Numerical Methods for system of integral equations

[Konstantin1]

Are there any standart ways to solve such systems?
\[ \begin{cases} m(t, x) - f(t, x)= \int_{0}^{t} q(\tau,x) \, d\tau \\ u(t,x) = \int_{-\infty}^{+\infty} \frac{1}{2 \sqrt{\pi s t}} e^{-\frac{(x-\xi)^2}{4st}} f(t,x-\xi) \, d\xi \end{cases} \]

Unknown functions are \( f(t,x) \) and \( q(t,x) \), \( t \in [0,T] \), \( s > 0 \).

Can Tikhonov regularization method be used to solve such system?

 

[gtoguy97]

Yes, Tikhonov regularization method can be used to solve such a system. This method is often used to solve ill-posed problems, where the number of unknowns is greater than the number of equations. The idea is to introduce a regularization term in the cost function so that the solution can be found by minimizing the cost function. This technique has been used to solve various types of partial differential equations and systems of equations.