Difference between revisions of "Systems of differential equations"

From JSXGraph Wiki
Jump to navigationJump to search
(New page: Display solutions of the ordinary differential equation :<math> y'= f(x,y)</math> with initial value <math>(x_0,y_0)</math>. <html> <form> f<sub>1</sub>(x,y)=<input type="text" id="odeinpu...)
 
 
(17 intermediate revisions by the same user not shown)
Line 1: Line 1:
 
Display solutions of the ordinary differential equation
 
Display solutions of the ordinary differential equation
:<math> y'= f(x,y)</math>
+
:<math> y_1'= f_1(x,y_1,y_2)</math>
with initial value <math>(x_0,y_0)</math>.
+
:<math> y_2'= f_2(x,y_1,y_2)</math>
 +
with initial values <math>(x_0,c_1)</math>, <math>(x_0,c_2)</math>.
 
<html>
 
<html>
 
<form>
 
<form>
f<sub>1</sub>(x,y)=<input type="text" id="odeinput1" value="(2-x)*y"><br />
+
f<sub>1</sub>(x,y1,y2)=<input type="text" id="odeinput1" value="y1+y2"><br />
f<sub>2</sub>(x,y)=<input type="text" id="odeinput2" value="(2-x)*y"><input type=button value="ok" onclick="doIt()">
+
f<sub>2</sub>(x,y1,y2)=<input type="text" id="odeinput2" value="y2+1"><input type=button value="ok" onclick="doIt()">
 
</form>
 
</form>
 
</html>
 
</html>
Line 11: Line 12:
 
var brd = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-11,11,11,-11]});
 
var brd = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-11,11,11,-11]});
 
var N = brd.create('slider',[[-7,9.5],[7,9.5],[-15,10,15]], {name:'N'});
 
var N = brd.create('slider',[[-7,9.5],[7,9.5],[-15,10,15]], {name:'N'});
var P1 = brd.create('point',[0,1], {name:'(x_0,y_1)'});
+
var P1 = brd.create('point',[1,-1], {name:'(x_0,c_1)'});
var line = brd.create('line',[function(){return -P1.X();},function(){return 1;},function(){return 0;}]);
+
var line = brd.create('line',[function(){return -P1.X();},function(){return 1;},function(){return 0;}],{visible:false});
var P2 = brd.create('glider',[0,2,line], {name:'(x_0,y_2)'});
+
var P2 = brd.create('glider',[1,-0.5,line], {name:'(x_0,c_2)'});
  
 
function doIt() {
 
function doIt() {
   var txt = JXG.GeonextParser.geonext2JS(document.getElementById("odeinput1").value);
+
   var txt1 = document.getElementById("odeinput1").value;
   f = new Function("x", "yy", "var y = yy[0]; var z = " + txt + "; return [z]");
+
  var txt2 = document.getElementById("odeinput2").value;
 +
 
 +
  var snip1 = brd.jc.snippet(txt1, true, 'x, y1, y2');
 +
  var snip2 = brd.jc.snippet(txt2, true, 'x, y1, y2');
 +
   f = function (x, yy) {
 +
      return [snip1(x, yy[0], yy[1]), snip2(x, yy[0], yy[1])];
 +
  }
 
   brd.update();
 
   brd.update();
 
}
 
}
  
 
function ode() {
 
function ode() {
   return JXG.Math.Numerics.rungeKutta(JXG.Math.Numerics.predefinedButcher.Heun, [P.Y()], [P.X(), P.X()+N.Value()], 200, f);
+
   return JXG.Math.Numerics.rungeKutta('heun', [P1.Y(),P2.Y()], [P1.X(), P1.X()+N.Value()], 200, f);
 
}
 
}
  
var g = brd.createElement('curve', [[0],[0]], {strokeColor:'red', strokeWidth:'2px'});
+
var g1 = brd.create('curve', [[0],[0]], {strokeColor:'red', strokeWidth:2, name:'y_1', withLabel:false});
g.updateDataArray = function() {
+
var g2 = brd.create('curve', [[0],[0]], {strokeColor:'black', strokeWidth:2, name:'y_2', withLabel:false});
 +
g1.updateDataArray = function() {
 
     var data = ode();
 
     var data = ode();
 
     var h = N.Value()/200;
 
     var h = N.Value()/200;
 +
    var i;
 +
 
     this.dataX = [];
 
     this.dataX = [];
 
     this.dataY = [];
 
     this.dataY = [];
     for(var i=0; i<data.length; i++) {
+
     for(i=0; i<data.length; i++) {
         this.dataX[i] = P.X()+i*h;
+
         this.dataX[i] = P1.X()+i*h;
 
         this.dataY[i] = data[i][0];
 
         this.dataY[i] = data[i][0];
 +
    }
 +
};
 +
g2.updateDataArray = function() {
 +
    var data = ode();
 +
    var h = N.Value()/200;
 +
    var i;
 +
 +
    this.dataX = [];
 +
    this.dataY = [];
 +
    for(i=0; i<data.length; i++) {
 +
        this.dataX[i] = P2.X()+i*h;
 +
        this.dataY[i] = data[i][1];
 
     }
 
     }
 
};
 
};
 
doIt();
 
doIt();
 
</jsxgraph>
 
</jsxgraph>
 +
 +
===See also===
 +
* [[Differential equations]]
 +
* [[Lotka-Volterra equations]]
 +
* [[Epidemiology: The SIR model]]
 +
* [[Population growth models]]
 +
* [[Autocatalytic process]]
 +
* [[Logistic process]]
 +
 +
===The underlying JavaScript code===
 +
<source lang="xml">
 +
<form>
 +
f<sub>1</sub>(x,y1,y2)=<input type="text" id="odeinput1" value="y1+y2"><br />
 +
f<sub>2</sub>(x,y1,y2)=<input type="text" id="odeinput2" value="y2+1"><input type=button value="ok" onclick="doIt()">
 +
</form>
 +
</source>
 +
<source lang="javascript">
 +
var brd = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-11,11,11,-11]});
 +
var N = brd.create('slider',[[-7,9.5],[7,9.5],[-15,10,15]], {name:'N'});
 +
var P1 = brd.create('point',[1,-1], {name:'(x_0,c_1)'});
 +
var line = brd.create('line',[function(){return -P1.X();},function(){return 1;},function(){return 0;}],{visible:false});
 +
var P2 = brd.create('glider',[1,-0.5,line], {name:'(x_0,c_2)'});
 +
 +
function doIt() {
 +
  var txt1 = document.getElementById("odeinput1").value;
 +
  var txt2 = document.getElementById("odeinput2").value;
 +
 +
  var snip1 = brd.jc.snippet(txt1, true, 'x, y1, y2');
 +
  var snip2 = brd.jc.snippet(txt2, true, 'x, y1, y2');
 +
  f = function (x, yy) {
 +
      return [snip1(x, yy[0], yy[1]), snip2(x, yy[0], yy[1])];
 +
  }
 +
  brd.update();
 +
}
 +
 +
function ode() {
 +
  return JXG.Math.Numerics.rungeKutta('heun', [P1.Y(),P2.Y()], [P1.X(), P1.X()+N.Value()], 200, f);
 +
}
 +
 +
var g1 = brd.create('curve', [[0],[0]], {strokeColor:'red', strokeWidth:2, name:'y_1', withLabel:false});
 +
var g2 = brd.create('curve', [[0],[0]], {strokeColor:'black', strokeWidth:2, name:'y_2', withLabel:false});
 +
g1.updateDataArray = function() {
 +
    var data = ode();
 +
    var h = N.Value()/200;
 +
    var i;
 +
 +
    this.dataX = [];
 +
    this.dataY = [];
 +
    for(i=0; i<data.length; i++) {
 +
        this.dataX[i] = P1.X()+i*h;
 +
        this.dataY[i] = data[i][0];
 +
    }
 +
};
 +
g2.updateDataArray = function() {
 +
    var data = ode();
 +
    var h = N.Value()/200;
 +
    var i;
 +
 +
    this.dataX = [];
 +
    this.dataY = [];
 +
    for(i=0; i<data.length; i++) {
 +
        this.dataX[i] = P2.X()+i*h;
 +
        this.dataY[i] = data[i][1];
 +
    }
 +
};
 +
doIt();
 +
</source>
 +
 +
[[Category:Examples]]
 +
[[Category:Calculus]]

Latest revision as of 13:34, 19 January 2017

Display solutions of the ordinary differential equation

[math] y_1'= f_1(x,y_1,y_2)[/math]
[math] y_2'= f_2(x,y_1,y_2)[/math]

with initial values [math](x_0,c_1)[/math], [math](x_0,c_2)[/math].

f1(x,y1,y2)=
f2(x,y1,y2)=

See also

The underlying JavaScript code

<form>
f<sub>1</sub>(x,y1,y2)=<input type="text" id="odeinput1" value="y1+y2"><br />
f<sub>2</sub>(x,y1,y2)=<input type="text" id="odeinput2" value="y2+1"><input type=button value="ok" onclick="doIt()">
</form>
var brd = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-11,11,11,-11]});
var N = brd.create('slider',[[-7,9.5],[7,9.5],[-15,10,15]], {name:'N'});
var P1 = brd.create('point',[1,-1], {name:'(x_0,c_1)'});
var line = brd.create('line',[function(){return -P1.X();},function(){return 1;},function(){return 0;}],{visible:false});
var P2 = brd.create('glider',[1,-0.5,line], {name:'(x_0,c_2)'});

function doIt() {
  var txt1 = document.getElementById("odeinput1").value;
  var txt2 = document.getElementById("odeinput2").value;

  var snip1 = brd.jc.snippet(txt1, true, 'x, y1, y2');
  var snip2 = brd.jc.snippet(txt2, true, 'x, y1, y2');
  f = function (x, yy) {
      return [snip1(x, yy[0], yy[1]), snip2(x, yy[0], yy[1])];
  }
  brd.update();
}

function ode() {
   return JXG.Math.Numerics.rungeKutta('heun', [P1.Y(),P2.Y()], [P1.X(), P1.X()+N.Value()], 200, f);
}

var g1 = brd.create('curve', [[0],[0]], {strokeColor:'red', strokeWidth:2, name:'y_1', withLabel:false});
var g2 = brd.create('curve', [[0],[0]], {strokeColor:'black', strokeWidth:2, name:'y_2', withLabel:false});
g1.updateDataArray = function() {
    var data = ode();
    var h = N.Value()/200;
    var i;

    this.dataX = [];
    this.dataY = [];
    for(i=0; i<data.length; i++) {
        this.dataX[i] = P1.X()+i*h;
        this.dataY[i] = data[i][0];
    }
};
g2.updateDataArray = function() {
    var data = ode();
    var h = N.Value()/200;
    var i;

    this.dataX = [];
    this.dataY = [];
    for(i=0; i<data.length; i++) {
        this.dataX[i] = P2.X()+i*h;
        this.dataY[i] = data[i][1];
    }
};
doIt();