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
G
p
X
X
G
X
G
X
G
X
G
G
X
201
__________________________________________________________________________
n
GG GG
d
dt
n
=
() (),..., (), ();
tFX t XtXtt
n
(1)
−
()
′
, (4.5.11)
with the unknown vector-function ( )
tG taking values in
n
. Suppose that all the derivatives in
^
(4.5.11) are approximated by the corresponding difference schemes. Thus, the differential
equation (4.5.11) can be rewritten in terms of finite differences:
+≈ +
() (),...,(),();
thchFX t XtXtt
00
The method of approximation is said to be of order
G
evaluated at
and approximated one
th+
0
GGG
(1)
nn
cXt c Xt nh
10 10
−
()
GG
++ −
( ) ... ( )
′
, (4.5.12)
n
−
, if difference between the exact solution
()
th+
satisfies the following asymptotic
0
relation
with
()()(), 0
Xt h Xt h Oh h
+− + = →
00
being one of the equivalent norms in
⋅
n
^ .
1
p
+
. (4.5.13)
GG
4.5.2. Explicit and implicit difference schemes
Basic concepts 4.5.1
I. First-order ODE
Let a differential equation be represented in a form
with the unknown vector-function
Applying one of difference schemes (4.5.1) - (4.5.3) to the left-hand side of Eq. (4.5.14)
at some
In obtaining (4.5.16) we used the forward difference scheme (4.5.2), but it does not
matter, which one to use. Now, it remains to denote the "next" point as
and rewrite Eq. (4.5.16)
tt=
, yields
i
d
() (),
tFXtt
dt
G
=
()
()
t
(4.5.14)
that satisfies the initial condition
G
()
tX=
00
. (4.5.15)
GG
()()
Xt h Xt
+−
ii
h
tth
1ii
+
FXtt
≈
=+
()
(4.5.17)
. (4.5.16)
(),
GGG
() (), ()
thFXttXt
≈+
1
ii
+
()
. (4.5.18)
Chapter 4. Ordinary differential equations
X
X
G
X
G
G
X
G
X
G
G
X
X
G
X
X
X
X
202
__________________________________________________________________________
Now, it is clear that to define value of the vector-function ( )
we should know its value
from the initial point
(4.5.15), we can find values in successive points
Eq. (4.5.18).
at which value
t
0
at the current point
t
()
i
t
i
is known from the initial condition
()
t
0
10 21
tG at the "next" point
and use Eq. (4.5.18). Starting
,,...tt ht th=+ =+
by applying
1it+
However, up to now, it is not clear, at what values
appearing in the right-hand side of (4.5.18), should be evaluated.
II. Second-order ODE
Let a differential equation be of the form
with the unknown vector-function
Applying one of difference schemes (4.5.4) - (4.5.6) to approximate the second
derivative and one of the schemes (4.5.1) - (4.5.3) for the first derivative, Eq. (4.5.19)
can be transformed to its finite difference counterpart:
()2()()
In obtaining (4.5.21) we assume that the first order derivative
side of this equation is substituted by a finite difference approximation. Denoting
Xt h Xt Xt h
2
GGG
d
dt
+− + −
iii
() (), ();
Xt F Xt Xt t
2
GGGG
=
() ; ()
tX XtV==
00 00
GG
2
2
h
(
that satisfies two initial conditions
()
t
FXt Xtt
≈
(
or
t
i
(4.5.19)
)
. (4.5.20)
G
(), ();
the function
1it+
. (4.5.21)
)
G
in the right-hand
t
()
(),FXtt
()
the Eq. (4.5.21) can be rewritten in a following form
The values
remaining question is at what values
GG
(), ();FXt Xtt
(
III. Higher order ODE
For a higher order ODE
with the unknown vector-function ( )
()2 (),(); 2() ()
thFXtXttXtXt
iii
+−
G
()
d
dt
tthtth
=+ =−
11
ii ii
+−
2
=+−
11
t
and
i
appearing in the right-hand side of (4.5.18), should be evaluated.
)
n
GG GG
n
(
G
=
() (),..., (), ();
tFX t XtXtt
,
GG G G
)
()
t
are assumed to be known from previous steps. The
1
i
−
(1)
−
n
()
tG that satisfies n initial conditions
, (4.5.22)
. (4.5.23)
and
t
i
′
or
1it−
(4.5.24)
and
1it+
the function
t
i
Chapter 4. Ordinary differential equations
X
X
X
X
X
G
X
203
__________________________________________________________________________
the construction procedure is similar to those described above.
Remark 4.5.2
The outlined procedures can be applied to ODEs of higher order, but from henceforth we
confine ourselves to systems of ODEs of the first-order, since any higher order equation can be
reduced to a system of the first-order equations, as it was demonstrated in Sec.4.1.
Definition 4.5.3 (Explicit difference scheme)
GGG GG G
== =
( ) , ( ) , ..., ( )
Xt X X t X X t X
00 01 0 1
′
(1)
n
−
n
−
, (4.5.25)
The difference scheme (4.5.18) is called explicit, if function
"previous" point
:
t
i
GGG
() (), ()
thFXttXt
=+
1
iiii
+
G
()
t
at which
is already known from previous steps.
i
Definition 4.5.4 (Implicit difference scheme)
The difference scheme (4.5.18) is called implicit, if function
"next" point
:
1it+
GG G
() (), ()
thFXtt Xt
=+
111
iiii
+++
G
at which
is not yet known. So, to obtain value
()
t
1
i
+
following equation
GG G
() (), ()
thFXtt Xt
−=
111
iiii
+++
(),FXttG is evaluated at the
()
()
()
()
, (4.5.26)
(),FXttG is evaluated at the
()
, (4.5.27)
it is needed to solve the
()
t
1
i
+
. (4.5.28)
Remarks 4.5.3
A.
Comparing explicit and implicit difference schemes reveals that at least theoretically, the
implicit scheme should be more time-consuming than explicit one. The advantage of the
explicit scheme over implicit becomes more prominent for non-linear equations. In this
case solution of non-linear Eq. (4.5.28) can lead to computational problems.
B.
The advantage of implicit methods is in their better stability in solving problems with stiff
equations; see Hairer and Wanner (1996).
Chapter 4. Ordinary differential equations
X
G
204
__________________________________________________________________________
4.5.3. Euler methods
Definition 4.5.5 (Explicit Euler method)
The explicit Euler method coincides with the method described in Definition 4.5.3. Sometimes
it is called as forward Euler method.
Definition 4.5.6 (Implicit Euler method)
The implicit Euler method coincides with the method described in Definition 4.5.4. Sometimes
it is called as backward Euler method.
Remark 4.5.4
According to Definition 4.5.2 both explicit and implicit variants of Euler method are of order 1.
4.5.4. Runge-Kutta methods
Definition 4.5.7 (Explicit Runge-Kutta method)
The classical explicit Runge-Kutta method applied to differential Eq. (4.5.14) can be written in
a form:
where
This is the fourth-order method
() 2 2 ()
tFFFFXt
ii
GG
h
=++++
11234
+
()
6
FFXtt
=
1
FFXt t
=++
2
FFXt t
=++
3
FFXt Fth
=++
43
(),
()
G
⎛⎞
() ,
⎜⎟
⎝⎠
G
⎛⎞
() ,
⎜⎟
⎝⎠
G
() ,
()
ii
F
1
ii
22
F
2
ii
22
ii
h
h
, (4.5.29)
. (4.5.30)
Definition 4.5.8 (Implicit Runge-Kutta method)
The classical implicit Runge-Kutta method applied to differential Eq. (4.5.14) can be written as
a two stage method, coinciding with the implicit Euler method on the first stage, that yields a
Chapter 4. Ordinary differential equations
X
X
X
G
x
γ
205
__________________________________________________________________________
rough approximation
computed
G
()
t
11
i
+
; then on the second stage a refined approximation
G
()
t
21
is
i
+
() 2 2 ()
tFFFFXt
21 1 2 3 4
ii
GG
h
=++++
+
()
6
in the form (4.5.29) ,but with the following , 1,..., 4
FFXt t
=
1111
FFXt t
=+−
211 1
=+−
FFXt t
311 1
(),
()
ii
++
G
⎛⎞
() ,
ii
++
⎜⎟
⎝⎠
G
⎛⎞
() ,
ii
++
⎜⎟
⎝⎠
G
FFXt Ft h
=+−
41131
() ,
()
ii
++
As its explicit counterpart, the implicit Runge-Kutta method is the fourth-order accurate
method.
Remarks 4.5.5 (Runge-Kutta-Fehlberg method)
A.
A great variety of different realizations of Runge-Kutta methods exist; One of the most
numerically efficient methods is the Runge-Kutta-Fehlberg method (denoted as rk45),
which combines fourth-order and fifth-order Runge-Kutta methods.
Fk=
k
F
1
h
22
F
2
h
22
, (4.5.31)
. (4.5.32)
B.
It should also be noted that all Runge-Kutta methods can be treated as methods with
intermediate steps, in contrast to the single step methods (Euler) and the multistep methods
that will be discussed in the following subsection.
Definition 4.5.9 (Stiff ODE)
The differential equation is called stiff, if its partial or general solution contains terms with
different scales. For example, the ODE with the general solution containing exponents
tt
with small ε and large
() e
tC Ce
should be regarded as stiff ODE.
ε−γ
=+
12
Remark 4.5.6 (Rosenbrock method for solving stiff ODE)
A variant of the Runge-Kutta method with different time-steps (small at time regions with
possibly fast variation of the solution, and large at slow variation) is known as the Rosenbrock
method.
, (4.5.33)
Chapter 4. Ordinary differential equations
G
X
206
__________________________________________________________________________
4.5.5. Multistep methods
Definition 4.5.10 (Explicit linear multistep method)
The numerical method for solving Cauchy problem of ODE is called a multistep method, if it
retains information on behavior of the solution at previous steps for obtaining the solution on
the next step. For example, the explicit (linear) n -step method for Eq. (4.5.14) can be defined
by the following numerical scheme
1
n
−
() ( ) (( ), )
Xt a Xt hbFXt t
=+
∑
1
i ik k ik ik
+−−−
k
GG
k
0
=
, (4.5.34)
where coefficients , , 0,..., 1
ab k n=− are determined by the corresponding multistep
k
k
method. Some of the coefficients can be zeros.
Definition 4.5.11 (Implicit linear multistep method)
The implicit (linear)
-step method for Eq. (4.5.14) can be defined by the following numerical
n
scheme
GG G
() ((), ) ( )
Xt hb FXt t a Xt
−=+
11 11
iiiik
+− ++ −
where coefficients
, , 0,..., 1
ab k n=−
k
k
and
multistep method. Some of the coefficients can be zeros.
Definition 4.5.12 (Adams-Bashforth methods)
These are explicit (linear) n-step methods. For Eq. (4.5.14) these methods can be defined by
the following numerical schemes:
1
n
−
∑
k
0
k
=
1
n
−
hb F X t t
∑
k
=
0
G
(( ), )
kikik
are determined by the corresponding
1b−
−−
, (4.5.35)
A. Two-step Adams-Bashforth method
tXthFXtthFXtt
=+ −
() () ((),) ((),)
111
ii ii ii
+−−
GG G G
31
22
. (4.5.36)
B. Three-step Adams-Bashforth method
GG G G
Xt Xt hFXt t hFXt t
=+ − +
() () ((),) ((),)
111
ii ii ii
+−−
23 4
22 3
12
5
G
hF X t t
(( ), )
ii
22
−−
. (4.5.37)
Chapter 4. Ordinary differential equations
X
207
__________________________________________________________________________
Remark 4.5.7
There are other variants of the Adams-Bashforth methods, containing more steps. Coefficients
in these methods correspond to coefficients of the interpolation polynomials; see Butcher
(2008).
Definition 4.5.13 (Adams-Moulton methods)
These are implicit (linear) n -step methods. For Eq. (4.5.14) these methods can be defined by
the following numerical schemes:
A. Two-step Adams-Moulton method
B. Three-step Adams-Moulton method
Remark 4.5.8
Similarly to Adams-Bashforth methods, there are variants of the Adams-Moulton method
containing more steps. Coefficients in these methods correspond to coefficients of the
interpolation polynomials; see Butcher (2008).
GG
() ((),)
Xt hFXt t
iii
+++
GG
Xt hFXt t
Xt hFXt t hFXt t hFXt t
−=
() ((), )
111
iii
+++
GG G G
19 5 1
+− +
() ((),) (( ), ) (( ), )
iiiii ii
5
−=
111
12
GG G
() ( (),) ( ( ), )
21
+−
thFXtt hFXtt
iii ii
312
3
8
11 2 2
24 24 24
−− − −
. (4.5.38)
11
−−
. (4.5.39)
Chapter 4. Ordinary differential equations
208
__________________________________________________________________________
Bibliography to Chapter 4
ABRAMOWITZ M. and STEGUN I.A.
Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. Dover
Publications. 1965. ISBN-13: 978-0486612720
A
NDRONOV A.A., VITT A.A., and KHAIKIN S.E.
Theory of Oscillators. Dover Publications. 1987. ISBN-13: 978- 0486655086
A
RNOLD V.I.
Mathematical Methods of Classical Mechanics (2
0387968902
A
RNOLD V.I.
Ordinary Differential Equations. Springer. 2006. ISBN-13: 978-3540345633
B
ENEST D. and FROESCHLE C. (Editors)
Analysis and Modelling of Discrete Dynamical Systems (Advances in Discrete Mathematics &
Applications Series, Vol. 1). CRC Press. 1998. ISBN-13: 978-9056996253
nd
edition). Springer, 1989. ISBN-13: 978-
B
IRKHOFF G. and ROTA G.-C.
Ordinary Differential Equations. John Wiley & Sons, New York, 1978. ISBN-13: 9783540345633
B
UTCHER J.C.
Numerical Methods for Ordinary Differential Equations, 2
978-0470723357
C
ODDINGTON E.A. and LEVINSON N.
Theory of Ordinary Differential Equations. Krieger Pub Co. 1984. ISBN-13: 978- 0898747553
D
E COSTA L.M.B.
Differential Equations with Applications to Vibrations and Waves (Mathematical-Physics for
Science and Technology). CRC Press. 2010. ISBN-13: 978-1420071153
D
EN HARTOG J.P.
Mechanical Vibrations. Craste Press. 2008. ISBN-13: 978-1443725361
D
RAZIN P.G.
Nonlinear Systems (Cambridge Texts in Applied Mathematics). Cambridge University Press.
1992. ISBN-13: 978-0521406680
G
ODUNOV S.K. AND RYABENKII V.S.
Difference Schemes: An Introduction to the Underlying Theory. North-Holland Publishing Co.,
New York, 1987.
nd
edition. Wiley. 2008. ISBN-13:
H
AIRER E. AND WANNER G.
Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (Springer
Series in Computational Mathematics). Springer. 1996. ISBN-13: 978-3642052200
H
ALE J. K.
Chapter 4. Ordinary differential equations
209
__________________________________________________________________________
Ordinary Differential Equations. Dover Publications. 2009. ISBN-13: 978- 0486472119
H
ARTMAN P.
Ordinary Differential Equations. Society for Industrial Mathematics. 2002. ISBN-13: 9780898715101
I
NMAN D.J.
Engineering Vibration (3d Edition). Prentice Hall. 2007. ISBN-13: 978-0132281737
K
AMBE T.
Geometrical Theory of Dynamical Systems and Fluid Flows. World Scientific Publishing
Company. 2009. ISBN-13: 978-9814282246
L
AMBERT J.D.
Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. Wiley. 1991.
ISBN-13: 978-0471929901
M
CLACHLAN N.W.
Ordinary non-linear differential equations in engineering and physical sciences. Clarendon
Press. 1958. ASIN: B0007IVEDW
M
ELIN P. and CASTILLO O.
Modelling, Simulation and Control of Non-linear Dynamical Systems: An Intelligent Approach
Using Soft Computing and Fractal Theory (Numerical Insights). CRC Press. 2001. ISBN-13:
978-0415272360
P
IERSOL A. and PAEZ T.
Harris’ Shock and Vibration Handbook (6
13: 978- 0071508193
P
OLYANIN A.D. ZAITSEV V.F.
Handbook of Exact Solutions for Ordinary Differential Equations. CRC, Boca Raton, 1995.
ISBN-13: 978-1584882978
R
EISSIG R., SANSONE G., and CONTI R.
Non-Linear Differential Equations of Higher Order. Springer. 1974. ISBN-13: 978-9001752705
R
OBERTS C.
Ordinary Differential Equations: Applications, Models, and Computing. Chapman & Hall.
2010. ISBN-13: 978-1439819081
S
TROGATZ S.H.
Nonlinear Dynamics fnd Chaos: With Applications to Physics, Biology, Chemistry, and
Engineering. Westview Press. 2001. ISBN-13: 978-0738204536
T
ENENBAUM M. and POLLARD H.
Ordinary Differential Equations. Dover Publications. 1985. ISBN-13: 978- 0486649405
T
IMOSHENKO S.P.
Vibration Problems in Engineering (2
1443731676
th
edition). McGraw-Hill Professional. 2009. ISBN-
nd
edition). Wolfenden Press. 2008. ISBN-13: 978-
Z
WILLINGER D.
Handbook of Differential Equations, Third Edition, Academic Press. 1997. ISBN-13: 9780127843964
Учебное издание
Кузнецов Сергей Владимирович, Кошелева Елена Леонидовна
ADDITIONAL CHAPTERS OF HIGHER MATHEMATICS
FOR MASTERS IN CIVIL AND GEOTECHNICAL ENGINEERING
Учебное пособие по дополнительным разделам высшей математики
для магистрантов по направлению «Строительство»
Редактор Н.А. Котова
Компьютерная верстка Н.А. Котовой
Дизайн обложки Н.А. Кильдишева
Подписано к печати 28.12.2012 г. Формат 60×84 1/8. Печать офсетная.
И-367. Объем 26,5 печ. л. Усл. печ. л. 24,7. Тираж 100 экз. Заказ № 584.
ФГБОУ ВПО «Московский государственный строительный университет».
Издательство МИСИ – МГСУ.
Тел. (495) 287-49-14, вн. 13-71, (499) 183-97-95,
e-mail: ric@mgsu.ru, rio@mgsu.ru
Отпечатано в типографии Издательства МИСИ – МГСУ.
Тел. (499) 183-91-90, (499) 183-67-92, (499) 183-91-44.
129337, Москва, Ярославское ш., 26
Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]