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Файл:Additional Chapters of Higher Mathematics for Masters in Civill and Geotechnical Engineering. Учебное пособие по дополнительным разделам высшей матема
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Chapter 4. Ordinary differential equations
X
X
G
X
X
X
191
__________________________________________________________________________
Definition 4.3.4 (Reducible system)
System of differential equations (4.3.8) (with a non-constant matrix G ) is called reducible, if
there is a Lyapunov transformation ( )tL , such that matrix
appearing in the right-hand side of (4.3.27) is a constant (not necessary non-degenerate) matrix.
Remark 4.3.4
For an arbitrary matrix
special cases finding Lyapunov transformation that reduces matrix
via transformation (4.3.25), can be possible.
()tG , nothing can be said about its reducibility. However, for some
11−−
≡⋅⋅−⋅AL GLL L
4.3.5. A special case of reducibility (Floquet theory
of equations with periodic coefficients)
Proposition 4.3.9 (Reducibility of the periodic system)
Let matrix
be continuous, non-degenerate, and periodic with period
()tG
()()tp t+=GG
(4.3.27)
into a constant matrix
()tG
:
0p >
, (4.3.28)
Proof
then matrix ( )tG is reducible.
It should be noted at first that due to periodicity of matrix
of homogeneous Eq. (4.3.8) generates another fundamental solution
of (4.3.28)
GGG
Thus,
with some (not necessary non-degenerate) constant matrix A. Relation (4.3.30) implies
reducibility of periodic matrix ( )tG .
G
tp+
()
d
()()()()()
tp tpXtp tXtp
dt
is also a fundamental solution. That ensures (see Hartman, 2002)
+= +⋅ += ⋅ +GG
GG
()()exp
tp Xt p+= ⋅ A
()
()tG , any fundamental solution ( )
()
. (4.3.29)
(4.3.30)
. Indeed, in view
tp+
tG
Chapter 4. Ordinary differential equations
x
x
x
x
x
x
x
192
__________________________________________________________________________
4.3.6. A special case of reducibility for autonomous
equations
Basic concept 4.3.1
The described methodology is applicable to both linear and non-linear autonomous
homogeneous equations
()
with no explicit dependence of the independent variable t.
Introducing a new variable
n
Fx x=
;...; 0
()
(4.3.31)
()
n
all the derivatives
derivatives with respect to
(4.3.32) with respect to
Similarly, combining (4.3.33) and (4.3.32), gives
The described procedure can be continued, giving (rather complicated) expressions for
successive derivatives
Thus, Eq. (4.3.31) can be reduced to a lower order equation
which appears as non-autonomous one, due to presence of the (assumed independent) variable
.
,..., /
xdxdt≡
variable. Indeed, taking the first derivative of both sides of
t variable, yields
22
du du du du
=+=+
()k
xxu u
()
22
dx dx
in terms of function u and its derivatives up to
Fu ux−=
can be expressed in terms of function ()ux and its
du du
==
dx dx
2
dx dx
(1)
n
()
1
()ux x=
, (4.3.32)
xu
;...; ; 0
. (4.3.33)
2
⎛⎞
2
⎜⎟
⎝⎠
, (4.3.35)
. (4.3.34)
-th order.
1k −
Sometimes the lower-order Eq. (4.3.35) can be explicitly integrating, yielding the solution
. On obtaining
()ux
eliminating
Integrating both sides of the right equation in (4.3.36), yields
from the left-hand side:
dt
, the desired solution
()ux
dx dx
ux dt
= ⇒ = . (4.3.36)
()
dt u x
dx
∫
ux
()
can be deduced from (4.3.32) by
()
t
()
t
=
. (4.3.37)
Chapter 4. Ordinary differential equations
x
193
__________________________________________________________________________
Now, after integrating the left-hand side of Eq. (4.3.37) it remains to solve this equation,
expressing
Remark 4.3.5
The describe method is not universal, since for only limited kind of non-autonomous equations
the solutions can be constructed explicitly. Moreover, as expressions (4.3.33) and (4.3.34)
show, it "spoils" the initially linear equation yielding the non-linear one. However, for initially
non-linear differential equations, the discussed method can lead to construction of the solution;
that will be discussed in the next section.
in terms of t.
4.4. Equation Chapter 4 Section 4 Closed form
solutions for non-linear differential equations
4.4.1. Preliminary results
For non-linear equations it is a fortune if a closed form solution exists and it is known. Several
methods can be used for obtaining a closed form solution, three of them are discussed herein.
Basic techniques 4.4.1 (Look-up technique)
For non-linear differential equations it is advisable to look at the well-known collections of
these equations for which the closed form solutions are known; see Abramowitz and Stegun
(1965), Birkhoff and Rota (1978), Polyanin and Zaitsev (1995), Zwillinger (1997).
Basic techniques 4.4.2 (Change of independent variable technique)
Sometimes, change of independent variable can lead to a much simpler non-linear equation that
admits a closed form solution. This technique will be discussed in the next section..
Basic techniques 4.4.3 (Autonomous equations – reducing order of equation)
That method coincides with the analogous method used for linear autonomous equations in Sec.
4.3.6; for non-linear equations such a method will be discussed in one of the following sections.
Chapter 4. Ordinary differential equations
x
p
x
x
194
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4.4.2. Change of independent variable technique
Such a technique is very often applied to solving non-linear Bernoulli equations along with
some other types of non-linear ODEs.
Definition 4.4.1 (Bernoulli equation)
The non-linear differential equation
d
() () () () ()
t ptxt qtxt
+= (4.4.1)
α
dt
with ( )
t and ( )qt arbitrary locally integrable functions and 0, 1α> α≠ is called the
Bernoulli equation.
Proposition 4.4.1 (Constructing partial solution for the Bernoulli equation)
The partial solution for the Bernoulli equation can be represented in a following form:
⎡⎤
⎛⎞⎛⎞
⎢⎥
=α−ττ −αττ
( ) exp ( 1) ( ) ( ) exp (1 ) ( )
xt pd qs pdds
⎜⎟⎜⎟
⎢⎥
⎜⎟⎜⎟
⎢⎥⎝⎠⎝⎠
⎣⎦
where it is assumed that at
tt s
∫∫ ∫
tt t
00 0
the corresponding initial condition is homogeneous:
t
0
⎛⎞
⎜⎟
⎜⎟
⎝⎠
Proof
Dividing both sides of Eq. (4.4.1) by ( )
d
−α −α
() () () () ()
txtptxtqt
α
t
(and assuming for a moment that () 0xt ≠ ), yields
+=. (4.4.3)
1
dt
1
α−
, (4.4.2)
() 0xt =
.
0
Now, introducing a new dependent variable
1
() ()ut xt
= , (4.4.4)
−α
Eq. (4.4.3) transforms into
−α + = . (4.4.5)
()
ut ptut qt
d
1
−
1()()()()
dt
The exact solution of the latter equation can be constructed by use of the integrating factors
(Zwillinger, 1997, p.356):
⎛⎞⎛⎞
( ) exp ( 1) ( ) ( )exp (1 ) ( )
ut pd qs pdds
⎜⎟⎜⎟
=α− ττ −αττ
⎜⎟⎜⎟
⎝⎠⎝⎠
tt s
∫∫ ∫
tt t
00 0
⎛⎞
⎜⎟
⎜⎟
, (4.4.6)
⎝⎠
from which expression (4.4.2) immediately follows.
Chapter 4. Ordinary differential equations
x
x
195
__________________________________________________________________________
Example 4.4.1
Consider equation
Introducing a new dependent variable
and applying the integrating factors technique, yields
Combining (4.4.8) and (4.4.9) produces the desired solution in a form
The constant C should be obtained from satisfying initial condition. It should be noted that if
constant
Example 4.4.2
Another interesting example delivers the following non-linear (and autonomous) Bernoulli
equation
d
() () sin() ()
txt txt
+= . (4.4.7)
3
dt
2
() ()ut xt
2
() cos 2sin
ut Ce t t=+ + . (4.4.9)
t
−
= , (4.4.8)
2
()
5
⎛⎞
() cos 2sin
xt Ce t t
=+ +
⎜⎟
⎝⎠
is negative, then the solution becomes complex.
C
2
2
t
()
5
1/2
−
. (4.4.10)
d
() () ()
txtxt
+=
1/3
. (4.4.11)
dt
Dividing both sides of Eq. (4.4.11) by
(we assume that
()xt
() 0xt ≠
) and introducing
1/3
according to (4.4.4) a new function
2/3
() ()ut xt=
, (4.4.12)
Equation (4.4.11) can be reduced to a linear equation with constant coefficients:
d
3
ut ut
() () 1
dt
2
+=
. (4.4.13)
The solution of Eq. (4.4.13) can be represented in the form
ut t
C
=−+
() exp
tt
21
⎛⎞
⎜⎟
3
⎝⎠
. (4.4.14)
Taking into account (4.4.12), the solution of Eq. (4.4.11) becomes
C
⎛⎞
xt t
() exp
=−+
⎜⎟
tt
⎝⎠
21
⎛⎞
⎜⎟
3
⎝⎠
3/2
. (4.4.15)
Chapter 4. Ordinary differential equations
x
B
x
196
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As in the previous example, the constant C should be obtained from satisfying initial condition.
It is interesting to note that the solution (4.4.15) is unbounded at
t →+∞ regardless of the constant C value.
Remark 4.4.1
Unfortunately, the considered techniques is not applicable to coupled systems of ordinary
equations of the first order, even if all the equations in a system are analogous to Bernoulli
equation.
0t →+
4.4.3. Reducing order of an autonomous non-linear
equation
Herein we give examples of applying one special technique for autonomous equations described
in Sec. 4.3.6.
and vanishes at
Example 4.4.3
Consider the following non-linear autonomous equation
Introducing a new variable (4.3.32), and applying Eq. (4.3.33), leads to a following nonautonomous equation of the first-order
Factorizing (4.4.17), yields
Equation (4.4.18) admits two solutions:
() 0ux = (4.4.19)
and
Equation (4.4.19) yields
hand side of Eq. (4.3.37):
(1 2 ) 0xxx−+ =
du
dx
ux
() (1 2)ux xdx x x C=+ =++∫. (4.4.20)
const=
(1 2 ) 0
−+ =. (4.4.17)
uxu
du
⎛⎞
⎜⎟
⎝⎠
(1 2 ) 0
−+ =
dx
, while on obtaining (4.4.20) it remains to integrate the left-
. (4.4.16)
. (4.4.18)
2
Now, according to (4.3.37) we need to solve the following transcendental equation
dx x
∫
2
++
xC
221
=+
41 41
arctan
−−
CC
+
. (4.4.21)
Chapter 4. Ordinary differential equations
B
197
__________________________________________________________________________
x
221
41 41
from which the solution takes the form
arctan
−−
CC
+
+=
t
, (4.4.22)
41 41 1
CC
The last thing is to rename constant
and finally rewrite Eq. (4.4.23)
It should be noted that in the considered case the trivial solution (4.4.19) became invalid, since
it leaded to the meaningless relation
Example 4.4.4 (partial analytical solution of Duffing equation)
Herein, again we consider a non-linear autonomous equation
() tan
=−−
xt t B
−−
22 2
() tan
xt A At B=−−
⎛⎞
⎜⎟
⎝⎠
41
CA−
2
()
()
const t=
()
= (4.4.24)
. (4.4.26)
3
0xxx+α +β =. (4.4.27)
. (4.4.23)
1
. (4.4.25)
2
Such an equation is known as the Duffing equation without dissipation terms. As in the
previous example, we introduce a new variable
order non-autonomous equation by applying transformation rule (4.3.33):
Integrating both sides of Eq. (4.4.28) yields
where
following one
Now, according to (4.3.37) we need to integrate
where it is denoted
is an independent constant. Equation (4.4.29) can now be transformed into the
C
udu x x dx=− α +β
1
224
uxxC
224
() 2 2
ux x x C
=− + +
∫
()
αβ
⎛⎞
=− + +
⎜⎟
⎝⎠
αβ
⎛⎞
24
⎜⎟
24
⎝⎠
dx
42
ax bx c
++
()ux x=
3
and transform Eq. (4.4.27) into a first-
. (4.4.28)
, (4.4.29)
. (4.4.30)
. (4.4.31)
t
=
Chapter 4. Ordinary differential equations
E
x
f
g
x
198
__________________________________________________________________________
After integration, the left-hand side of Eq. (4.4.31) becomes
dx
∫
42
ax bx c
++
⎛⎞
ix b b c i ca b b b c
EllipticF
⎜⎟
⎜⎟
⎜⎟
⎝⎠
/2; ; 2abcC=−β =−α = . (4.4.32)
2242
(2)(4)
ibx c xb c
−+−−
=×
⎛⎞
24
c b b c ax bx c
⎜⎟
⎝⎠
−− −− −
22
242
−− ++
222
42 4
cc
()
, (4.4.33)
;
where
And at last,
Remark 4.4.2
One interesting result corresponds to a case, when the radicand in the left-hand side of (4.4.33)
is negative of the perfect square:
then the integral in the right-hand side of (4.4.33) becomes
In obtaining (4.4.35) we assumed that
desired partial solution in a following form:
Further analysis of solution (4.4.36) reveals that two different cases can occur: if the product
0gf > , then the solution (4.4.36) yields to a non-cyclic process; if 0gf < , then the solution
becomes periodic, as tangent function. In this case the solution is discontinuous with following
points of discontinuity:
llipticF
denotes the incomplete elliptic integral F.
should be expressed in terms of t, by resolving Eq. (4.4.31).
42 22
++=− +, (4.4.34)
dx x
∫
2
ifx g
()
=
+
() tanh
tig tgf=
()ax bx c
x
1
arctan
ifg g
. Now, Eq. (4.3.37) can be resolved, yielding the
0g ≠
()
⎛⎞
⎜⎟
⎜⎟
⎝⎠
. (4.4.35)
. (4.4.36)
π
It remains to note that points of discontinuity admit an interesting physical interpretation
associated with zones of orbital stability.
tgfnn
=− +π =
d
⎛⎞
⎜⎟
2
⎝⎠
,0,1,2,...
(4.4.37)
Chapter 4. Ordinary differential equations
x
x
x
x
199
__________________________________________________________________________
4.5. Equation Chapter 4 Section 5 Numerical
methods for solving Cauchy problem
of ordinary differential equations
Herein several numerical algorithms for solving Cauchy problem of both linear and
non-linear differential equations are considered.
4.5.1. Basic definitions
Definition 4.5.1 (Difference schemes)
A difference scheme approximates in some sense derivatives of a differential equation by finite
difference expressions that involve values of a function in points belonging to a vicinity of the
given point. There can be different difference schemes.
I. Difference schemes for first derivative
A.
Central difference scheme. That is symmetric about the given point
scheme:
()()
thxth
Forward difference scheme. That is a one sided difference scheme:
B.
Backward difference scheme. That is again a one sided difference scheme:
C.
All the difference schemes presented in (4.5.1) - (4.5.3) approximate first derivative. If higher
derivatives need to be approximated, more elaborated difference schemes are to be considered.
Central difference scheme. That is symmetric about the given point
D.
scheme obtained by considering two successive forward (4.5.2) or backward (4.5.3)
difference schemes :
d
()
xt
≈
0
dt h
d
()
xt
0
dt h
d
()
xt
0
dt h
II. Difference schemes for second derivative
2
d
()
xt
22
dt h
()2()()
≈ . (4.5.4)
0
+− −
00
2
()()
thxt
+−
00
≈
() ( )
txth
−−
00
≈
t h xt xt h
+− + −
000
2
. (4.5.1)
. (4.5.2)
. (4.5.3)
difference
t
0
difference
t
0
Forward difference scheme. That is a one sided difference scheme (with respect to point
E.
):
t
0
Chapter 4. Ordinary differential equations
x
x
x
x
x
x
200
__________________________________________________________________________
2
d
()
F.
Backward difference scheme. That is again a one sided difference scheme:
xt
22
dt h
(2)2( )()
thxthxt
+− ++
000
≈ . (4.5.5)
0
2
2
d
()
xt
22
dt h
() 2( ) ( 2)
txthxth
−−+−
00 0
≈ . (4.5.6)
0
2
III. Difference schemes for third derivative
Central difference scheme. That is an example of symmetric about the given point
G.
difference scheme obtained by considering two successive central difference schemes
(4.5.4):
3
Forward difference scheme. That is a one sided difference scheme (with respect to point
H.
I.
Backward difference scheme. That is a one sided difference scheme (with respect to
d
()
xt
33
dt h
) obtained by applying two successive forward difference schemes (4.5.5):
t
0
3
d
xt
33
dt h
point
) obtained by applying two successive backward difference schemes (4.5.6):
t
0
3
d
xt
33
dt h
(2)2( )2( )(2)
t h xt h xt h xt h
+− ++ −− −
0000
≈
0
. (4.5.7)
4
(3)3(2)3( )()
t h xt h xt h xt
+− ++ +−
()
0
0000
≈ . (4.5.8)
2
( ) 3( ) 3( 2) ( 3)
txthxthxth
−−+−−−
()
0
00 0 0
≈ . (4.5.9)
2
t
0
Remark 4.5.1
There can be constructed other types of difference schemes for second order derivative. For
example, it is possible to consider combination of two successive central difference schemes
(4.5.1), yielding
2
d
()
xt
22
dt h
(2)2()(2)
t h xt xt h
+− + −
000
≈ . (4.5.10)
0
4
For higher order derivatives number of different difference schemes increases; see Godunov
and Ryabenkii (1987) and Lambert (1991).
Definition 4.5.2 (Order of approximation of the numerical method)
Let a differential equation be written in the form
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