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How to obtain the exact solution of a partial differential equation?


I know that Mathematica can solve a PDE numerically, but I wonder if it is possible to obtain the exact solution. For example, consider the heat equation


ut=κuxx


Is it possible to solve it with a set of initial and boundary conditions to calculate the exact equation of u


u=f(x,t)

and u(x=0)=f(t)


I don't need numeral solution or the graph but the general equations.


EXAMPLE


One dimensional heat flow in an slab, one side is insulated and the other side at a constant flux of heat


u(x,0)=Uux(0,t)=0ux(L,t)=T



The solution is available from the textbooks. I just wonder, if Mathematica can give us the solution, as we can slightly alter the conditions to find new solutions.




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