Can you please suggest the text book or journal paper(s) of how to solve two dimensional partial differential equation with four boundary conditions and two initial conditions? The applications are temperature distribution in metal cutting, heat transfer rate, etc. Can you please suggest the software to solve the same.
First it is better to know domain over which your PDE which is an initial boundary value problem is defined. Since you have two initial conditions which I assume as u(x,y,0) and ut(x,y,0), then it is a hyperbolic equation.If your initial conditions are different than these, you are solving a different which most likely a two phase problem. Any way these problems can be solved with the help of ADI (Alternating Direction Implicit) method for a rectangular domain. You may see G. D. Smith or M. K. Jain or A. R. Mitchel book on the subject.
Is it time dependent; then you must look for crank- niclson method; if it is not; you need to look for finite element method ; you will see many programs if you look at internet; solvinge pde by matlab or c++ or c and by fortran
https://code.msdn.microsoft.com/Crank-Nicolson-Method-7bf91783
http://www.nada.kth.se/~jjalap/numme/FDheat.pdf
Article Crank Nicolson Method for Solving Parabolic Partial Differen...
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