# [?] Obtain a symbolic solution of a PDE?

Posted 2 years ago
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 Resolve the following PDE: pde = {D[u[x, t], {x, 2}] == D[u[x, t], t]}; inc = {u[-L, t] == u[L, t], Derivative[1, 0][u][-L, t] == Derivative[1, 0][u][L, t], u[x, 0] == Sin[x]}; DSolve[Join[pde, inc], u[x, t], {x, t}, Assumptions -> L > 0] I have already seen in a textbook a symbolic solution to this PDE. I got the numerical solution with NDSolve. Is there any way to find the symbolic solution with DSolve?
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Posted 2 years ago
 Thanks Neil for the response and attention.
 Sinval,You should use the code formatting tool (the first button) so your code posts correctly. Your equation is the heat equation. The problem is with your boundary conditions -- I believe that you specified no information with your first and second initial conditions because the solution is symmetric in x. So for example, This works: heqn = D[u[x, t], t] == D[u[x, t], {x, 2}]; bc = {u[0, t] == 20, u[L, t] == 50}; ic = u[x, 0] == Sin[x]; sol = DSolve[{heqn, bc, ic}, u[x, t], {x, t}, Assumptions -> L > 0] But this will not: (I changed the boundary condition to -L and made them equal) heqn = D[u[x, t], t] == D[u[x, t], {x, 2}]; bc = {u[-L, t] == 20, u[L, t] == 20}; ic = u[x, 0] == Sin[x]; sol = DSolve[{heqn, bc, ic}, u[x, t], {x, t}, Assumptions -> L > 0] You can also use a derivative boundary condition for example: heqn = D[u[x, t], t] == D[u[x, t], {x, 2}]; bc = {u[L, t] == 20, Derivative[1, 0][u][0, t] == 0}; ic = u[x, 0] == Sin[x]; sol = DSolve[{heqn, bc, ic}, u[x, t], {x, t}, Assumptions -> L > 0] There is more information about this PDE in the help for DSolve (search for "Model the flow of heat in a bar of length 1 using the heat equation:").Regards,Neil