applications of partial differential equations in engineering ppt
Tài liệu Integration of Ordinary Differential Equations part 2 pptx
Ngày tải lên :
15/12/2013, 04:15
... discussion of the pitfalls in
constructing a good Runge-Kutta code is given in
[3]
.
Here is the routine for carrying out one classical Runge-Kutta step on a set
of n differential equations. You input ... 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 ... derive from this basic
712
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 (C)...
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Tài liệu Partial Differential Equations part 3 pptx
Ngày tải lên :
15/12/2013, 04:15
... 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 a differencing scheme ... form again and in practice usually retains
the stability advantages of fully implicit differencing.
Schr
ă
odinger Equation
Sometimes the physical problem being solved imposes constraints on ... evolve through of order λ
2
/(∆x)
2
steps before things start to happen on the
scale of interest. This number of steps is usually prohibitive. We must therefore
find a stable way of taking timesteps...
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Tài liệu Partial Differential Equations part 4 ppt
Ngày tải lên :
15/12/2013, 04:15
... underlying PDEs, perhaps allowing second-order
spatial differencing for first-order -in- space PDEs. When you increase the order of
a differencing method to greater than the order of the original ... Recipes Software.
Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine-
readable files (including this ... America).
are using is known to be extremely stable, we do not recommend anything higher
than second-order in time (for sets of first-order equations) . For spatial differencing,
we recommend the order of the...
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Tài liệu Partial Differential Equations part 5 ppt
Ngày tải lên :
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 ... y-values on these
x-lines. Then fill in the intermediate x-lines as in the original CR algorithm.
The trick is to choose the number of levels of CR so as to minimize the total
number of arithmetic operations. ... mentioned in Đ19.0, relaxation methods involve splitting the sparse
matrix that arises from finite differencing and then iterating until a solution is found.
There is another way of thinking about...
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Tài liệu Integration of Ordinary Differential Equations part 4 ppt
Ngày tải lên :
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 (C) ... hmin) nrerror("Step size too small in odeint");
h=hnext;
}
nrerror("Too many steps in routine odeint");
}
CITED REFERENCES AND FURTHER READING:
Gear, C.W. 1971,
Numerical Initial ... step h instead of the two required by second-order Runge-Kutta. Perhaps there
are applications where the simplicity of (16.3.2), easily coded in- line in some other
program, recommends it. In general,...
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Tài liệu Integration of Ordinary Differential Equations part 7 pptx
Ngày tải lên :
24/12/2013, 12:16
... as
in the original Bulirsch-Stoer method.
The starting point is an implicit form of the midpoint 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 ... calculates
dydx
.
{
void lubksb(float **a, int n, int *indx, float b[]);
void ludcmp(float **a, int n, int *indx, float *d);
int i,j,nn,*indx;
float d,h,x,**a,*del,*ytemp;
indx=ivector(1,n);
a=matrix(1,n,1,n);
del=vector(1,n);
ytemp=vector(1,n);
h=htot/nstep; ... methods have been, we think, squeezed
740
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...
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Tài liệu AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS ppt
Ngày tải lên :
16/02/2014, 15:20
... computation of the Jacobian at points located on the initial curve , using
24 First-order equations
2.2 Quasilinear equations
We consider first a special class of nonlinear equations where the nonlinearity ... unknown
function and its partial derivatives. PDEs appear frequently in all areas of physics
and engineering. Moreover, in recent years we have seen a dramatic increase in the
use of PDEs in areas such ... fourth-order equation.
r
Linear equations
Another classification is into two groups: linear versus nonlinear equations. An equation is
called linear if in (1.1), F is a linear function of the unknown function...
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Tài liệu Boundary Value Problems, Sixth Edition: and Partial Differential Equations pptx
Ngày tải lên :
17/02/2014, 14:20
... other kinds of linear,
homogeneous equations. Later, we will be using the same principle on partial
differential equations. To be able to satisfy an unrestricted initial condition, we
need two linearly ... follow the derivations of the heat and wave
equations.
The principal objective of the book is solving boundary value problems
involving partial differential equations. Separation of variables receives ... where the
differential equation is linear and of second order. However, problems in elas-
ticity often involve fourth-order equations.
In contrast to initial value problems, even the most innocent...
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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 ... 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 and ... 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...
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Partial Differential Equations part 2
Ngày tải lên :
07/11/2013, 19:15
... various ways of improving the accuracy of first-order upwind
differencing. In the continuum equation, material originally a distance v∆t away
840
Chapter 19. Partial Differential Equations
Sample ... own domain of dependency determined by the choice
of points on one time slice (shown as connected solid dots) whose values are used in determining a new
point (shown connected by dashed lines). ... stability of a differencing scheme. The solution of a hyperbolic
problem at a point depends on information within some domain of dependency to the past, shown here
shaded. The differencing scheme...
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