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1 @c Copyright (C) 1996, 1997 John W. Eaton |
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2 @c This is part of the Octave manual. |
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3 @c For copying conditions, see the file gpl.texi. |
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4 |
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5 @node Differential Equations |
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6 @chapter Differential Equations |
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7 |
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8 Octave has two built-in functions for solving differential equations. |
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9 Both are based on reliable ODE solvers written in Fortran. |
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10 |
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11 @menu |
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12 * Ordinary Differential Equations:: |
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13 * Differential-Algebraic Equations:: |
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14 @end menu |
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15 |
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16 @cindex Differential Equations |
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17 @cindex ODE |
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18 @cindex DAE |
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19 |
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20 @node Ordinary Differential Equations |
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21 @section Ordinary Differential Equations |
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22 |
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23 The function @code{lsode} can be used to solve ODEs of the form |
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24 @iftex |
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25 @tex |
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26 $$ |
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27 {dx\over dt} = f (x, t) |
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28 $$ |
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29 @end tex |
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30 @end iftex |
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31 @ifinfo |
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32 |
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33 @example |
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34 dx |
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35 -- = f (x, t) |
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36 dt |
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37 @end example |
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38 @end ifinfo |
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39 |
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40 @noindent |
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41 using Hindmarsh's ODE solver @sc{Lsode}. |
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42 |
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43 @DOCSTRING(lsode) |
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44 |
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45 @DOCSTRING(lsode_options) |
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46 |
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47 Here is an example of solving a set of three differential equations using |
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48 @code{lsode}. Given the function |
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49 |
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50 @cindex oregonator |
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51 |
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52 @example |
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53 @group |
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54 function xdot = f (x, t) |
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55 |
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56 xdot = zeros (3,1); |
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57 |
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58 xdot(1) = 77.27 * (x(2) - x(1)*x(2) + x(1) \ |
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59 - 8.375e-06*x(1)^2); |
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60 xdot(2) = (x(3) - x(1)*x(2) - x(2)) / 77.27; |
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61 xdot(3) = 0.161*(x(1) - x(3)); |
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62 |
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63 endfunction |
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64 @end group |
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65 @end example |
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66 |
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67 @noindent |
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68 and the initial condition @code{x0 = [ 4; 1.1; 4 ]}, the set of |
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69 equations can be integrated using the command |
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70 |
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71 @example |
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72 @group |
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73 t = linspace (0, 500, 1000); |
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74 |
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75 y = lsode ("f", x0, t); |
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76 @end group |
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77 @end example |
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78 |
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79 If you try this, you will see that the value of the result changes |
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80 dramatically between @var{t} = 0 and 5, and again around @var{t} = 305. |
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81 A more efficient set of output points might be |
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82 |
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83 @example |
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84 @group |
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85 t = [0, logspace (-1, log10(303), 150), \ |
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86 logspace (log10(304), log10(500), 150)]; |
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87 @end group |
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88 @end example |
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89 |
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90 See Alan C. Hindmarsh, @cite{ODEPACK, A Systematized Collection of ODE |
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91 Solvers}, in Scientific Computing, R. S. Stepleman, editor, (1983) for |
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92 more information about the inner workings of @code{lsode}. |
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93 |
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94 @node Differential-Algebraic Equations |
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95 @section Differential-Algebraic Equations |
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96 |
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97 The function @code{daspk} can be used to solve DAEs of the form |
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98 @iftex |
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99 @tex |
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100 $$ |
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101 0 = f (\dot{x}, x, t), \qquad x(t=0) = x_0, \dot{x}(t=0) = \dot{x}_0 |
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102 $$ |
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103 @end tex |
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104 @end iftex |
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105 @ifinfo |
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106 |
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107 @example |
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108 0 = f (x-dot, x, t), x(t=0) = x_0, x-dot(t=0) = x-dot_0 |
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109 @end example |
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110 @end ifinfo |
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111 |
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112 @noindent |
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113 using Petzold's DAE solver @sc{Daspk}. |
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114 |
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115 @DOCSTRING(daspk) |
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116 |
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117 @DOCSTRING(daspk_options) |
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118 |
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119 Octave also includes @sc{Dassl}, an earlier version of @var{Daspk}, |
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120 and @var{dasrt}, which can be used to solve DAEs with constraints |
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121 (stopping conditions). |
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122 |
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123 @DOCSTRING(dasrt) |
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124 |
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125 @DOCSTRING(dasrt_options) |
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126 |
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127 See K. E. Brenan, et al., @cite{Numerical Solution of Initial-Value |
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128 Problems in Differential-Algebraic Equations}, North-Holland (1989) for |
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129 more information about the implementation of @sc{Dassl}. |
