Systems of differential equations: Difference between revisions

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Line 2: Line 2:
:<math> y_1'= f_1(x,y_1,y_2)</math>
:<math> y_1'= f_1(x,y_1,y_2)</math>
:<math> y_2'= f_2(x,y_1,y_2)</math>
:<math> y_2'= f_2(x,y_1,y_2)</math>
with initial values <math>(x_0,y_1)</math>, <math>(x_0,y_2)</math>.
with initial values <math>(x_0,c_1)</math>, <math>(x_0,c_2)</math>.
<html>
<html>
<form>
<form>
Line 12: 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;}],{visible:false});
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 txt1 = JXG.GeonextParser.geonext2JS(document.getElementById("odeinput1").value);
   var txt1 = document.getElementById("odeinput1").value;
   var txt2 = JXG.GeonextParser.geonext2JS(document.getElementById("odeinput2").value);
   var txt2 = document.getElementById("odeinput2").value;
   f = new Function("x", "yy", "var y1 = yy[0], y1 = yy[1];  var z1 = " + txt1 + "; var z2 = " + txt2 + "; return [z1,z2];");
 
  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, [P1.Y(),P2.Y()], [P1.X(), P1.X()+N.Value()], 200, f);
   return JXG.Math.Numerics.rungeKutta('heun', [P1.Y(),P2.Y()], [P1.X(), P1.X()+N.Value()], 200, f);
}
}


var g1 = 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});
var g2 = brd.createElement('curve', [[0],[0]], {strokeColor:'black', strokeWidth:'2px'});
var g2 = brd.create('curve', [[0],[0]], {strokeColor:'black', strokeWidth:2, name:'y_2', withLabel:false});
g1.updateDataArray = function() {
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] = P1.X()+i*h;
         this.dataX[i] = P1.X()+i*h;
         this.dataY[i] = data[i][0];
         this.dataY[i] = data[i][0];
Line 42: Line 49:
     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] = P2.X()+i*h;
         this.dataX[i] = P2.X()+i*h;
         this.dataY[i] = data[i][1];
         this.dataY[i] = data[i][1];
Line 51: Line 60:
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 11:34, 19 January 2017

Display solutions of the ordinary differential equation

[math]\displaystyle{ y_1'= f_1(x,y_1,y_2) }[/math]
[math]\displaystyle{ y_2'= f_2(x,y_1,y_2) }[/math]

with initial values [math]\displaystyle{ (x_0,c_1) }[/math], [math]\displaystyle{ (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();