Geometric way of interpreting equations¶
Let's consider the following non-linear ordinary differential equation $$\frac{d}{dt}x = \sin x$$
The solution to this equation is given by:
$$t = \ln \frac{\csc x_0 + \cot x_0}{\csc x + \cot x}$$
Let us look at how the solution looks like. We make an implicit plot of the following form: $$\exp{t}\times (\csc x + \cot x) = (\csc x_0 + \cot x_0)$$
In [1]:
set(0, "defaultlinelinewidth", 2);
set (0, "defaulttextfontname", "TimesNewRoman")
set (0, "defaulttextfontsize", 20)
set (0, "DefaultAxesFontName", "TimesNewRoman")
set(0, 'DefaultAxesFontSize', 20)
close all
clear h
graphics_toolkit qt
h.ax = axes ("position", [0.05 0.4 0.5 0.5]);
function update_plot (obj, init = false)
## gcbo holds the handle of the control
h = guidata (obj);
replot = false;
recalc = false;
switch (gcbo)
case {h.print_pushbutton}
fn = uiputfile ("*.png");
print (fn);
case {h.grid_checkbox}
v = get (gcbo, "value");
grid (merge (v, "on", "off"));
case {h.minor_grid_toggle}
v = get (gcbo, "value");
grid ("minor", merge (v, "on", "off"));
case {h.plot_title_edit}
v = get (gcbo, "string");
set (get (h.ax, "title"), "string", v);
case {h.isovalue_slider}
recalc = true;
end
if (recalc || init)
x0 = 0.1+ 6.1*get (h.isovalue_slider, "value");
x = linspace(1e-6, 2*pi, 100);
t = linspace(1e-6, 10, 100);
[X, T] = meshgrid(x, t);
y = exp(T).*(1./sin(X) + 1./tan(X));
isoval = (1./sin(x0) + 1./tan(x0));
h.plot = contour(T, X, y, [isoval isoval]);
hold on;
h.plot = contour(T, X, X, [pi/2 3*pi/2]);
hold off;
set (h.isovalue_label, "string", sprintf ("isovalue: %.1f", x0));
guidata (obj, h);
else
#set (h.plot, "ydata", y);
end
end
## plot title
h.plot_title_label = uicontrol ("style", "text",
"units", "normalized",
"string", "plot title: (text)",
"horizontalalignment", "left",
"position", [0.6 0.85 0.35 0.08]);
h.plot_title_edit = uicontrol ("style", "edit",
"units", "normalized",
"string", "Please fill me! (edit)",
"callback", @update_plot,
"position", [0.6 0.80 0.35 0.06]);
## grid
h.grid_checkbox = uicontrol ("style", "checkbox",
"units", "normalized",
"string", "show grid\n(checkbox)",
"value", 0,
"callback", @update_plot,
"position", [0.6 0.65 0.35 0.09]);
h.minor_grid_toggle = uicontrol ("style", "togglebutton",
"units", "normalized",
"string", "minor\n(togglebutton)",
"callback", @update_plot,
"value", 0,
"position", [0.77 0.65 0.18 0.09]);
## print figure
h.print_pushbutton = uicontrol ("style", "pushbutton",
"units", "normalized",
"string", "print plot\n(pushbutton)",
"callback", @update_plot,
"position", [0.6 0.45 0.35 0.09]);
## isovalue
h.isovalue_label = uicontrol ("style", "text",
"units", "normalized",
"string", "isovalue:",
"horizontalalignment", "left",
"position", [0.05 0.25 0.35 0.08]);
h.isovalue_slider = uicontrol ("style", "slider",
"units", "normalized",
"string", "slider",
"callback", @update_plot,
"value", 0.2,
"position", [0.05 0.20 0.35 0.06]);
set (gcf, "color", get(0, "defaultuicontrolbackgroundcolor"))
guidata (gcf, h)
update_plot (gcf, true);
In [2]:
x = linspace(0, 2*pi);
dxdt = sin(x);
plot(x, dxdt);
hold on;
plot(x, 0*x);
hold off;
In [3]:
set(0, "defaultlinelinewidth", 5);
set (0, "defaulttextfontname", "TimesNewRoman")
set (0, "defaulttextfontsize", 20)
set (0, "DefaultAxesFontName", "TimesNewRoman")
set(0, 'DefaultAxesFontSize', 20)
function y = f(N, t, K, r)
y = r*N*(1-N/K);
end
K = 3;
r = 0.3;
N0 = K/3;
t=linspace(0,20,100)';
Ndot = @(N,t) f(N, t, K, r);
sol = lsode(Ndot, N0, t);
plot(t, sol, "-k", t, K*ones(size(t)), '--b');
xlabel("t"); ylabel("N(t)");
In [4]:
clear;
clc; close all;
h.ax = axes ("position", [0.05 0.4 0.5 0.5]); #reduce plot windows size to accomodate sliders
function y = f(N, t, K, r)
y = r*N*(1-N/K);
end
function update_plot (obj, init = false)
## gcbo holds the handle of the control
h = guidata (obj);
replot = false;
recalc = false;
switch (gcbo) # If we make any change then we replot
case {h.print_pushbutton}
fn = uiputfile ("*.png");
print (fn);
case {h.slider1}
recalc = true;
case {h.slider2}
recalc = true;
case {h.slider3}
recalc = true;
case {h.slider4}
recalc = true;
end
if (recalc || init)
k = 1+ 9*get (h.slider1, "value");
r = .1 + .85*get (h.slider2, "value");
N0 = 0.1 + 20*get (h.slider3, "value");
t_end = 10 + 40*get(h.slider4, "value");
set (h.label1, "string", sprintf ("k: %.1f", k));
set (h.label2, "string", sprintf ("r: %.1f", r));
set (h.label3, "string", sprintf ("N0: %.1f", N0));
set (h.label4, "string", sprintf ("t_end: %.1f", t_end));
t=linspace(0,t_end,100)';
Ndot = @(N,t) f(N, t, k, r);
sol = lsode(Ndot, N0, t);
plot(t, sol, "-k", t, k*ones(size(t)), '--b', t, k/2*ones(size(t)), '--r');
xlabel("t"); ylabel("N(t)");
guidata (obj, h);
else
set (h.plot, "ydata", y);
end
end
# specify the GUI plot properties like sliders and label formatting
## print figure
h.print_pushbutton = uicontrol ("style", "pushbutton",
"units", "normalized",
"string", "print plot\n(pushbutton)",
"callback", @update_plot,
"position", [0.6 0.45 0.35 0.09]);
## guess
h.label1 = uicontrol ("style", "text",
"units", "normalized",
"string", "Guess:",
"horizontalalignment", "left",
"position", [0.05 0.25 0.35 0.08]);
h.slider1 = uicontrol ("style", "slider",
"units", "normalized",
"string", "slider",
"callback", @update_plot,
"value", 0.1,
"position", [0.05 0.20 0.35 0.06]);
h.label2 = uicontrol ("style", "text",
"units", "normalized",
"string", "Iteration:",
"horizontalalignment", "left",
"position", [0.05 0.15 0.35 0.08]);
h.slider2 = uicontrol ("style", "slider",
"units", "normalized",
"string", "slider",
"callback", @update_plot,
"value", 0.5,
"position", [0.05 0.10 0.35 0.06]);
h.label3 = uicontrol ("style", "text",
"units", "normalized",
"string", "Guess:",
"horizontalalignment", "left",
"position", [0.6 0.25 0.35 0.08]);
h.slider3 = uicontrol ("style", "slider",
"units", "normalized",
"string", "slider",
"callback", @update_plot,
"value", 0.1,
"position", [0.6 0.20 0.35 0.06]);
h.label4 = uicontrol ("style", "text",
"units", "normalized",
"string", "Iteration:",
"horizontalalignment", "left",
"position", [0.6 0.15 0.35 0.08]);
h.slider4 = uicontrol ("style", "slider",
"units", "normalized",
"string", "slider",
"callback", @update_plot,
"value", 0.5,
"position", [0.6 0.10 0.35 0.06]);
set (gcf, "color", get(0, "defaultuicontrolbackgroundcolor"))
guidata (gcf, h)
update_plot (gcf, true);
