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I have been trying to set up a system of ODEs that are ultimately a solution to Burgers equation with a source term, and it boils down to:

x' = 11v

v' = f(H,H_x,s,s_x)

where x = x(t), H = H(x,t), s = s(x) and H_x,s_x are the partial derivatives wrtx.

The problem comes that I do not have an explicit formula for H, all I have is an equation for H_t, and the knowledge that (int H)' = int s

By the chain rule,

H' = x'H_x + H_t

I know everything on the RHS except the H_x, so I thought since H also depends on time this needs to go into the system, giving

x' = 11v

v' = f(H,H_x,s,s_x)

H' = 11vH_x + H_t

which solves the problem of having H in the v' equation, but I am stumped as to how to deal with the H_x equation.

I am looking to be able to solve this with ODE45 on matlab, which I have never used before.

Any ideas?