Movable Singularities of Solutions of Ordinary Differential Equations.. Singular points of solutions to ordinary differential equations can be fixed or movable.. The coordinates of fixed
Trang 1Further, differentiating (12.3.3.1) yields
y
xxx = f x (x, y, y x ) + f y (x, y, y x )y x + f y
x (x, y, y x )y xx (12.3.3.5)
On substituting x = x0, the initial conditions (12.3.3.2), and the expression of y xx (x0)
of (12.3.3.4) into the right-hand side of equation (12.3.3.5), we calculate the value of the third derivative:
y
xxx (x0) = f x (x0, y0, y1) + f y (x0, y0, y1)y1+ f (x0, y0, y1)f y x (x0, y0, y1)
The subsequent derivatives of the unknown are determined likewise
The thus obtained solution (12.3.3.3) can only be used in a small neighborhood of the
point x = x0
Example 1 Consider the following Cauchy problem for a second-order nonlinear equation:
y(0) = y
Substituting the initial values of the unknown and its derivative (12.3.3.7) into equation (12.3.3.6) yields the initial value of the second derivative:
Differentiating equation (12.3.3.6) gives
y xxx = yy xx + (y x )2+ 3y 2y
Substituting here the initial values from (12.3.3.7) and (12.3.3.8), we obtain the initial condition for the third derivative:
Differentiating (12.3.3.9) followed by substituting (12.3.3.7), (12.3.3.8), and (12.3.3.10), we find that
y xxxx (0) = 24 (12.3.3.11)
On substituting the initial data (12.3.3.7), (12.3.3.8), (12.3.3.10), and (12.3.3.11) into (12.3.3.3), we arrive at
the Taylor series expansion of the solution about x =0 :
y= 1+ x + x2+ x3+ x4+· · · (12.3.3.12) This geometric series is convergent only for |x| < 1.
12.3.3-2 Pad´e approximants
Suppose the k +1 leading coefficients in the Taylor series expansion of a solution to
a differential equation about the point x = 0 are obtained by the method presented in Paragraph 12.3.3-1, so that
y k+1(x) = a0+ a1x+· · · + a k x k. (12.3.3.13)
The partial sum (12.3.3.13) pretty well approximates the solution at small x but is poor for intermediate and large values of x, since the series can be slowly convergent or even divergent This is also related to the fact that y k → ∞ as x → ∞, while the exact solution
can well be bounded
In many cases, instead of the expansion (12.3.3.13), it is reasonable to consider a
Pad´e approximant P M N (x), which is the ratio of two polynomials of degree N and M ,
specifically,
P N
M (x) = A0+ A1x+· · · + A N x N
1+ B1x+· · · + B M x M , where N + M = k. (12.3.3.14)
Trang 2The coefficients A1, , A N and B1, , B M are selected so that the k +1 leading terms in the Taylor series expansion of (12.3.3.14) coincide with the respective terms of the expansion (12.3.3.13) In other words, the expansions (12.3.3.13) and (12.3.3.14) must be
asymptotically equivalent as x →0
In practice, one usually takes N = M (the diagonal sequence) It often turns out that formula (12.3.3.14) pretty well approximates the exact solution on the entire range of x (for sufficiently large N ).
Example 2 Consider the Cauchy problem (12.3.3.6)–(12.3.3.7) again The Taylor series expansion of
the solution about x =0 has the form (12.3.3.12) This geometric series is convergent only for |x| < 1 The diagonal sequence of Pad´e approximants corresponding to series (12.3.3.12) is
P1(x) = 1
1– x, P
2(x) = 1
1– x, P
3(x) = 1
1– x. (12.3.3.15)
It is not difficult to verify that the function y(x) = 1
1– x is the exact solution of the Cauchy
prob-lem (12.3.3.6)–(12.3.3.7) Hence, in this case, the diagonal sequence of Pad´e approximants recovers the exact solution from only a few terms in the Taylor series.
Example 3 Consider the Cauchy problem for a second-order nonlinear equation:
y xx= 2yy
x; y(0) =0, y x (0) = 1 (12.3.3.16) Following the method presented in Paragraph 12.3.3-1, we obtain the Taylor series expansion of the solution
to problem (12.3.3.16) in the form
y(x) = x + 13x3+152x5+31517x7+· · · (12.3.3.17)
The exact solution of problem (12.3.3.16) is given by y(x) = tan x Hence it has singularities at x =
1
2 (2n + 1)π However, any finite segment of the Taylor series (12.3.3.17) does not have any singularities With series (12.3.3.17), we construct the diagonal sequence of Pad´e approximants:
P2(x) = 3x
3– x2 , P3(x) = x(x
2 – 15) 3(2x 2 – 5), P4(x) =
5x(21 – 2x 2 )
x4– 45x 2 + 105. (12.3.3.18) These Pad´e approximants have singularities (at the points where the denominators vanish):
x 1.571 and x 6.522 for P4(x).
It is apparent that the Pad´e approximants are attempting to recover the singularities of the exact solution at
x = π/2 and x = 3π/2.
In Fig 12.2, the solid line shows the exact solution of problem (12.3.3.16), the dashed line corresponds
to the four-term Taylor series solution (12.3.3.17), and the dot-and-dash line depicts the Pad´e approximants
(12.3.3.18) It is evident that the Pad´e approximant P4(x) gives an accurate numerical approximation of the
exact solution on the interval |x| ≤ 2 ; everywhere the error is less than 1%, except for a very small neighborhood
of the point x = π/2 (the error is 1% for x = 1.535and 0.84% for x = 2).
12.3.4 Movable Singularities of Solutions of Ordinary Differential
Equations Painlev ´e Transcendents
12.3.4-1 Preliminary remarks Singular points of solutions
1◦ Singular points of solutions to ordinary differential equations can be fixed or movable.
The coordinates of fixed singular points remain the same for different solutions of an equation.* The coordinates of movable singular points vary depending on the particular solution selected (i.e., they depend on the initial conditions)
* Solutions of linear ordinary differential equations can only have fixed singular points, and their positions are determined by the singularities of the equation coefficients.
Trang 30.5 1 1.5 2 1
4 8 12
1.5 2
4 8
12
y x( ) = tanx
y x( ) =P x( )
y x( ) =P x( )
y x ( ) = x+ x + 1 3
2
3
2
3
3
y
x O
Figure 12.2 Comparison of the exact solution to problem (12.3.3.16) with the approximate truncated series
solution (12.3.3.17) and associated Pad´e approximants (12.3.3.18).
Listed below are simple examples of first-order ordinary differential equations and their solutions having movable singularities:
y
z = –y2 y=1/(z – z0) movable pole
y
z =1/y y=2√ z – z0 algebraic branch point
y
z = e–y y = ln(z – z0) logarithmic branch point
y
z = –y ln2y y= exp[1/(z – z0)] essential singularity
Algebraic branch points, logarithmic branch points, and essential singularities are called
movable critical points.
2◦ The Painlev´e equations arise from the classification of the following second-order
differential equations over the complex plane:
y
zz = R(z, y, y z),
where R = R(z, y, w) is a function rational in y and w and analytic in z It was shown
by P Painlev´e (1897–1902) and B Gambier (1910) that all equations of this type whose solutions do not have movable critical points (but are allowed to have fixed singular points and movable poles) can be reduced to 50 classes of equations Moreover, 44 classes out of them are integrable by quadrature or admit reduction of order The remaining 6 equations
are irreducible; these are known as the Painlev´e equations or Painlev´e transcendents, and their solutions are known as the Painlev´e transcendental functions.
The canonical forms of the Painlev´e transcendents are given below in Paragraphs 12.3.4-2 through 12.3.4-6 Solutions of the first, second, and fourth Painlev´e transcendents have movable poles (no fixed singular points) Solutions of the third and fifth Painlev´e
transcendents have two fixed logarithmic branch points, z =0 and z = ∞ Solutions of the sixth Painlev´e transcendent have three fixed logarithmic branch points, z =0, z =1, and
z=∞.
It is significant that the Painlev´e equations often arise in mathematical physics
Trang 412.3.4-2 First Painlev´e transcendent.
1◦ The first Painlev´e transcendent has the form
y
The solutions of the first Painlev´e transcendent are single-valued functions of z.
The solutions of equation (12.3.4.1) can be presented, in the vicinity of movable pole zp,
in terms of the series
y= 1
(z – zp)2 +
∞
n=2
a n (z – zp)n,
a2 = –101zp, a3 = –16, a4= C, a5=0, a6= 3001 z2
p,
where zp and C are arbitrary constants; the coefficients a j (j ≥ 7) are uniquely defined
in terms of zp and C.
2◦ In a neighborhood of a fixed point z = z0, the solution of the Cauchy problem for
the first Painlev´e transcendent (12.3.4.1) can be represented by the Taylor series (see Paragraph 12.3.3-1):
y = A+B(z –z0)+12(6A2+z
0)(z –z0)2+16(12AB+1)(z –z0)3+12(6A3+B2+Az
0)(z –z0)4+· · · , where A and B are initial data of the Cauchy problem, so that y|z=z0= A and y z|z=z0 = B.
Remark The solutions of the Cauchy problems for the second and fourth Painlev´e transcendents can be expressed likewise (fixed singular points should be excluded from consideration for the remaining Painlev´e transcendents).
3◦ For large values of |z|, the following asymptotic formula holds:
y ∼ z1 2℘ 4
5z5 4– a; 12, b
,
where the elliptic Weierstrass function ℘(ζ;12, b) is defined implicitly by the integral
ζ= d℘
4℘3–12℘ – b;
a and b are some constants.
12.3.4-3 Second Painlev´e transcendent
1◦ The second Painlev´e transcendent has the form
y
zz =2y3+ zy + α. (12.3.4.2)
The solutions of the second Painlev´e transcendent are single-valued functions of z.
The solutions of equation (12.3.4.2) can be represented, in the vicinity of a movable
pole zp, in terms of the series
y= m
z – zp +
∞
n=1
b n (z – zp)n,
b1= –16mzp, b2= –14(m + α), b3= C, b4 = 721zp(m +3α),
b5= 30241
(27+81α2–2z3
p)m +108α–216Czp
,
where zp and C are arbitrary constants, m = 1, and the coefficients b n (n ≥ 6) are
uniquely defined in terms of zp and C.
Trang 52◦ For fixed α, denote the solution by y(z, α) Then the following relation holds:
y(z, –α) = –y(z, α), (12.3.4.3)
while the solutions y(z, α) and y(z, α –1) are related by the B¨acklund transformations:
y(z, α –1) = –y(z, α) + 2α–1
2y
z (z, α) –2y2(z, α) – z,
y(z, α) = –y(z, α –1) – 2α–1
2y
z (z, α –1) +2y2(z, α –1) + z.
(12.3.4.4)
Therefore, in order to study the general solution of equation (12.3.4.2) with arbitrary α, it
is sufficient to construct the solution for all α out of the band 0 ≤Re α < 12
Three solutions corresponding to α and α 1 are related by the rational formulas
y α+1= –(y α–1+ y α)(4y3
α+2zy α+2α+1) + (2α–1)y α
2(y α–1+ y α)(2y2
α + z) +2α–1 ,
where y α stands for y(z, α).
The solutions y(z, α) and y(z, –α –1) are related by the B¨acklund transformations:
y(z, –α –1) = y(z, α) + 2α+1
2y
z (z, α) +2y2(z, α) + z,
y(z, α) = y(z, –α –1) – 2α+1
2y
z (z, –α –1) +2y2(z, –α –1) + z.
3◦ For α =0, equation (12.3.4.2) has the trivial solution y =0 Taking into account this fact and relations (12.3.4.3) and (12.3.4.4), we find that the second Painlev´e transcendent
with α = 1, 2, has the rational particular solutions
y(z, 1) =
1
z, y(z, 2) = 1
z – 3z2
z3+4
, For α = 12, equation (12.3.4.2) admits the one-parameter family of solutions:
y(z,12) = –w z
w , where w=√
z
*
C1J1 3
√
2
3 z3 2
+ C2Y1 3
√
2
3 z3 2
+
(12.3.4.5)
(Here, the function w is a solution of the Airy equation, w zz + 12zw = 0.) It follows
from (12.3.4.3)–(12.3.4.5) that the second Painlev´e transcendent for all α = n + 12 with
n=0, 1, 2, has a one-parameter family of solutions that can be expressed in terms
of Bessel functions
12.3.4-4 Third Painlev´e transcendent
1◦ The third Painlev´e transcendent has the form
y
zz = (y
z)2
y – y z
z + 1
z (αy2+ β) + γy3+ δ
y (12.3.4.6)
In terms of the new independent variable ζ defined by z = e ζ, the solutions of the
transformed equation will be single-valued functions of ζ.
In some special cases, equation (12.3.4.6) can be integrated by quadrature
Trang 62◦ Any solution of the Riccati equation
y
z = ky2+ α kz – k y + c, (12.3.4.7)
where k2= γ, c2= –δ, kβ + c(α –2k) =0, is a solution of equation (12.3.4.6) Substituting
z = λτ , y = – u z
ku , where λ2 = 1
kc, into (12.3.4.7), we obtain a linear equation
u
ττ+ k kτ – α u τ + u =0, whose general solution is expressed in terms of Bessel functions:
u = τ2α k*
C1J α
2k (τ ) + C2Y2α k (τ )
+
12.3.4-5 Fourth Painlev´e transcendent
1◦ The fourth Painlev´e transcendent has the form
y
zz = (y
z)2
2y + 3
2y3+4zy2+2(z2– α)y +
β
y (12.3.4.8)
The solutions of the fourth Painlev´e transcendent are single-valued functions of z.
The Laurent-series expansion of the solution of equation (12.3.4.8) in the vicinity of a
movable pole zp is given by
y = m
z – zp – zp–
m
3 (zp2+2α–4m)(z – zp) + C(z – zp)2+
∞
j=3
a j (z – zp) ,
where m = 1; zp and C are arbitrary constants; and the a j (j ≥ 3) are uniquely defined
in terms of α, β, zp, and C.
2◦ Two solutions of equation (12.3.4.8) corresponding to different values of the parameters
α and β are related to each other by the B¨acklund transformations:
2y = 21
sy (y z – q –2szy – sy2), q2= –2β,
y= – 1
2s 2y(2y z – p +2sz 2y + s2y2), p2= –22β,
2β= –(2αs –1– 12p)2, 4α= –2s–22α –3sp, where y = y(z, α, β), 2y = 2y(z, 2α, 2β), and s is an arbitrary parameter.
3◦ If the condition β +2(1+ αm)2 =0, where m = 1, is satisfied, then every solution
of the Riccati equation
y
z = my2+2mzy–2(1+ αm)
is simultaneously a solution of the fourth Painlev´e equation (12.3.4.8)
Trang 712.3.4-6 Fifth and sixth Painlev´e transcendents.
1◦ The fifth Painlev´e transcendent has the form
y
zz = 3y–1
2y(y –1)(y
z)2–
y z
z + (y –1)2
z2
αy+ β
y
+ γ y
z + δy(y +1)
y–1 .
2◦ The sixth Painlev´e transcendent has the form
y
zz = 1
2
1
y + 1
y–1 +
1
y – z
(y z)2–
1
z + 1
z–1 +
1
y – z
y z
+ y(y –1)(y – z)
z2(z –1)2
*
α + β z
y2 + γ
z–1
(y –1)2 + δ
z(z –1)
(y – z)2
+
For details about these equations, see the list of references given at the end of the current chapter
12.3.5 Perturbation Methods of Mechanics and Physics
12.3.5-1 Preliminary remarks A summary table of basic methods
Perturbation methods are widely used in nonlinear mechanics and theoretical physics for
solving problems that are described by differential equations with a small parameter ε.
The primary purpose of these methods is to obtain an approximate solution that would be equally suitable at all (small, intermediate, and large) values of the independent variable as
ε →0
Equations with a small parameter can be classified according to the following:
(i) the order of the equation remains the same at ε =0;
(ii) the order of the equation reduces at ε =0
For the first type of equations, solutions of related problems* are sufficiently smooth (little
varying as ε decreases) The second type of equation is said to be degenerate at ε = 0,
or singularly perturbed In related problems, thin boundary layers usually arise whose
thickness is significantly dependent on ε; such boundary layers are characterized by high
gradients of the unknown
All perturbation methods have a limited domain of applicability; the possibility of using one or another method depends on the type of equations or problems involved The most commonly used methods are summarized in Table 12.3 (the method of regular series expansions is set out in Paragraph 12.3.5-2) In subsequent paragraphs, additional remarks and specific examples are given for some of the methods In practice, one usually confines oneself to few leading terms of the asymptotic expansion
In many problems of nonlinear mechanics and theoretical physics, the independent
variable is dimensionless time t Therefore, in this subsection we use the conventional t
(0 ≤t<∞) instead of x.
* Further on, we assume that the initial and/or boundary conditions are independent of the parameter ε.
...Therefore, in order to study the general solution of equation (12.3.4.2) with arbitrary α, it
is sufficient to construct the solution for all α out of the band ≤Re...
The solutions of the fourth Painlev´e transcendent are single-valued functions of z.
The Laurent-series expansion of the solution of equation (12.3.4.8) in the vicinity of a
movable...
+
For details about these equations, see the list of references given at the end of the current chapter
12.3.5 Perturbation Methods of Mechanics and Physics
12.3.5-1