To solve the Partial Differential Equations you can use MATLAB which has a special toolbox for PDF(Partial Differential Equations), but if you not familiar with that and want to solve your problem manually, I'm sharing some MATLAB files(.m) with you, you can use to solve with the help of them, note that MATLAB has more accuracy than the other softwares.
If you want EXACT solutions to constant-coefficients linear second-order partial differential equations (heat, wave and Laplace equations, say) under very general boundary conditions, obtained by Fourier methods, you can try our Maxima's package "pdefourier". You can see some examples here:
https://github.com/emmanuelroque/pdefourier
and there is an accompanying paper:
Preprint Automated solving of constant-coefficients second-order line...
There are some general software around that deal with PDEs like Matlab PDE tool box , comsol, femlab, etc. But, this depends on the problem you want to solve and the domain. Your familiarity with the numerical method used by the software is important.
Of the PDE solver's out there I believe COMSOL Multiphysics is one of the more straight forward solvers. It doesn't have a tremendous learning curve. On this thread there are several excellent suggestions. I am not super computer savy and I was/am able to use COMSOL effectively which is why I recommend it. But definitely investigate all the suggestions that have been given.