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annotate doc/interpreter/diffeq.txi @ 15879:f69530e3600d
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| author | John W. Eaton <jwe@octave.org> |
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| date | Wed, 02 Jan 2013 19:18:15 -0500 |
| parents | c3fd61c59e9c |
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| rev | line source |
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1 @c Copyright (C) 1996-2012 John W. Eaton |
| 7018 | 2 @c |
| 3 @c This file is part of Octave. | |
| 4 @c | |
| 5 @c Octave is free software; you can redistribute it and/or modify it | |
| 6 @c under the terms of the GNU General Public License as published by the | |
| 7 @c Free Software Foundation; either version 3 of the License, or (at | |
| 8 @c your option) any later version. | |
| 9 @c | |
| 10 @c Octave is distributed in the hope that it will be useful, but WITHOUT | |
| 11 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or | |
| 12 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License | |
| 13 @c for more details. | |
| 14 @c | |
| 15 @c You should have received a copy of the GNU General Public License | |
| 16 @c along with Octave; see the file COPYING. If not, see | |
| 17 @c <http://www.gnu.org/licenses/>. | |
| 3294 | 18 |
| 4167 | 19 @node Differential Equations |
| 3294 | 20 @chapter Differential Equations |
| 21 | |
| 6700 | 22 Octave has built-in functions for solving ordinary differential equations, |
| 23 and differential-algebraic equations. | |
| 8828 | 24 All solvers are based on reliable ODE routines written in Fortran. |
| 3294 | 25 |
| 26 @menu | |
| 27 * Ordinary Differential Equations:: | |
| 28 * Differential-Algebraic Equations:: | |
| 29 @end menu | |
| 30 | |
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31 @cindex differential equations |
| 3294 | 32 @cindex ODE |
| 33 @cindex DAE | |
| 34 | |
| 4167 | 35 @node Ordinary Differential Equations |
| 3294 | 36 @section Ordinary Differential Equations |
| 37 | |
| 38 The function @code{lsode} can be used to solve ODEs of the form | |
| 39 @tex | |
| 40 $$ | |
| 41 {dx\over dt} = f (x, t) | |
| 42 $$ | |
| 43 @end tex | |
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44 @ifnottex |
| 3294 | 45 |
| 46 @example | |
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47 @group |
| 3294 | 48 dx |
| 49 -- = f (x, t) | |
| 50 dt | |
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51 @end group |
| 3294 | 52 @end example |
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53 |
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54 @end ifnottex |
| 3294 | 55 |
| 56 @noindent | |
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57 using Hindmarsh's ODE solver @sc{lsode}. |
| 3294 | 58 |
| 6700 | 59 |
| 60 | |
| 3373 | 61 @DOCSTRING(lsode) |
| 3294 | 62 |
| 4115 | 63 @DOCSTRING(lsode_options) |
| 64 | |
| 3294 | 65 Here is an example of solving a set of three differential equations using |
| 66 @code{lsode}. Given the function | |
| 67 | |
| 68 @cindex oregonator | |
| 69 | |
| 70 @example | |
| 71 @group | |
| 72 function xdot = f (x, t) | |
| 73 | |
| 74 xdot = zeros (3,1); | |
| 75 | |
| 76 xdot(1) = 77.27 * (x(2) - x(1)*x(2) + x(1) \ | |
| 77 - 8.375e-06*x(1)^2); | |
| 78 xdot(2) = (x(3) - x(1)*x(2) - x(2)) / 77.27; | |
| 79 xdot(3) = 0.161*(x(1) - x(3)); | |
| 80 | |
| 81 endfunction | |
| 82 @end group | |
| 83 @end example | |
| 84 | |
| 85 @noindent | |
| 86 and the initial condition @code{x0 = [ 4; 1.1; 4 ]}, the set of | |
| 87 equations can be integrated using the command | |
| 88 | |
| 89 @example | |
| 90 @group | |
| 91 t = linspace (0, 500, 1000); | |
| 92 | |
| 93 y = lsode ("f", x0, t); | |
| 94 @end group | |
| 95 @end example | |
| 96 | |
| 97 If you try this, you will see that the value of the result changes | |
| 98 dramatically between @var{t} = 0 and 5, and again around @var{t} = 305. | |
| 99 A more efficient set of output points might be | |
| 100 | |
| 101 @example | |
| 102 @group | |
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103 t = [0, logspace(-1, log10(303), 150), \ |
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104 logspace(log10(304), log10(500), 150)]; |
| 3294 | 105 @end group |
| 106 @end example | |
| 107 | |
| 108 See Alan C. Hindmarsh, @cite{ODEPACK, A Systematized Collection of ODE | |
| 109 Solvers}, in Scientific Computing, R. S. Stepleman, editor, (1983) for | |
| 110 more information about the inner workings of @code{lsode}. | |
| 111 | |
| 4167 | 112 @node Differential-Algebraic Equations |
| 3294 | 113 @section Differential-Algebraic Equations |
| 114 | |
| 4115 | 115 The function @code{daspk} can be used to solve DAEs of the form |
| 3294 | 116 @tex |
| 117 $$ | |
| 118 0 = f (\dot{x}, x, t), \qquad x(t=0) = x_0, \dot{x}(t=0) = \dot{x}_0 | |
| 119 $$ | |
| 120 @end tex | |
| 6700 | 121 @ifnottex |
| 3294 | 122 |
| 123 @example | |
| 124 0 = f (x-dot, x, t), x(t=0) = x_0, x-dot(t=0) = x-dot_0 | |
| 125 @end example | |
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126 |
| 6700 | 127 @end ifnottex |
| 3294 | 128 |
| 129 @noindent | |
| 6700 | 130 where |
| 131 @tex | |
| 132 $\dot{x} = {dx \over dt}$ | |
| 133 @end tex | |
| 134 @ifnottex | |
| 135 @math{x-dot} | |
| 136 @end ifnottex | |
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137 is the derivative of @math{x}. The equation is solved using Petzold's |
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138 DAE solver @sc{daspk}. |
| 4115 | 139 |
| 140 @DOCSTRING(daspk) | |
| 141 | |
| 142 @DOCSTRING(daspk_options) | |
| 3294 | 143 |
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144 Octave also includes @sc{dassl}, an earlier version of @sc{daspk}, |
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145 and @sc{dasrt}, which can be used to solve DAEs with constraints |
| 4115 | 146 (stopping conditions). |
| 3294 | 147 |
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148 @DOCSTRING(dassl) |
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149 |
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150 @DOCSTRING(dassl_options) |
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151 |
| 4115 | 152 @DOCSTRING(dasrt) |
| 153 | |
| 154 @DOCSTRING(dasrt_options) | |
| 3294 | 155 |
| 156 See K. E. Brenan, et al., @cite{Numerical Solution of Initial-Value | |
| 157 Problems in Differential-Algebraic Equations}, North-Holland (1989) for | |
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158 more information about the implementation of @sc{dassl}. |