[?] 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
 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