coupled ordinary differential equations examples
Tài liệu Integration of Ordinary Differential Equations part 1 doc
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... Integration of Ordinary
Differential Equations
16.0 Introduction
Problems involving ordinary differential equations (ODEs) can always be
reduced to the study of sets of first-order differential equations. ... auxiliary variables.
The generic problem in ordinary differential equations is thus reduced to the
study of a set of N coupled first-order differential equations for the functions
y
i
,i=1,2, ,N, ... 1973,
Computational Methods in Ordinary Differential Equations
(New York: Wiley).
Lapidus, L., and Seinfeld, J. 1971,
Numerical Solution of Ordinary Differential Equations
(New
York: Academic...
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Tài liệu Integration of Ordinary Differential Equations part 2 pptx
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... 1973,
Computational Methods in Ordinary Differential Equations
(New York: Wiley).
Lapidus, L., and Seinfeld, J. 1971,
Numerical Solution of Ordinary Differential Equations
(New
York: Academic ... REFERENCES AND FURTHER READING:
Gear, C.W. 1971,
Numerical Initial Value Problems in Ordinary Differential Equations
(Englewood
Cliffs, NJ: Prentice-Hall).
Acton, F.S. 1970,
Numerical Methods ... 710
Chapter 16. Integration of Ordinary Differential Equations
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Tài liệu Integration of Ordinary Differential Equations part 3 doc
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... 722
Chapter 16. Integration of Ordinary Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... REFERENCES AND FURTHER READING:
Gear, C.W. 1971,
Numerical Initial Value Problems in Ordinary Differential Equations
(Englewood
Cliffs, NJ: Prentice-Hall). [1]
Cash, J.R., and Karp, A.H. 1990,
ACM ... ,n−1
y(x+H)≈y
n
≡
1
2
[z
n
+z
n−1
+hf(x + H, z
n
)]
(16.3.2)
714
Chapter 16. Integration of Ordinary Differential Equations
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Tài liệu Integration of Ordinary Differential Equations part 4 ppt
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... 722
Chapter 16. Integration of Ordinary Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... REFERENCES AND FURTHER READING:
Gear, C.W. 1971,
Numerical Initial Value Problems in Ordinary Differential Equations
(Englewood
Cliffs, NJ: Prentice-Hall). [1]
Cash, J.R., and Karp, A.H. 1990,
ACM ... modified midpointmethod tothe Bulirsch-Stoertechnique
(§16.4) derives from a “deep” result about equations (16.3.2), due to Gragg. It turns
out that the error of (16.3.2), expressed as a power...
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Tài liệu Integration of Ordinary Differential Equations part 5 pdf
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... solutions
to ordinary differential equations with minimal computational effort. (A possible
exception, infrequently encountered in practice, is discussed in §16.7.)
726
Chapter 16. Integration of Ordinary ... REFERENCES AND FURTHER READING:
Gear, C.W. 1971,
Numerical Initial Value Problems in Ordinary Differential Equations
(Englewood
Cliffs, NJ: Prentice-Hall),
§
6.1.4.
Stoer, J., and Bulirsch, R. ... Method
The techniques described in this section are not for differential equations
containing nonsmooth functions. For example, you might have a differential
equation whose right-handside involves a...
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Tài liệu Integration of Ordinary Differential Equations part 6 pdf
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... Second-Order Conservative Equations
Usually when you have a system of high-order differential equations to solve it is best
to reformulate them as a system of first-order equations, as discussed ... 27, pp. 505–535.
16.6 Stiff Sets of Equations
As soon as one deals with more than one first-order differential equation, the
possibility of a stiff set of equations arises. Stiffness occurs in ... reason for this is explained below). This is so even
732
Chapter 16. Integration of Ordinary Differential Equations
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Tài liệu Integration of Ordinary Differential Equations part 7 pptx
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... general case implicit methods give better stability.
742
Chapter 16. Integration of Ordinary Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... rule:
y
n+1
− y
n−1
=2hf
y
n+1
+ y
n−1
2
(16.6.29)
738
Chapter 16. Integration of Ordinary Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... REFERENCES AND FURTHER READING:
Gear, C.W. 1971,
Numerical Initial Value Problems in Ordinary Differential Equations
(Englewood
Cliffs, NJ: Prentice-Hall). [1]
Kaps, P., and Rentrop, P. 1979,
Numerische...
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Tài liệu Integration of Ordinary Differential Equations part 8 pdf
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... Problems in Ordinary Differential Equations
(Englewood
Cliffs, NJ: Prentice-Hall), Chapter 9. [1]
Shampine, L.F., and Gordon, M.K. 1975,
Computer Solution of Ordinary Differential Equations.
The ... REFERENCES AND FURTHER READING:
Gear, C.W. 1971,
Numerical Initial Value Problems in Ordinary Differential Equations
(Englewood
Cliffs, NJ: Prentice-Hall). [1]
Kaps, P., and Rentrop, P. 1979,
Numerische ... Predictor-corrector methods have been, we think, squeezed
752
Chapter 16. Integration of Ordinary Differential Equations
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Partial Differential Equations part 1
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... America).
Chapter 19. Partial Differential
Equations
19.0 Introduction
The numerical treatment of partial differential equations is, by itself, a vast
subject. Partial differential equations are at the ... Recipes dealing with partial differential equations alone. (The
references
[1-4]
provide, of course, available alternatives.)
In most mathematics books, partial differential equations (PDEs) are classified
into ... possible choices for
ellipticsecond-order equations, but morecomplicated boundaryconditions
can also be encountered.)
834
Chapter 19. Partial Differential Equations
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Partial Differential Equations part 2
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... schemes, which require us to solve implicit equations coupling
the u
n+1
j
for various j. (Explicit and implicit methods for ordinary differential
equations were discussed in §16.6.) The FTCS ... relation
F(u)=
0 −v
−v 0
·u (19.1.5)
(The physicist-reader may recognize equations (19.1.3) as analogous to Maxwell’s
equations for one-dimensional propagation of electromagnetic waves.)
We will ... stability
requirement that ever turns up in PDEs. It keeps doing so in our model examples
just because those examples are so simple in form. The method of analysis is,
however, general.
Except...
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Tài liệu Partial Differential Equations part 3 pptx
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... 848
Chapter 19. Partial Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... are not as easy. The replacement (19.2.22) with n → n +1leaves
us with a nasty set of coupled nonlinear equations to solve at each timestep. Often
there is an easier way: If the form of D(u) ... given the initial wavepacket, ψ(x, t =0), together with boundary
852
Chapter 19. Partial Differential Equations
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Tài liệu Partial Differential Equations part 4 ppt
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... 856
Chapter 19. Partial Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... U
2
(u
n+(1/m)
, ∆t)
···
u
n+1
= U
m
(u
n+(m−1)/m
, ∆t)
(19.3.20)
854
Chapter 19. Partial Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... again!
CITED REFERENCES AND FURTHER READING:
Ames, W.F. 1977,
Numerical Methods for Partial Differential Equations
, 2nd ed. (New York:
Academic Press), Chapter 2.
Goldberg, A., Schey, H.M., and...
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Tài liệu Partial Differential Equations part 5 ppt
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... systems.
In practice, equations (19.4.33) should be rewritten to avoid numerical instabil-
ity. For these and other practical details, refer to
[2]
.
860
Chapter 19. Partial Differential Equations
Sample ... with
T
(1)
=21−T
2
g
(1)
j
=∆
2
(g
j−1
−T·g
j
+g
j+1
)
(19.4.33)
After one level of CR, we have reduced the number of equations by a factor of
two. Since the resulting equations are of the same form as the original equation, we
can repeat ... 862
Chapter 19. Partial Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN...
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