Inverse source problem for a time-fractional diffusion equation with nonlocal boundary conditions

•The time-fractional heat equation with nonlocal boundary conditions is considered.•The time-dependent source term is determined by energy measurement.•The non-self adjoint auxiliary spectral problem is derived.•The well-posedness of the inverse problem is showed by generalized Fourier method.

In this paper, an inverse problem of determining a time-dependent source term in a one-dimensional time-fractional diffusion equation from the energy measurement is studied. This problem is obtained from a classical diffusion problem by replacing the time derivative with a fractional derivative. The well-posedness of the inverse problem is shown by using eigenfunction expansion of a non-self adjoint spectral problem along the generalized Fourier method.