Please recommend some books/links for numerical solution of inverse source problem

Question

Equation: $$ \frac{\partial f}{\partial t} = \frac{\partial^2 f}{\partial x^2} + k(x,t)\frac{\partial f}{\partial x} + \psi(t)\theta(x) $$ $$ x \in [0, 1], t \in [0, T] $$ boundary conditions: $$ f(0,t)=f_1(t) $$ $$ f(1,t)=f_2(t) $$ initial condition: $$ f(x,0)=f_3(x) $$ and additional condition: $$ f(x,T)=f_4(x) $$

Suppose we know $ \theta(x) $ and $k(x,t)$. Are there any numerical methods for determining pair of functions $f(x,t)$ and $\psi(t)$ using information above. I will be grateful for any links to books and/or papers.

Answer 1

You may have a look to the following book, especially Chapter 3:

A. H. Hasanoğlu and V. G. Romanov, Introduction to Inverse Problems for Differential Equations, Springer, Cham, 2017.