Partial symmetry of initial value problems

It is generally believed that the symmetry of initial value problems (IVP) must leave both the partial differential equations (PDEs) and initial conditions invariant. In this paper, we propose partial symmetry of IVP which needs less restrictive conditions on the PDEs and initial conditions and only leaves the IVP invariant on some non-empty subset of the whole solution set of the governing PDEs. Thus some symmetries which either only leave the PDEs invariant or only leave initial conditions invariant are partial symmetries of IVP. Then considering from two different starting points, we define two types of partial symmetry and give the corresponding algorithms where one starts with the PDEs and the other is from the initial conditions. Applications to Boussinesq equation and a class of nonlinear heat equation are performed.