The energy spectra and composition of ultra-high energy cosmic rays are changing in a course of propagation in the expanding Universe filled with background radiation. We use a numerical code for solution of inverse problem for cosmic-ray transport equations that enables the determination of source spectra of different nuclei from the cosmic ray spectra observed at the Earth. The source spectrum is calculated without ad hoc assumptions about its' shape. The spectra of protons and iron nuclei injected from the extragalactic sources are found. The effects of cosmological source evolution and the finite distance to the nearest source are studied. The data from the Auger experiment are used to illustrate the method.