DE with Dirac/Delta Impulse

  • #1
Hey experts,

I have the following differential equation with an impulse described by the Dirac delta function

dx/dt (t) = - a⋅x(t) + d⋅delta(t) x(0)=0

with a,d scalars. My problem is this delta function in the right hand side.

Is there an equivalent formulation without the delta function, e.g. putting the d into the initial value?

How could I solve this numerically, e.g. in Matlab?

Many thanks for any suggestions.
 

Answers and Replies

  • #3
HallsofIvy
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The "delta function" is defined in terms of its integral properties so we want to be able to write a solution in terms of an integral. This is a linear equation so it is easy to find an "integrating factor". Write the equation as [itex]dx/dt+ ax= \delta(t)[/itex]. We want to find a function u(t) such that multiplying by it, to get u dx/dt+ axu, gives an "exact derivative", d(ux)/dt. Of course, d(ux)/dt= u(dx/dt)+ (du/dt)x so, setting those equal, u(dx/dt}+ (du/dt)x= u (dx/dt)+ axu, we must have (du/dx)x= axu or just du/dt= au which has [itex]u= e^{at}[/itex] as a solution.

That is, our equation is just [itex]e^{at}(dx/dt)+ axe^{at}= d(e^{at}x)/dt= e^{at}\delta(t)[/itex] and we solve by integrating both sides, integrating from negative infinity to t:
The integral of [itex]e^{at}\delta(t)[/itex] is 0 if the integration does not include 0, [itex]e^{a(0)}= 1[/itex] if it does. [itex]e^{at}x(t)= C[/itex] or [itex]x(t)= Ce^{-at}[/itex] if t<0, [itex]e^{at}x(t)= 1+ C[/itex] or [itex]x(t)= (1+ C)e^{-at}[/itex] if [itex]t\ge 0[/itex]. Since we must have x(0)= 0, we must have x(0)= 1+ C= 0, C= -1.

[itex]x(t)= -e^{-at}[/itex] if t< 0, x(t)= 0 if [itex]t\ge 0[/itex].
 
  • #4
Thanks guys,

Unfortunately, the final system I am interested in is much more complicated and contains the following non-linear equation (michaelis menten) as a sub unit.

dx/dt = - (a*x(t))/(b+x(t)) + d *delta(t) x(0) = 0

The example above was just a simple example. So it is more proper to write

dx/dt = f(x(t)) + d*delta(t) x(0)=x0

where x and d are n-dimensional. Hence, I cannot solve this analytically and therefore, I am looking for an equivalent formulation without delta (if it exists) or some numerically techniques to integrate to whole system with delta.

I hardly remember a talk about impulsive differential equations. Is this a IDE?

Any ideas?
 
  • #5
Simon Bridge
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One approach to the numerical method is, depending on your method, to substitute for the Dirac delta function, an impulse that takes a finite amount of time but with similar properties under your numerical integration method.

i.e. if you method involves dividing the time axis into lots of smaller times ##\Delta t## then the ##\Delta t## block at t=0 corresponding to ##A\delta(t)## has value ##A/\Delta t## so the integral behaves itself.

I'm pretty sure HallsofIvy knows a better one though.
 
  • #6
DrDu
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So this is enzyme kinetics? I think it is useful with ill-defined mathematical problems to discuss what they describe physically to pick the right solution. What does the delta term stand for? Where does integration start (t=0 or t=-infinity)?
 

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