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me5180_numerical/intro_to_numerical_integration.ipynb

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"pkg load odepkg"
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"## Intro to functions and m-files"
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"## Building our own functions."
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"Functions are saved in memory (or better yet) in a folder in your path or current directory\n",
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"Example of storing a function in memory\n",
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"$f(x,y) = (xy^{3}-x^{3}y)$"
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"f =\n",
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"@(x, y) (x .* y .^ 3 - x .^ 3 .* y)\n",
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"ans = 6\n"
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"f= @(x,y) (x.*y.^3-x.^3.*y)\n",
"f(1,2)"
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"x=linspace(-1,1,10); % 10 data points from -1 to 1\n",
"y=linspace(-1,1,10); % 10 data points from -1 to 1\n",
"[X,Y]=meshgrid(x,y); % 100 data points from -1 to 1 on the x- and y-axes\n",
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"Here we will save a function called `my_ode` as `my_ode.m` For a single-DOF forced spring-mass-damper:\n",
"\n",
"$\\ddot{x}+2\\dot{x}+9x=\\cos(t/2)$"
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"function dr = my_ode(t,r)\n",
" % Help documentation for \"my_ode.m\"\n",
" % input is time, t (s) and r=[position (m); velocity (m/s)] and time\n",
" % output is dr=[velocity (m/s); acceleration (m/s/s)] at time, t\n",
" % the ODE is defined by:\n",
" %\n",
" % a = -2*v-9*x+cos(t/2)\n",
" \n",
" dr=zeros(size(r));\n",
" dr(1)=r(2);\n",
" dr(2)=-2*r(2)-9*r(1)+cos(2*t);\n",
"\n",
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"'my_ode' is a command-line function\n",
"\n",
" Help documentation for \"my_ode.m\"\n",
" input is time, t (s) and r=[position (m); velocity (m/s)] and time\n",
" output is dr=[velocity (m/s); acceleration (m/s/s)] at time, t\n",
" the ODE is defined by:\n",
"\n",
" a = -2*v-9*x+cos(t/2)\n",
"\n",
"\n",
"Additional help for built-in functions and operators is\n",
"available in the online version of the manual. Use the command\n",
"'doc <topic>' to search the manual index.\n",
"\n",
"Help and information about Octave is also available on the WWW\n",
"at http://www.octave.org and via the help@octave.org\n",
"mailing list.\n"
]
}
],
"source": [
"help my_ode"
]
},
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"source": [
"## Euler approximation\n",
"\n",
"$r_{i+1}=r_{i}+\\frac{dr_{i}}{dt}\\Delta t$"
]
},
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"dt=0.1;\n",
"t=[0:dt:6*pi/2];\n",
"x0=0.1;\n",
"v0=0;\n",
"r=zeros(length(t),2);\n",
"r(1,:)=[x0;v0];\n",
"for i=2:length(t)\n",
" dr=my_ode(t(i-1),r(i-1,:));\n",
" r(i,:)=r(i-1,:)+dr*dt;\n",
"end\n",
" "
]
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"## Runge-Kutta\n",
"\n",
"\n",
"Using `ode23` and `ode45` have the same syntax\n",
"\n",
"Numerical integration algorithms for differential equations:"
]
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"text/plain": [
"<IPython.core.display.SVG object>"
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},
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}
],
"source": [
"[t23,r23]=ode23(@(r,t) my_ode(r,t),[t(1),t(end)],[x0 v0]);\n",
"[t45,r45]=ode45(@(r,t) my_ode(r,t),[t(1),t(end)],[x0 v0]);\n",
"plot(t23,r23(:,1),'-',t45,r45(:,1),'.')\n",
"legend('ode23','ode45')\n",
"axis([0 10 -0.2 0.3])\n",
"xlabel('time (s)')\n",
"ylabel('position (m)')"
]
}
],
"metadata": {
"celltoolbar": "Slideshow",
"kernelspec": {
"display_name": "Octave",
"language": "octave",
"name": "octave"
},
"language_info": {
"file_extension": ".m",
"help_links": [
{
"text": "MetaKernel Magics",
"url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
}
],
"mimetype": "text/x-octave",
"name": "octave",
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