Convert differential equation to finite difference equation

[Hypatio]

I have the differential equation

\frac{dM}{dt}=4\pi \rho(r,t)r(t)^2\frac{dr}{dt}<br />

which is the first term from

M(t)=4\pi\int_0^{r(t)}C(r,t)r(t)^2dr

This describes the change in mass (M) of a sphere from a change in radius (r) given a density (rho) that depends on radius and time (t).

My problem is somewhat simple. I tried to convert this equation into a finite difference formula as follows:

M_1-M_0=4\pi \rho(r,t)r^2(r_1-r_0)

where subscript 1 indicates the value at a new timestep.

I must be doing something wrong because the volume of a sphere requires a 1/3 to come from somewhere on the right hand side..

 

[arildno]

Hmm..
1. Haven't you forgotten the integral term in dM/dt that sums up the rate of mass change due to the local rate of change of the density function?

 

[Hypatio]

Hmm..
1. Haven't you forgotten the integral term in dM/dt that sums up the rate of mass change due to the local rate of change of the density function?

Yes the full form is

\frac{dM}{dt}=4\pi \rho(r,t)r(t)^2\frac{dr}{dt}+4\pi\int_0^{r(t)}\frac{\partial \rho}{\partial t}x^2 dx

but I don't see how the 1/3 for the first term can come from the additional term. My problem with the finite-difference conversion would remain if I assumed \rho did not depend on time (thus removing the last term).

 

[arildno]

Yes the full form is

\frac{dM}{dt}=4\pi \rho(r,t)r(t)^2\frac{dr}{dt}+4\pi\int_0^{r(t)}\frac{\partial \rho}{\partial t}x^2 dx

but I don't see how the 1/3 for the first term can come from the additional term. My problem with the finite-difference conversion would remain if I assumed \rho did not depend on time (thus removing the last term).

As it should be.
Let's review the case in which the density is constant.

We then have:
M(t+\bigtriangleup{t})=\frac{4\pi\rho}{3}r(t+\bigtriangleup{t})^{3}\approx\frac{4\pi\rho}{3}(r(t)+\frac{dr}{dt}\bigtriangleup{t})^{3}\approx\frac{4\pi\rho}{3}r(t)^{3}+{4\pi}r(t)^{2}\frac{dr}{dt}\bigtriangleup{t}=M(t)+{4\pi}r(t)^{2}\frac{dr}{dt}\bigtriangleup{t}
when we discard non-linear terms in the time interval.

 

[Hypatio]

I still don't see where 1/3 and the r^3 are coming from. It looks like it comes out of an integration over time but the dr/dt in such an integral seems to make it go to r^4.

 

[Chestermiller]

Yes the full form is

\frac{dM}{dt}=4\pi \rho(r,t)r(t)^2\frac{dr}{dt}+4\pi\int_0^{r(t)}\frac{\partial \rho}{\partial t}x^2 dx

but I don't see how the 1/3 for the first term can come from the additional term. My problem with the finite-difference conversion would remain if I assumed \rho did not depend on time (thus removing the last term).

\frac{dM}{dt}=\frac{4\pi}{3} \rho(r^3,t)\frac{dr^3}{dt}+\frac{4\pi}{3}\int_0^{r^3(t)}\frac{\partial \rho}{\partial t}dx^3