In solving our PDE(in particular take heat equation or wave equation and $n=1$) if we have given our initial condition on the half plane then we use method of reflection to extend the initial condition to the whole domain $\mathbf R$, but i am confused where to use method of reflection by odd function and where to use method of reflection by even function to start solving the PDE?
Any type of help will be appreciated. Thanks in advance.
You need to use a reflection that will enforce your boundary condition. Odd functions are zero at the origin while even (differentiable) functions have zero derivative; so odd reflections enforce a Dirichlet condition while even reflections enforce a Neumann condition.
If your problem doesn't have a boundary condition then the two types of reflection will produce two different solutions with the same initial data, which is a manifestation of the fact that you need boundary conditions to get well-posed problems on bounded domains.