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An inverse source problem associated with a semi-linear transport or one-way wave equation in one

spatial dimension is considered. It is shown an analytic solution to the inverse problem can be

given and furthermore, that this inverse problem of determination of a source function is

ill-posed, and must be regularised. A novel regularisation scheme which combines least squares

monotone approximation and mollification of the noisy data is used to provide this regularisation.

Proof of convergence of this regularisation scheme of \emph{monotone smoothing} is given. Numerical

solutions from the inverse problems are presented showing that the method is robust to noisy

signals.



The solution of this inverse problem is also shown to illustrate the behaviour of more complex

problems from electromagnetism and nonlinear optics. The mathematical techniques that are developed

are therefore applicable to other sets of nonlinear first order equations. The method is therefore

model independent.