Abstract: In this paper, under assumption that tempeature is linearly distributed along the thickness of the shell, we deal with problems as indicated in the title and obtain general solutions of them which are expressed in analytic form.In the first part, we investigate free vibration of circular shallow spherical shells with circular holes at the center under usual arbitrary boundary conditions. As an example, we calculate fundamental natural frequency of a circular shallow spherical shell whose edge is fixed (m=0). Results we get are expressed in analytic form and check well with E. Reissner’s [1]. Method for calculating frequency equation is recently suggested by Chien Wei-zang and is to be introduced in appendix 3.In the second part, we investigate forced vibration of shells as indicated in the title under arbitrary harmonic temperature field and arbitrary harmonic dynamic normal load.In the third part, we investigate forced vibration of the above mentioned shells with initial conditions under arbitrary unsteady temperature field and arbitrary normal load.In appendix 1 and 2, we discuss how to express displacement boundary conditions with stress function and boundary conditions in the case m=1.

Key words: graph, factor, factorization, orthogonal factorization