First order integro differential equation

[Wisam]

Can anyone help me to solve a differential equation?
I want to solve

∂v(p,t)/∂t=-p^2 v(p,t)-sqrt(2/pi)∫v(p,t)[1-δ(t)R(t)]dp+sqrt(2/pi)[δ(t)R^2(t) C]
with initial data v(p,0)=0

where C is constant and the integration from zero to infinty
Any suggestion please?

Solution by volterra integral equation??

 

[BvU]

Do I read this right ? You want to solve $${\partial v(p, t) \over \partial t} = - p^2 v(p, t) - \sqrt{2\over \pi} \int_0^\infty \ v(p,t)\ \left [ 1 - \delta(t) R(t) \right ] \, dp \ \ + \sqrt{2\over \pi} C\, \delta(t) \,R^2(t) \ \ ? $$with $\ v(p, t) = 0\$ and $R(t)$ a given function of time ?

(where does it come from ? what do the symbols stand for ?)

--

Any link with earlier posts (that seem to have petered out somewhat ) ?

 

[Wisam]

Do I read this right ? You want to solve $${\partial v(p, t) \over \partial t} = - p^2 v(p, t) - \sqrt{2\over \pi} \int_0^\infty \ v(p,t)\ \left [ 1 - \delta(t) R(t) \right ] \, dp \ \ + \sqrt{2\over \pi} C\, \delta(t) \,R^2(t) \ \ ? $$with $\ v(p, t) = 0\$ and $R(t)$ a given function of time ?

(where does it come from ? what so the symbols stand for ?)

--

Any link with earlier posts (that seem to have petered out somewhat ) ?

Dear BvU,
Thank you for your replay, yes the equation is right.
The field equation is diffusion equation with 2 free boundary conditions
I applied the Fourier transform for the diffusion and the boundary conditions and finally i got this first ODE
I stuck on it ?
any idea please?

 

[BvU]

(Sorry for mistyping $\ v(p, t) = 0\$ -- should of course have been $\ v(p, 0) = 0\$ as you wrote).

Pretty hefty ! And does the $\delta(t)$ represent a time-dependent coefficient or is it the Kronecker delta function (in which case the term with the fector C is a bit problematic) ?

I hope someone more knowledgeable reads this and helps out, for me it's not obvious how to start with such a thing...

 

[Wisam]

(Sorry for mistyping $\ v(p, t) = 0\$ -- should of course have been $\ v(p, 0) = 0\$ as you wrote).

Pretty hefty ! And does the $\delta(t)$ represent a time-dependent coefficient or is it the Kronecker delta function (in which case the term with the fector C is a bit problematic) ?

I hope someone more knowledgeable reads this and helps out, for me it's not obvious how to start with such a thing...

Thank you BvU and $\delta(t)$ is represent a time-dependent.
I hope someone can help me in this...

 

[BvU]

I hope so too. My recollection of diffusion is that it gives equations like $${\partial u(x, t) \over \partial t} = {\partial^2 u\over \partial x^2}$$ so I have a hard time putting your equation into a context. But, as you say in your post #3, it is an intermediate situation in a solution procedure that involves Fourier transforms. I'll have to read up on that (little time for that ) and even then you probably have to spell out what you are doing from the beginning before I can be of any use, so we'll have to wait for help...

Oh, and

$\delta(t)$ is represent a time-dependent.

doesn't tell me much.

 

[BvU]

If you can't wait that long, here's what I'm reading. Particularly pages 110 and further