Добавил:
ivanov666
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Additional Chapters of Higher Mathematics for Masters in Civill and Geotechnical Engineering. Учебное пособие по дополнительным разделам высшей матема
.pdf
Chapter 4. Ordinary differential equations
x
x
x
x
x
x
161
__________________________________________________________________________
Remark 4.1.4
It should be noted that in the resulting system (4.1.15) of n ODEs of the first order all
equations, but (possibly) last, are linear.
Remark 4.1.5
Sometimes, reduction of Eq. (4.1.2) to a system of equations of the first order (4.1.15) is
associated with the Hamiltonian generalized formalism.
⎧
⎪
⎪
⎪
Fxx
≡− =
110
Fxx
≡−=
221
⎨
⎪
Fxx
≡− =
112
nnn
⎪
⎪
FFtxxx x pt
nnn
⎩
−−−
( ; , ,... , ) ( )
≡=
01 1 1
.......
′
0
′
0
′
0
−−
′
. (4.1.15)
Example 4.1.3
Consider the following second-order differential equation:
where dots over
(4.1.16) is of great importance in mechanics, as well as in other areas of mathematical physics.
It describes forced vibrations of the system with damping. According to Proposition 4.1.1 to
reduce Eq. (4.1.16) to a first-order system, we denote
x=
10
Quite often in mechanical and physical applications, symbols
is denoted as v resembling that it actually defines a velocity, and
1
in these notations Eq. (4.1.17) after some regrouping becomes
, (4.1.16)
()mx cx kx p t++=
stand for the corresponding derivatives with respect to time t. Equation
as
and introduce a new variable
0
, then we arrive at the following system of ODEs of the first-order:
≡−=
⎧
110
⎨
F mxcxkx p
⎩
≡++=
2110
xv
=
⎧
⎪
⎨
=− − +
vxvp
⎪
⎩
0Fxx
kc
mm
. (4.1.17)
and
F
1
F
2
is again reverted to x,
0
. (4.1.18)
are omitted, variable
Proposition 4.1.2 (the inverse to the preceding proposition)
Any system of n ODE of the first order ODE in the form (4.1.15) can be reduced to a single
ODE of the
n -th order (4.1.1).
Chapter 4. Ordinary differential equations
x
x
x
x
f
f
162
__________________________________________________________________________
Proof
The proof is similar to the proof of the preceding proposition.
Proposition 4.1.3
Any non-autonomous, but homogeneous equation of the form (4.1.2), can be made autonomous.
Proof
At first, we reduce a non-autonomous homogeneous Eq. (4.1.2) into a system of the first order
(4.1.15). Now, introducing a new variable
and a new equation for it
n
we can replace the independent variable
(4.1.19) is inhomogeneous, but with the admissible kind of a constant inhomogeneity.
Basic concepts 4.1.1
System of the first order (4.1.15) can formally be rewritten in a form:
where the last equation in (4.1.15) was resolved with respect to derivative
theorem of implicit function, that can be done, at least locally, if function
sufficiently smooth. Taking derivatives of the function
variables
,,...
01 1
these variables) we can construct a linear approximation to
( , ,... )
aa a
≡∈a \
01 1
xx
n
−
−
n
x =
, (4.1.19)
'1
n
in (4.1.15) by
t
. Now, it remains to note that Eq.
n
4.1.2. Jacobian
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
′
=−
1011
nn
−−
⎩
(herein, we again, assume that function f is differentiable with respect
n
:
′
xx
=
01
′
xx
=
12
.......
′
xx
=
21
nn
−−
ftx x x pt
( ; , ,... ) ( )
., (4.1.20)
′
. Due to the
1nx−
in (4.1.1) is
F
( ; , ,... )
ftx x x
01 1
with respect to
n
−
near some point
where
( , ,... )
xx x
≡∈x \
01 1
n
( ; ,..., ) ( )
ftx x f
01
n
−
and symbol
≈∇ ⋅ −xa
n
−
∇
()
stands for gradient with respect to x . Expression
(4.1.21) can be treated as linear approximation to function
often point
, where the derivatives are calculated, is chosen at the origin:
a
(4.1.21)
in the vicinity of point a . Quite
. In this
=
a0
situation system (4.1.20) can be represented by the following approximation:
Chapter 4. Ordinary differential equations
xf
p
j
j
x
f
163
__________________________________________________________________________
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
n
⎩
Now, Eq. (4.1.22) can be rewritten in the form:
01
xx
'
⎛⎞ ⎛⎞
⎜⎟ ⎜⎟
xx
'
⎜⎟ ⎜⎟
⎜⎟ ⎜⎟
... ...
⎜⎟ ⎜⎟
xx
'
nn
⎜⎟ ⎜⎟
⎜⎟ ⎜⎟
xx
'
nn
⎝⎠ ⎝⎠
⎛⎞
00
⎜⎟
11
⎜⎟
⎜⎟
=⋅−
⎜⎟
−−
22
⎜⎟
⎜⎟
∂∂ ∂
ff f
−−
11
xx x
01 1
⎝⎠
Another, more concise form of Eq. (4.1.23) is as follows:
′
=
xx
01
′
=
xx
12
.......
′
=
xx
nn
−−
21
′
=∇ ⋅ −
−
1
x
...
...
1
...
n
−
. (4.1.22)
pt
()
0
⎛⎞
⎜⎟
0
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎜⎟
−
⎝⎠
(4.1.23)
...
0
t
()
where
, G is the matrix in the right-hand side of (4.1.24), and p is the loading vector
≡xx
d
dt
appeared in the right-hand of (4.1.24).
Definition 4.1.7 (Jacobian)
Matrix
in (4.1.24) is called the Jacobian of system (4.1.20).
G
Remark 4.1.6
Sometimes, if function f from Eqs. (4.1.20), is a polynomial with separable variables, a
different definition for the Jacobian is used. In such a case the Jacobian is defined as
G , (4.1.25)
=
=⋅−xGxp
01
⎛⎞
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎜⎟
fk fk fk
∂∂ ∂
/ ! / ! ... ... / !
xx xn
⎝⎠
01 1
01 1
(4.1.24)
1
...
1
n
−
−
Proposition 4.1.4
where
is the order of the variable
k
If function
from Eqs. (4.1.20), is a polynomial with separable variables, then Eq. (4.1.24) is
in the polynomial f.
exact: it coincides with Eq. (4.1.20), provided the Jacobian is defined by Eq. (4.1.25).
Chapter 4. Ordinary differential equations
f
164
__________________________________________________________________________
Proof
Is obvious, since in the considered case
x
⎛⎞
0
Example 4.1.4
For Eq. (4.1.16) and the corresponding system (4.1.18), the Jacobian defined by Eq. (4.1.25) is
System (4.1.18) can be written in a form of (4.1.24) in terms of the Jacobian (4.1.27)
∂∂⋅=
/ ! ... / ! ...
()
kfk f
xxn
01
01
G
n
−
01
⎛⎞
=
⎜⎟
//km cm
−−
⎝⎠
⎜⎟
−
⎜⎟
⎜⎟
x
1
n
−
⎝⎠
. (4.1.27)
(4.1.26)
xx
⎛⎞ ⎛ ⎞⎛⎞ ⎛ ⎞
⎜⎟ ⎜ ⎟⎜⎟ ⎜ ⎟
vkmcmvp
⎝⎠ ⎝ ⎠⎝⎠ ⎝ ⎠
01 0
=⋅+
−−
//
4.1.3. Fundamental matrix
Along with the Jacobian, another useful tool for solving systems of linear ODEs is
the fundamental matrix
Definition 4.1.8 (Fundamental (exponential) matrix)
The fundamental (exponential) matrix
first order with Jacobian
Remark 4.1.6
The matrix exponent appearing in the right-hand side of (4.1.29) can be defined by Taylor’s
series for the exponent:
G is defined as
of a linear and autonomous system of ODEs of the
F
exp t≡FG
. (4.1.28)
(4.1.29)
()
exp ... ...
tk≡+ + + + +
GI
()
GG G
22
tt t
kk
(4.1.30)
1! 2! !
While formula (4.1.30) is convenient for defining the exponent, it is almost never used for
calculating matrix exponent because of a lot of computations needed.
From computational point of view it is much more efficient to reduce matrix
G
by
multiplying it from both sides by a suitably chosen non-degenerate matrix W and its inverse
(see Proposition 3.4.2) into the Jordan canonical form
Chapter 4. Ordinary differential equations
165
__________________________________________________________________________
1−
=⋅⋅GW DW, (4.1.31)
where matrix
D is called the Jordan canonical form of matrix G . Substituting decomposition
(4.1.31) into right-hand side of (4.1.30) yields
⎛⎞
1
exp ... ...
t
GW I W
()
−
≡⋅++ ++ +⋅
DD D
⎜⎟
⎜⎟
1! 2! !
⎝⎠
22
tt t
D
exp
()
kk
(4.1.32)
k
t
Now, assuming that G is a semisimple matrix (and thus does not have Jordan blocks), then D
is a diagonal matrix, so
k
⎛⎞
d
1
⎜⎟
where
are eigenvalues of
,...,
dd
1
n
D
k
=
...
⎜⎟
⎜⎟
⎝⎠
(4.1.33)
k
d
n
G . Now, comparing (4.1.32) and (4.1.33) it becomes clear
that
exp( )
dt
⎛⎞
exp ...
()
⎜⎟
=
D
t
⎜⎟
⎜⎟
⎝⎠
1
. (4.1.34)
exp( )
dt
n
It should be noted that expression (4.1.34) is valid for any diagonal matrix. Now, combining
(4.1.32) and (4.1.34) we arrive at the following expression for the fundamental matrix
where
exp tD
In a case when matrix is not semisimple, the Jordan canonical matrix
Jordan blocks. In this situation expression (4.1.35) is still valid, but the matrix exponential
exp tD
can be obtained by a slightly modified procedure.
()
Proposition 4.1.5
The fundamental exponential matrix is non-degenerate regardless of the corresponding Jacobian
matrix.
Proof
In virtue of expressions (4.1.35) and (4.1.34) it becomes clear that the matrix exponent is nondegenerate for at least a semisimple Jacobian. If the Jacobian is not semisimple, the proof
becomes more complicated, but the final result is still valid.
Proposition 4.1.6
1
exp exptt
() ()
is defined by an easily computed formula (4.1.34).
()
−
≡⋅ ⋅GW DW
, (4.1.35)
will contain
D
Any (square) matrix G and its matrix exponent (4.1.30) commute.
Chapter 4. Ordinary differential equations
166
__________________________________________________________________________
Proof
Is obvious, since multiplying
same result in virtue of the right-hand side of (4.1.30).
Corollary
The fundamental matrix commutes with the corresponding Jacobian.
Proposition 4.1.7 (Jacobian identity)
exp tG
by G from either left- or right-hand sides produces the
()
Proof
For any square matrix
property
Results for a semisimple matrix
of a semisimple matrix its Jordan normal form is
where
diagonal matrix containing eigenvalues of
computation of the exponent matrix, yields
But, the determinant of the right-hand side of (4.1.38) is
Now, since
to
is a non-singular matrix composed of eigenvectors of matrix G , and D is the
W
11
det exp det det exp dettt
exp tD
−−
()()
()
, the determinant of its matrix exponent possesses the following
G
det exp exp trtt=GG. (4.1.36)
exp exptt
⋅⋅=WDWW DW
()
is the diagonal matrix, the right-hand side of (4.1.39) can be transformed
() ()
rely on results on methods developed in Sec. 3.6. In a case
G
=⋅⋅GW DW, (4.1.37)
=⋅ ⋅GW DW. (4.1.38)
() ()
()
1−
. Now, applying decomposition (4.1.37) to
G
1
−
()
()
()
. (4.1.39)
det det exp det
WDW
()
exp ... exp exp exp tr
Proof of the case, when
a similar deduction.
tttt
λ×× λ = λ =
()
1
()
G is not a semisimple matrix, relies on results of Sec. 3.7 and leads to
t
()
()
()
nk
=
n
⎛⎞
⎜⎟
∑
⎜⎟
k
1
=
⎝⎠
. (4.1.40)
G
()
()
1
−
Chapter 4. Ordinary differential equations
167
__________________________________________________________________________
4.1.4. Integral curves, phase trajectories,
and phase portraits
Definition 4.1.9 (Phase plane; Phase hyper plane)
Phase (hyper-) plane (sometimes phase plane is also called as the phase space) is the 1n − -
dimensional space, where solution of the differential equation is analyzed. The equation can be
represented in either form of (4.1.1) or (4.1.20).
Definition 4.1.10 (Integral curve)
Any solution of the initial value problem passing through the point
is called the integral curve in the n-dimensional region
tt=
0
admissible values for parameters
infinite time interval.
Remark 4.1.7
It is clear that notions of the integral curve and the solution of the initial value problem are
coincident. If the considered differential equation has a unique solution satisfying the specified
initial value problem, then the corresponding integral curve cannot have points of intersection.
Definition 4.1.11 (Phase portrait)
Projection of the integral curve onto
Remark 4.1.8
While for differential equations having unique solutions the integral curves cannot intersect, the
corresponding phase trajectories can; see Fig.4.1.1, where both integral curves have no points of
intersection, while the corresponding phase trajectories (trajectories onto the bottom plane) can
intersect. For example, projection of the right integral curve has infinite number of points of
intersection, since its phase trajectory is a circle.
at some time
n
−
,
,...,
aa
()
01
n
Ω∈\ or
−
1n−
,...,
aa
()
01
, where Ω is a region of
TΩ×
Ω∈^ and T is possibly
1n −
Ω is called the phase trajectory or phase portrait.
Chapter 4. Ordinary differential equations
xf
x
f
168
__________________________________________________________________________
Figure 4.1.1. Integral curves and the corresponding phase trajectories onto the bottom plane
Proposition 4.1.8 (differential equation for the phase trajectory of a second-order autonomous
differential equation)
Let a second-order differential equation be resolved over its second derivative:
, (4.1.41)
(,)
xx=
then the following first-order differential equation stands for the phase trajectory
(, )dv f v x
=
, (4.1.42)
dx v
where
v denotes
.
Proof
In terms of a new variable v the initial equation (4.1.41) takes the form
Dividing both sides of (4.1.43) by
yields
v
=
. (4.1.43)
(, )v
vx=
(, )vfvx
. (4.1.44)
vv
Now, we transform the left-hand side of (4.1.44) by recalling that
dv
vdv
dt
==
dx
vdx
dt
. (4.1.45)
vx=
Combining (4.1.44) and (4.1.45) completes the proof.
Chapter 4. Ordinary differential equations
x
x
x
169
__________________________________________________________________________
Remark 4.1.9
On practice, even for autonomous differential equations of the second order, it can be more
convenient to construct phase trajectories by either numerical integration of the initial equation
(4.1.41), and then plotting the corresponding phase trajectory curve
or using analytical solution for
(), ()vt xt and then plotting a parametric plot. These two more
(), ()
tvt
general methods are applicable for non-autonomous equations, as well.
4.2. Equation Chapter 4 Section 2 Linear
differential equations with constant
coefficients
4.2.1. Preliminary results
on the
v -plane;
Definition 4.2.1 (Linear ODE with constant coefficients)
The linear ODE with constant coefficients of the order
where at least
nn
ax a x ax ax pt
.
0
a ≠
n
++++=
nn
−
110
−
... ( )
() ( 1)
Remark 4.2.1
In the theory of linear ODE it is custom to impose some restrictions on the right-hand side of
(4.2.1). That will be done later on, when the so called partial solution will be constructed.
Definition 4.2.2 (Characteristic polynomial)
The characteristic polynomial associated with Eq. (4.2.1) is
( ) ...
Pa a aa
λ≡ λ + λ + + λ+
nn
nn
1
−
110
−
Remark 4.2.2
can be written in a form:
n
′
, (4.2.1)
. (4.2.2)
Introducing the characteristic polynomial is attributed to Euler, who constructed the general
solution by substituting a test function with yet unknown exponent multiplier
t
λ
()
te
(4.2.3)
=
into Eq. (4.2.1)
λ
Chapter 4. Ordinary differential equations
x
p
x
p
170
__________________________________________________________________________
The function (4.2.3) is now known as the Euler’s representation.
Remark 4.2.3
Herein we will discuss constructing the general solution by Euler’s representation. Another
powerful method related to the Hamiltonian generalized formalism will be discussed in the next
subsection.
Proposition 4.2.1
a) If all the roots of the characteristic polynomial (4.2.2) are aliquant, then the general solution
of Eq. (4.2.1) has the form
t
...
λ
n
n
(they are called roots of multiplicity
where
CC are arbitrary constants defined by the initial (or Sturm-Liouville)
conditions, and
1
,...,
n
λλ
1
() ...
tCe Ce
=++ , (4.2.4)
are all the roots (possibly complex) of the characteristic
,...,
n
polynomial (4.2.2).
b)
If some of the roots are multiple, say
), then the general solution has the form
1
+
t
λ
1
1
λ= =λ
kkp+
Proof
tt t t
t
λ
Case of aliquant roots. Substituting any of the terms , 1,...,
a)
( ) ... ... ...
tCe Ce Cte Cte Ce
1
= ++ + ++ ++
11
λλ λ λ
kk k n
kk kp n
++
++
CC t C te
(
kk kp
...
++
1
p
p
λ
t
k
)
t
λ
q
eq n
= , where
. (4.2.5)
λ is the
q
corresponding root, into (homogeneous) Eq. (4.2.1) reveals the equation is satisfied. By
linearity of the equation, it follows that any linear combination of these terms will also be a
solution.
Let
b)
1
q
−
te q p
...
λ= =λ
kkp+
t
λ
k
,1,...,
=
qq
⎡⎤
tP q t P
⎢⎥
⎢ −× − ×× λ⎥
⎣⎦
In proving (4.2.6) it remains to note that if
roots also null all the successive derivatives of the order up to
be multiple roots, then direct verification shows that any of the terms
satisfies homogeneous Eq. (4.2.1):
12
−−
λ+ − λ+
()( 1) '()...
kk
(1)
q
(1)(2)... ()
qq P
−
k
...
λ= =λ
kkp+
t
λ
k
. (4.2.6)
=
0
e
are multiple roots, then these
of the polynomial.
Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]