coupled linear differential equations example
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 ... conceptually, to the
solutionof large numbers of simultaneous algebraic equations. When such equations
are nonlinear, they are usually solved by linearization and iteration; so without much
loss of generality...
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Partial Differential Equations part 2
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... error is one associated with nonlinear hyperbolic equations and
is therefore sometimes called nonlinearinstability. For example, a piece of the Euler
or Navier-Stokes equations for fluid flow looks ... solve implicit equations coupling
the u
n+1
j
for various j. (Explicit and implicit methods for ordinary differential
equations were discussed in Đ16.6.) The FTCS algorithm is also an example of
a ... s
n−1/2
and r
n
being needed to advance
the solution.
For equations more complicated than our simple model equation, especially
nonlinear equations, the leapfrog method usually becomes unstable...
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Tài liệu Integration of Ordinary Differential Equations part 1 doc
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... 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. For example ... 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, having ... by the differential equations. Boundary conditionscan be as simple as
requiring that certain variables have certain numerical values, or as complicated as
a set of nonlinear algebraic equations...
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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 Press).
16.1 ... 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 That ... 710
Chapter 16. Integration of Ordinary Differential Equations
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Tài liệu Partial Differential Equations part 3 pptx
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... complication one can consider is a nonlinear diffusion problem,
for example where D = D(u). Explicit schemes can be generalized in the obvious
way. For example, in equation (19.2.19) write
D
j+1/2
=
1
2
D(u
n
j+1
)+D(u
n
j
)
(19.2.22)
Implicit ... 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) ... 848
Chapter 19. Partial Differential Equations
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Tài liệu Partial Differential Equations part 4 ppt
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... value problems (elliptic equations, for
example) reduce to solving large sparse linear systems of the form
A· u = b (19.4.1)
either once, for boundary value equations that are linear, or iteratively, ... form
∂u
∂t
= Lu (19.3.17)
where L is some operator. While L is not necessarily linear, suppose that it can at
least be written as a linear sum of m pieces, which act additively on u,
Lu = L
1
u + L
2
u ... 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
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Tài liệu Solution of Linear Algebraic Equations part 1 docx
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... overdetermined linear problem reduces to
a (usually) solvable linear problem, called the
ã Linear least-squares problem.
Thereduced set of equationsto be solved can bewritten as the N ìN set of equations
(A
T
à ... trade@cup.cam.ac.uk (outside North America).
Chapter 2. Solution of Linear
Algebraic Equations
2.0 Introduction
A set of linear algebraic equations looks like this:
a
11
x
1
+ a
12
x
2
+ a
13
x
3
+ ···+a
1N
x
N
=b
1
a
21
x
1
+ ... linear combinations of each other, some of the equations
may be so close to linearly dependent that roundoff errors in the machine
render them linearly dependent at some stage in the solution...
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Tài liệu Solution of Linear Algebraic Equations part 2 ppt
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... about as efficient as any
other method. For solving sets of linear equations, Gauss-Jordan elimination
produces both the solution of the equations for one or more right-hand side vectors
b, and ... writeout equations
only for the case of four equations and four unknowns, and with three different right-
hand side vectors that are known in advance. You can write bigger matrices and
extend the equations ... to writing the same set of linear equations
in a different order.
ã Likewise, the solution set is unchanged and in no way scrambled if we
replace any row in A by a linear combination of itself...
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Tài liệu Solution of Linear Algebraic Equations part 11 ppt
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... not used for typical systems of linear equations. However, we will
meet special cases where QR is the method of choice.
100
Chapter 2. Solution of Linear Algebraic Equations
Sample page from NUMERICAL ... solve linear systems. In many applications only the
part (2.10.4) of the algorithm is needed, so we separate it off into its own routine rsolv.
98
Chapter 2. Solution of Linear Algebraic Equations
Sample ... factorizations we have met (LU, SVD, Cholesky), QR decompo-
sition can be used to solve systems of linear equations. To solve
A · x = b (2.10.3)
first form Q
T
· b and then solve
R · x = Q
T
· b (2.10.4)
by...
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Tài liệu Partial Differential Equations part 5 ppt
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... problems (elliptic equations, for
example) reduce to solving large sparse linear systems of the form
A · u = b (19.4.1)
either once, for boundary value equations that are linear, or iteratively, ... 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 ... 862
Chapter 19. Partial Differential Equations
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