power series solution of partial differential equations
Partial Differential Equations part 1
Ngày tải lên :
28/10/2013, 22:15
... 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 heart of ... entiresecond
volume of Numerical Recipes dealing with partial differential equations alone. (The
references
[1-4]
provide, of course, available alternatives.)
In most mathematics books, partial differential equations ... What are the variables?
ã What equations are satised in the interior of the region of interest?
ã What equations are satised by points on the boundary of the region of
interest? (Here Dirichlet...
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Partial Differential Equations part 2
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07/11/2013, 19:15
... of the
difference equations are so slowly varying as to be considered constant in space
and time. In that case, the independent solutions, or eigenmodes, of the difference
equations are all of ... differencing based on Taylor
series and includes the nonlinearity of the equations explicitly. There is an analytic
solutionforthe evolutionof two uniformstates of a fluid separated by a discontinuity,
the ... by a
large number of cells of uniform states, and piece them together using the Riemann
solution. There have been many generalizations of Godunov’s approach, of which
the most powerful is probably...
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Tài liệu Integration of Ordinary Differential Equations part 1 doc
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15/12/2013, 04:15
... 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, having ... 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...
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Tài liệu Integration of Ordinary Differential Equations part 2 pptx
Ngày tải lên :
15/12/2013, 04:15
... 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 ... 710
Chapter 16. Integration of Ordinary Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright ... (see Figure 16.1.1). That means (and you can verify
by expansion in power series) that the step’s error is only one power of h smaller
than the correction, i.e O(h
2
) added to (16.1.1).
There...
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Tài liệu Partial Differential Equations part 3 pptx
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15/12/2013, 04:15
... −
∂
2
∂x
2
+ V (x)(19.2.31)
The formal solution of equation (19.2.30) is
ψ(x, t)=e
−iHt
ψ(x, 0) (19.2.32)
where the exponential of the operator is defined by its power series expansion.
The unstable explicit ... Unless the
solution you are looking for is known to be smooth, and the high-order method you
850
Chapter 19. Partial Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC ... amplitudes, so that the evolution of the larger-scale features
of interest takes place superposed with a kind of “frozen in” (though fluctuating)
background of small-scale stuff. This answer gives...
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Tài liệu Partial Differential Equations part 4 ppt
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15/12/2013, 04:15
... 856
Chapter 19. Partial Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright (C) ... δ
2
y
u
n+1
j,l
(19.3.16)
The advantage of this method is that each substep requires only the solution of a
simple tridiagonal system.
Operator Splitting Methods Generally
The basic idea of operator splitting, ... ∆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 0-521-43108-5)
Copyright (C)...
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Tài liệu Solution of Linear Algebraic Equations part 1 docx
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15/12/2013, 04:15
... as many equations as unknowns, and there is a good
chance of solving for a unique solution set of x
j
’s. Analytically, there can fail to
be a unique solution if one or more of the M equations ... set of equationsto be solved can bewritten as the N ìN set of equations
(A
T
à A) à x =(A
T
Ãb)(2.0.4)
where A
T
denotes the transpose of the matrix A. Equations (2.0.4) are called the
normal equations ... columns
of the matrix inverse of A (Đ2.1 and Đ2.3).
ã Calculation of the determinant of a square matrix A (Đ2.3).
If M<N,orifM=Nbut the equations are degenerate, then there are
effectively fewer equations...
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Tài liệu Solution of Linear Algebraic Equations part 2 ppt
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15/12/2013, 04:15
... of N ì N matrices, with M sets of right-hand
side vectors, in completely analogous fashion. The routine implemented below
is, of course, general.
38
Chapter 2. Solution of Linear Algebraic Equations
Sample ... rows of A and the corresponding rows of the b’s
and of 1, does not change (or scramble in any way) the solution x’s and
Y. Rather, it just corresponds to writing the same set of linear equations
in ... column
we are about to eliminate. Partial pivoting is easier than full pivoting, because we
don’t have to keep track of the permutation of the solution vector. Partial pivoting
makes available...
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Tài liệu Solution of Linear Algebraic Equations part 11 ppt
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15/12/2013, 04:15
... (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 page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC ... 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 ... America).
x[i]=sum/p[i];
}
}
A typicaluseof choldcand cholslis in theinversionof covariancematrices describing
the fit of data to a model; see, e.g., Đ15.6. In this, and many other applications,one often needs
L
−1
....
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Tài liệu Partial Differential Equations part 5 ppt
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15/12/2013, 04:15
... 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 the process. Taking the number of ... 862
Chapter 19. Partial Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright (C) ... 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...
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Tài liệu Partial Differential Equations part 6 doc
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15/12/2013, 04:15
... ease of programming outweighs expense of computer
time. Occasionally, the sparse matrix methods of Đ2.7 are useful for solving a set
of difference equations directly. For production solution of ... adjust the various components of the algorithm
within this framework to solve your specific problem. We can only give a brief
868
Chapter 19. Partial Differential Equations
Sample page from NUMERICAL ... America).
The beauty of Chebyshev acceleration is that the norm of the error always decreases
with each iteration. (This is the norm of the actual error in u
j,l
. The norm of
the residual ξ
j,l
need...
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Tài liệu Integration of Ordinary Differential Equations part 3 doc
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24/12/2013, 12:16
... generally useful stepper
routine is this: One of the arguments of the routine will of course be the vector of
dependent variables at the beginning of a proposed step. Call that y[1 n].Let
us require ... ,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
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright ... efficiency are not mere tens of percents or factors of two;
they can sometimes be factors of ten, a hundred, or more. Sometimes accuracy
may be demanded not directly in the solution itself, but in...
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Tài liệu Integration of Ordinary Differential Equations part 4 ppt
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24/12/2013, 12:16
... 722
Chapter 16. Integration of Ordinary Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright ... usefulness ofthe modied midpointmethod tothe Bulirsch-Stoertechnique
(Đ16.4) 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 ... power series in h, the stepsize, contains
only even powers of h,
y
n
− y(x + H)=
∞
i=1
α
i
h
2i
(16.3.3)
where H is held constant, but h changes by varying n in (16.3.1). The importance
of this...
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Tài liệu Integration of Ordinary Differential Equations part 5 pdf
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24/12/2013, 12:16
... extrapolate each
component of a vector of quantities.
728
Chapter 16. Integration of Ordinary Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... method
a degree of robustness for problems with discontinuities.
Let us remind you once again that scaling of the variables is often crucial for
successful integration of differential equations. The ... method does an excellent job of feeling its way through rocky or discontinuous
terrain. It is also an excellent choice for quick-and-dirty, low-accuracy solution
of a set of equations. A second warning...
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