I have a problem to solve the following propagation equation :
$$I(s,t)=\int_0^s \frac{∂f(x,t)}{∂t}*\sin(f(x,t)) \,dx$$
or $$I(s,t)=-\frac{∂}{∂t}\int_0^s \cos(f(x,t)) \,dx$$
where $f(x,t)$ is a continuous function describing the concentration of the dye at position x of the pipe of length $L=1$ at time t, and $I(s,t)$ represents the measured intensity along the pipe at time t. $I(s,t)$ is known experimentally for different times and by this equation the local concentration along the pipe can be extracted for these different times, but how ?
Assuming an initial distribution of values for $f (t=0)$, the problem is to determine the function f for the different times $(t+dt, t+2dt, \ldots)$. I tried to numerically evaluate the first equation with equally spaced xi points by using a finite difference approximation for the time derivative
$$\frac{∂f(x,t)}{∂t}=\frac{f(x,t+dt)-f(x,t)}{dt}$$
and the composite trapezoidal rule for the spatial integral. The value of f(xi,t+dt) can be obtained from these combined numerical schemes :
$$f(xi,t+dt)=f(xi,t)+\frac{2dt}{dx}\frac{I(xi,t)}{\sin(f(xi,t))}-2\sum_{j=2}^{i-1} \frac{\sin(f(xj,t))}{\sin(f(xi,t))}$$
but unfortunately spurious oscillations appeared, and even predictor-corrector schemes do not avoid these oscillations. I would greatly appreciated if someone could enlighten me on other numerical schemes that could be used to numerically solve this equation without oscillations. Thanks in advance for your time.