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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, Optimization, Quadrature, Top |
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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, Differential-Algebraic Equations, Differential Equations, 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 @deftypefn {Loadable Function} {} lsode (@var{fcn}, @var{x0}, @var{t}, @var{t_crit}) |
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44 Return a matrix of @var{x} as a function of @var{t}, given the initial |
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45 state of the system @var{x0}. Each row in the result matrix corresponds |
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46 to one of the elements in the vector @var{t}. The first element of |
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47 @var{t} corresponds to the initial state @var{x0}, so that the first row |
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48 of the output is @var{x0}. |
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49 |
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50 The first argument, @var{fcn}, is a string that names the function to |
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51 call to compute the vector of right hand sides for the set of equations. |
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52 It must have the form |
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53 |
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54 @example |
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55 @var{xdot} = f (@var{x}, @var{t}) |
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56 @end example |
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57 |
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58 @noindent |
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59 where @var{xdot} and @var{x} are vectors and @var{t} is a scalar. |
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60 |
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61 The fourth argument is optional, and may be used to specify a set of |
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62 times that the ODE solver should not integrate past. It is useful for |
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63 avoiding difficulties with singularities and points where there is a |
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64 discontinuity in the derivative. |
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65 @end deftypefn |
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66 |
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67 Here is an example of solving a set of three differential equations using |
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68 @code{lsode}. Given the function |
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69 |
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70 @cindex oregonator |
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71 |
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72 @example |
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73 @group |
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74 function xdot = f (x, t) |
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75 |
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76 xdot = zeros (3,1); |
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77 |
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78 xdot(1) = 77.27 * (x(2) - x(1)*x(2) + x(1) \ |
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79 - 8.375e-06*x(1)^2); |
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80 xdot(2) = (x(3) - x(1)*x(2) - x(2)) / 77.27; |
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81 xdot(3) = 0.161*(x(1) - x(3)); |
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82 |
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83 endfunction |
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84 @end group |
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85 @end example |
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86 |
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87 @noindent |
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88 and the initial condition @code{x0 = [ 4; 1.1; 4 ]}, the set of |
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89 equations can be integrated using the command |
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90 |
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91 @example |
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92 @group |
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93 t = linspace (0, 500, 1000); |
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94 |
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95 y = lsode ("f", x0, t); |
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96 @end group |
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97 @end example |
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98 |
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99 If you try this, you will see that the value of the result changes |
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100 dramatically between @var{t} = 0 and 5, and again around @var{t} = 305. |
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101 A more efficient set of output points might be |
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102 |
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103 @example |
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104 @group |
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105 t = [0, logspace (-1, log10(303), 150), \ |
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106 logspace (log10(304), log10(500), 150)]; |
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107 @end group |
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108 @end example |
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109 |
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110 @deftypefn {Loadable Function} {} lsode_options (@var{opt}, @var{val}) |
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111 When called with two arguments, this function allows you set options |
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112 parameters for the function @code{lsode}. Given one argument, |
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113 @code{lsode_options} returns the value of the corresponding option. If |
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114 no arguments are supplied, the names of all the available options and |
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115 their current values are displayed. |
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116 @end deftypefn |
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117 |
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118 See Alan C. Hindmarsh, @cite{ODEPACK, A Systematized Collection of ODE |
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119 Solvers}, in Scientific Computing, R. S. Stepleman, editor, (1983) for |
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120 more information about the inner workings of @code{lsode}. |
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121 |
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122 @node Differential-Algebraic Equations, , Ordinary Differential Equations, Differential Equations |
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123 @section Differential-Algebraic Equations |
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124 |
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125 The function @code{dassl} can be used to solve DAEs of the form |
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126 @iftex |
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127 @tex |
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128 $$ |
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129 0 = f (\dot{x}, x, t), \qquad x(t=0) = x_0, \dot{x}(t=0) = \dot{x}_0 |
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130 $$ |
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131 @end tex |
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132 @end iftex |
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133 @ifinfo |
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134 |
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135 @example |
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136 0 = f (x-dot, x, t), x(t=0) = x_0, x-dot(t=0) = x-dot_0 |
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137 @end example |
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138 @end ifinfo |
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139 |
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140 @noindent |
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141 using Petzold's DAE solver @sc{Dassl}. |
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142 |
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143 @deftypefn {Loadable Function} {[@var{x}, @var{xdot}] =} dassl (@var{fcn}, @var{x0}, @var{xdot0}, @var{t}, @var{t_crit}) |
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144 Return a matrix of states and their first derivatives with respect to |
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145 @var{t}. Each row in the result matrices correspond to one of the |
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146 elements in the vector @var{t}. The first element of @var{t} |
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147 corresponds to the initial state @var{x0} and derivative @var{xdot0}, so |
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148 that the first row of the output @var{x} is @var{x0} and the first row |
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149 of the output @var{xdot} is @var{xdot0}. |
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150 |
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151 The first argument, @var{fcn}, is a string that names the function to |
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152 call to compute the vector of residuals for the set of equations. |
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153 It must have the form |
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154 |
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155 @example |
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156 @var{res} = f (@var{x}, @var{xdot}, @var{t}) |
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157 @end example |
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158 |
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159 @noindent |
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160 where @var{x}, @var{xdot}, and @var{res} are vectors, and @var{t} is a |
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161 scalar. |
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162 |
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163 The second and third arguments to @code{dassl} specify the initial |
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164 condition of the states and their derivatives, and the fourth argument |
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165 specifies a vector of output times at which the solution is desired, |
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166 including the time corresponding to the initial condition. |
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167 |
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168 The set of initial states and derivatives are not strictly required to |
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169 be consistent. In practice, however, @sc{Dassl} is not very good at |
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170 determining a consistent set for you, so it is best if you ensure that |
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171 the initial values result in the function evaluating to zero. |
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172 |
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173 The fifth argument is optional, and may be used to specify a set of |
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174 times that the DAE solver should not integrate past. It is useful for |
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175 avoiding difficulties with singularities and points where there is a |
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176 discontinuity in the derivative. |
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177 @end deftypefn |
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178 |
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179 @deftypefn {Loadable Function} {} dassl_options (@var{opt}, @var{val}) |
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180 When called with two arguments, this function allows you set options |
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181 parameters for the function @code{lsode}. Given one argument, |
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182 @code{dassl_options} returns the value of the corresponding option. If |
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183 no arguments are supplied, the names of all the available options and |
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184 their current values are displayed. |
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185 @end deftypefn |
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186 |
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187 See K. E. Brenan, et al., @cite{Numerical Solution of Initial-Value |
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188 Problems in Differential-Algebraic Equations}, North-Holland (1989) for |
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189 more information about the implementation of @sc{Dassl}. |
