Differential-Equations | Jupyter notebooks on differential equations | Machine Learning library

IDESolver is a general-purpose numerical integro-differential equation solver created by Josh Karpel. Its latest version allows the user to solve multidimensional, coupled IDEs. From the examples provided, an IDE like

with analytical solution (sin x, cos x), can be solved using the following piece of code:

Ok, so here is a somewhat simple numerical scheme, that shows conceptual properties of your system. It is analogous to (explicit) Euler's method. It can be easily generalized to an analogous implicit Euler-like method.

You are given:

The functions h(x), f(x) , tau_sx(x, t), tau_sy(x, t) and tau_by(x, t)

The constants g and rho

You are looking for :

The functions V(x, t) and eta(x, t) that satisfy the pair of differential equations above.

To be able to find solutions to this problem, you need to be given:

V(x, 0) = V0(x) and eta(0, t) = eta0(t)

Assume your domain is [0, L] X [0, T], where x in [0, L] and t in [0, T]. Discretize the domain as follows: choose M and N positive integers and let dx = L / M and dt = T / N. Then consider only the finite set of points x = m dx and t = n dt for any integers m = 0, 1, ..., M and n = 0, 1, ..., N.

I am going to restrict all functions on the finite set of points defined above and use the following notation for an arbitrary function funct:

funct(x, t) = funct[m, n] and funct(x) = funct[m] for any x = m dx and t = n dt.

Then, the system of differential equations can be discretized as

g*(h[m] + eta[m,n])*(eta[m+1, n] - eta[m,n])/dx = f[m]*(h[m] + eta[m,n])*V[m,n] + tau_sx[m,n]/rho

(V[m, n+1] - V[m,n])/dt = (tau_sy[m,n] - tau_by[m,n])/(rho*(h[m] + eta[m,n]))

Solve for eta[m+1,n] and V[m,n+1]

eta[m+1,n] = eta[m,n] + f[m]*V[m,n]*dx/g + tau_sx[m,n]*dx/(g*rho*(h[m] + eta[m,n]))

V[m,n+1] = V[m,n] + (tau_sy[m,n] - tau_by[m,n])*dt/(rho*(h[m] + eta[m,n]))

For simplicity, I am going to abbreviate the right hand sides of the equations above as

eta[m+1,n] = F_eta(m, n, eta[m,n], V[m,n])

V[m,n+1] = F_V(m, n, eta[m,n], V[m,n])

that is, something like

As a StackOverflow solution, you can use ForwardDiffSensitivity(convert_tspan=false) to work around this. Working code: