Direct and inverse problems for a nonlinear time fractional equation are
studied. It is proved that the direct problem has a unique weak solution
and the solution depends continuously on the nonlinear coefficient. Then
it is shown that the inverse problem has a quasi-solution. The direct
problem is solved by method of lines using an operator approach. A
quasi-Newton optimization method is used for the numerical solution to
the inverse problem. Tikhonov regularization is used to overcome the
ill-posedness of the inverse problem. Numerical examples with noise free
and noisy data illustrate applicability and accuracy of the proposed
method to some extent.