1,
y
x + k2y = a2 at x = x2.
(12.2.4.4)
4◦ The third boundary value problem: Find a solution of equation (12.2.4.1) satisfying the
boundary conditions
y = a1 at x = x1, y
x = a2 at x = x2. (12.2.4.5) (The unknown itself is prescribed at one point, and its derivative at another point.)
Conditions (12.2.4.2), (12.2.4.3), (12.2.4.4), and (12.2.4.5) are called homogeneous if
a1 = a2=0
12.2.4-2 Simplification of boundary conditions The self-adjoint form of equations
1◦ Nonhomogeneous boundary conditions can be reduced to homogeneous ones by the
change of variable z = A2x2+A
1x +A0+y (the constants A2, A1, and A0 are selected using the method of undetermined coefficients) In particular, the nonhomogeneous boundary conditions of the first kind (12.2.4.2) can be reduced to homogeneous boundary conditions
by the linear change of variable
z = y – a2– a1
x2– x1(x – x1) – a1.
2◦ On multiplying by p(x) = exp*
f (x) dx+
, one reduces equation (12.2.4.1) to the self-adjoint form:
[p(x)y x ] x + q(x)y = r(x). (12.2.4.6) Without loss of generality, we can further consider equation (12.2.4.6) instead of
(12.2.4.1) We assume that the functions p, p x , q, and r are continuous on the inter-val x1≤x≤x2, and p is positive.
12.2.4-3 Green’s function Linear problems for nonhomogeneous equations
The Green’s function of the first boundary value problem for equation (12.2.4.6) with homogeneous boundary conditions (12.2.4.2) is a function of two variables G(x, s) that
satisfies the following conditions:
1◦ G(x, s) is continuous in x for fixed s, with x1≤x ≤x2 and x1≤s≤x2.
2◦ G(x, s) is a solution of the homogeneous equation (12.2.4.6), with r = 0, for all
x1< x < x2 exclusive of the point x = s.
3◦ G(x, s) satisfies the homogeneous boundary conditions G(x1, s) = G(x2, s) =0
4◦ The derivative G
x (x, s) has a jump of 1/p (s) at the point x = s, that is,
G
x (x, s)
x→s, x>s – G x (x, s)
x→s, x<s=
1
p (s).
For the second, third, and mixed boundary value problems, the Green’s function is de-fined likewise except that in3◦the homogeneous boundary conditions (12.2.4.3), (12.2.4.4),
and (12.2.4.5), with a1 = a2=0, are adopted, respectively
Trang 4The solution of the nonhomogeneous equation (12.2.4.6) subject to appropriate homo-geneous boundary conditions is expressed in terms of the Green’s function as follows:*
y (x) = x2
x1 G (x, s)r(s) ds.
12.2.4-4 Representation of the Green’s function in terms of particular solutions
We consider the first boundary value problem Let y1= y1(x) and y2 = y2(x) be linearly independent particular solutions of the homogeneous equation (12.2.4.6), with r =0, that satisfy the conditions
y1(x1) =0, y2(x2) =0 (Each of the solutions satisfies one of the homogeneous boundary conditions.)
The Green’s function is expressed in terms of solutions of the homogeneous equation
as follows:
G (x, s) =
⎧
⎪
⎪
⎩
y1(x)y2(s)
p (s)W (s) for x1≤x≤s,
y1(s)y2(x)
p (s)W (s) for s≤x≤x2,
(12.2.4.7)
where W (x) = y1(x)y2 (x) – y1 (x)y2(x) is the Wronskian determinant.
Remark Formula (12.2.4.7) can also be used to construct the Green’s functions for the second, third, and
mixed boundary value problems To this end, one should find two linearly independent solutions, y1 = y1(x) and y2= y2(x), of the homogeneous equation; the former satisfies the corresponding homogeneous boundary condition at x = x1and the latter satisfies the one at x = x2.
12.2.5 Eigenvalue Problems
12.2.5-1 Sturm–Liouville problem
Consider the second-order homogeneous linear differential equation
[p(x)y x ] x + [λs(x) – q(x)]y =0 (12.2.5.1) subject to linear boundary conditions of the general form
α1y
x + β1y=0 at x = x1,
α2y
x + β2y=0 at x = x2 (12.2.5.2)
It is assumed that the functions p, p x , s, and q are continuous, and p and s are positive
on an interval x1 ≤x≤x2 It is also assumed that |α1|+|β1|>0 and |α2|+|β2|>0
The Sturm–Liouville problem: Find the values λ n of the parameter λ at which problem (12.2.5.1), (12.2.5.2) has a nontrivial solution Such λ n are called eigenvalues and the cor-responding solutions y n = y n (x) are called eigenfunctions of the Sturm–Liouville problem
(12.2.5.1), (12.2.5.2)
* The homogeneous boundary value problem—with r(x) =0and a1= a2= 0—is assumed to have only the trivial solution.
Trang 512.2.5-2 General properties of the Sturm–Liouville problem (12.2.5.1), (12.2.5.2).
1◦ There are infinitely (countably) many eigenvalues All eigenvalues can be ordered so
that λ1< λ2< λ3<· · · Moreover, λ n → ∞ as n → ∞; hence, there can only be a finite
number of negative eigenvalues Each eigenvalue has multiplicity 1
2◦ The eigenfunctions are defined up to a constant factor Each eigenfunction y
n (x) has
precisely n –1zeros on the open interval (x1, x2)
3◦ Any two eigenfunctions y
n (x) and y m (x), n≠m , are orthogonal with weight s(x)
on the interval x1≤x≤x2:
x2
x1 s (x)y n (x)y m (x) dx =0 if n≠m
4◦ An arbitrary function F (x) that has a continuous derivative and satisfies the boundary
conditions of the Sturm–Liouville problem can be decomposed into an absolutely and uniformly convergent series in the eigenfunctions
F (x) =
∞
n=1
F n y n (x),
where the Fourier coefficients F n of F (x) are calculated by
F n= 1
y n 2
x2
x1 s (x)F (x)y n (x) dx, y n 2= x2
x1 s (x)y2n (x) dx.
5◦ If the conditions
q (x)≥ 0, α1β1≤ 0, α2β2 ≥ 0 (12.2.5.3)
hold true, there are no negative eigenvalues If q≡ 0 and β1= β2=0, the least eigenvalue
is λ1=0, to which there corresponds an eigenfunction y1= const In the other cases where conditions (12.2.5.3) are satisfied, all eigenvalues are positive
6◦ The following asymptotic formula is valid for eigenvalues as n → ∞:
λ n= π
2n2
Δ2 + O(1), Δ = x2
x1
s (x)
p (x) dx. (12.2.5.4) Paragraphs 12.2.5-3 through 12.2.5-6 will describe special properties of the Sturm– Liouville problem that depend on the specific form of the boundary conditions
Remark 1. Equation (12.2.5.1) can be reduced to the case where p(x)≡ 1 and s(x)≡ 1 by the change
of variables
ζ=
s (x)
p (x) dx, u (ζ) =
p (x)s(x) 1/4
y (x).
In this case, the boundary conditions are transformed to boundary conditions of similar form.
Remark 2 The second-order linear equation
ϕ2(x)y xx + ϕ1(x)y x + [λ + ϕ0(x)]y =0
can be represented in the form of equation (12.2.5.1) where p(x), s(x), and q(x) are given by
p (x) = exp
* ϕ
1(x)
ϕ2(x) dx
+
, s(x) = 1
ϕ2(x) exp
* ϕ
1(x)
ϕ2(x) dx
+
, q(x) = – ϕ0(x)
ϕ2(x) exp
* ϕ
1(x)
ϕ2(x) dx
+
.
Trang 6TABLE 12.2
Example estimates of the first eigenvalue λ1 in Sturm–Liouville problems with boundary conditions of the first
kind y(0) = y(1) =0 obtained using the Rayleigh–Ritz principle [the right-hand side of relation (12.2.5.6)]
Equation Test function λ1, approximate λ1 , exact
y xx + λ(1 + x2)–2y= 0 z = sin πx 15.337 15.0
y xx + λ(4 – x2) – 2y= 0 z = sin πx 135.317 134.837 [(1+ x)–1y x] x + λy =0 z = sin πx 7.003 6.772
1+ x y x
y xx + λ(1 + sin πx)y =0 z z = x(1 = sin πx – x) 0.54105π2
0.55204π2
0.54032π2
0.54032π2
12.2.5-3 Problems with boundary conditions of the first kind
Let us note some special properties of the Sturm–Liouville problem that is the first boundary value problem for equation (12.2.5.1) with the boundary conditions
y =0 at x = x1, y =0 at x = x2 (12.2.5.5)
1◦ For n → ∞, the asymptotic relation (12.2.5.4) can be used to estimate the
eigenval-ues λ n In this case, the asymptotic formula
y n (x)
y n =
Δ2p (x)s(x)
1 4
sin
πn
Δ
x
x1
s (x)
p (x) dx
+ O1
n
, Δ = x2
x1
s (x)
p (x) dx holds true for the eigenfunctions y n (x).
2◦ If q ≥ 0, the following upper estimate holds for the least eigenvalue (Rayleigh–Ritz
principle):
λ1≤
x2
x1
p (x)(z x )2+ q(x)z2
dx
x2
x1 s (x)z2dx , (12.2.5.6)
where z = z(x) is any twice differentiable function that satisfies the conditions z(x1) =
z (x2) =0 The equality in (12.2.5.6) is attained if z = y1(x), where y1(x) is the eigenfunction corresponding to the eigenvalue λ1 One can take z = (x–x1)(x2–x) or z = sin*π (x – x1)
x2– x1
+
in (12.2.5.6) to obtain specific estimates
It is significant to note that the left-hand side of (12.2.5.6) usually gives a fairly precise estimate of the first eigenvalue (see Table 12.2)
3◦ The extension of the interval [x1, x2] leads to decreasing in eigenvalues.
4◦ Let the inequalities
0< pmin≤p (x)≤pmax, 0< smin≤s (x)≤smax, 0< qmin≤q (x)≤qmax
be satisfied Then the following bilateral estimates hold:
pmin
smax
π2n2
(x2– x1)2 +
qmin
smax ≤λ n≤ pmax
smin
π2n2
(x2– x1)2 +
qmax
smin
Trang 75◦ In engineering calculations for eigenvalues, the approximate formula
λ n= π
2n2
Δ2 +
1
x2– x1
x2
x1
q (x)
s (x) dx, Δ = x2
x1
s (x)
p (x) dx (12.2.5.7)
may be quite useful This formula provides an exact result if p(x)s(x) = const and
q (x)/s(x) = const (in particular, for constant equation coefficients, p = p0, q = q0, and
s = s0) and gives a correct asymptotic behavior of (12.2.5.4) for any p(x), q(x), and s(x).
In addition, relation (12.2.5.7) gives two correct leading asymptotic terms as n → ∞ if
p (x) = const and s(x) = const [and also if p(x)s(x) = const].
6◦ Suppose p(x) = s(x) = 1 and the function q = q(x) has a continuous derivative The following asymptotic relations hold for eigenvalues λ n and eigenfunctions y n (x) as
n → ∞:
λ n = πn
x2– x1 +
1
πn Q (x1, x2) + O 1
n2
,
y n (x) = sin πn (x – x1)
x2– x1 –
1
πn
*
(x1– x)Q(x, x2) + (x2– x)Q(x1, x)
+ cosπn (x – x1)
x2– x1 + O
1
n2
, where
Q (u, v) = 1
2
v
u q (x) dx. (12.2.5.8)
7◦ Let us consider the eigenvalue problem for the equation with a small parameter
y
xx + [λ + εq(x)]y =0 (ε →0)
subject to the boundary conditions (12.2.5.5) with x1 = 0 and x2 = 1 We assume that
q (x) = q(–x).
This problem has the following eigenvalues and eigenfunctions:
λ n = π2n2– εA nn+ ε
2
π2
k n
A2
nk
n2– k2 + O(ε3), A nk =2 1
0 q (x) sin(πnx) sin(πkx) dx;
y n (x) = √
2 sin(πnx) – ε
√
2
π2
k n
A nk
n2– k2 sin(πkx) + O(ε2).
Here, the summation is carried out over k from1 to∞ The next term in the expansion
of y ncan be found in Nayfeh (1973)
12.2.5-4 Problems with boundary conditions of the second kind
Let us note some special properties of the Sturm–Liouville problem that is the second boundary value problem for equation (12.2.5.1) with the boundary conditions
y
x =0 at x = x1, y
x =0 at x = x2
1◦ If q >0, the upper estimate (12.2.5.6) is valid for the least eigenvalue, with z = z(x) being any twice-differentiable function that satisfies the conditions z x (x1) = z x (x2) = 0
The equality in (12.2.5.6) is attained if z = y1(x), where y1(x) is the eigenfunction corresponding to the eigenvalue λ1
... ε →0Then the leading terms of the asymptotic expansions of the fundamental system of solutions
of equation (12.2.3.4) are given by the formulas
y1=... p0, q = q0, and
s = s0) and gives a correct asymptotic behavior of (12.2.5.4) for any p(x), q(x), and s(x).
In addition, relation... problems for nonhomogeneous equations
The Green’s function of the first boundary value problem for equation (12.2.4.6) with homogeneous boundary conditions (12.2.4.2) is a function of two