Special Examples, Methods, Algorithms, Displays, etc. (under construction)

Section One: Differential Algebraic Equations (DAE's)

Plot of Solution:

Jsim is able to solve some types of DAE's directly. An example of such a system is given by:

$-{\frac {d}{dt}}x+2\,y{\frac {d}{dt}}y=0$

$y{\frac {d}{dt}}x+2\,xt{\frac {d}{dt}}y=2\,x+2\,{t}^{2}y$

$x{\frac {d}{dt}}z+zy \left( {\frac {d}{dt}}y \right) ^{3}=0$

$r{\frac {d}{dt}}y+stz={t}^{-1}+4\,t$

$rx-sty=-t \left( 2\,{t}^{2}-1 \right)$

${\it Boundary\ Conditions}= \left\{ x \left( 0 \right) =1,y \left( 0 \right) =1,z \left( 0 \right) =1 \right\}$

The exact solution is contained in the comments of the code.

Section Two: Stochastic Models: The Brusselator equations

Plot of Solution:

Stochastic Models can be written in JSim's Mathematical Modeling Language. This model has been written both as an ODE model and a stochastic model so the user can learn how to write a stochastic model. Stochastic models are used when the number of molecules interacting is small and the reaction cannot be represented by continuum mechanics. Consider the following reactions:

A -> X, k1
2X+Y -> 3X, k2
B+X -> Y + C, k3
X -> D, k4

where A, B, C, D are constant and only X and Y vary.

The governing equations are

${\frac {d}{dt}}X={\it k1}\,A+{\it k2}\,{X}^{2}Y-{\it k3}\,BX-{\it k4}\,X$

and

${\frac {d}{dt}}Y=-{\it k2}\,{X}^{2}Y+{\it k3}\,BX$