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I just confuse why ***u[0,t]==Sin[\omega t] cannot be used as a variable error was showing

\[Beta] = 1;
\[Alpha] = 1;
ic = {u[y, 0] == 0};
bc = {u[0, t] == Sin[\[Omega]*t], (D[u[y, t], t] /. t -> 0) == 0};
pde1 = D[u[y, t], t] + \[Alpha]*D[u[y, t], {t, 2}] == 
   D[u[y, t], {y, 2}] + \[Beta]*D[u[y, t], t, y, y];
NDSolve[pde1, ic, bc, u, {y, 0, 1}, {t, 0, 1}]```
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    $\begingroup$ First you are using NDSolve so everything you aren't solving for needs to have been assigned a numeric value and your Omega has no numeric value. If I insert Omega=2; that makes that error go away. Second you get new errors. The docs show NDSolve[eqns,funs,vars], not NDSolve[equs,condition1,condition2,funs,vars] If you change your input to Beta=1;Alpha=1;Omega=1; NDSolve[{D[u[y,t],t]+Alpha*D[u[y,t],{t,2}]==D[u[y,t],{y,2}]+Beta*D[u[y,t],t,y,y],u[y,0]==0, u[0,t]==Sin[Omega*t],(D[u[y,t],t]/.t->0)==0},u,{y,0,1},{t,0,1}] then it complains inconsistent conditions and needs more eqns $\endgroup$
    – Bill
    Aug 7, 2023 at 17:31
  • $\begingroup$ @Bill Question remains, which "Method" and missing "BoundaryConditions" the Mathematica solution uses. It isn't "FiniteElement" or "MethodOfLines". $\endgroup$ Aug 8, 2023 at 7:21
  • $\begingroup$ \[Omega] = 10; \[Beta] = 0.2; \[Alpha] = 0.1; ic = {u[y, 0] == 0, (D[u[y, t], t] /. t -> 0) == 0}; bc = {u[0, t] == Cos[\[Omega]*t], u[1, t] == 0}; pde1 = D[u[y, t], t] + \[Alpha]*D[u[y, t], {t, 2}] == D[u[y, t], {y, 2}] + \[Beta]*D[u[y, t], t, y, y]; NDSolve[{pde1, ic, bc}, u, {y, 0, 1}, {t, 0, 1}, Method -> {"MethodOfLines", "SpatialDiscretization" -> {"TensorProductGrid", "MinPoints" -> 50, "MaxPoints" -> 50}, Method -> {"IDA", "ImplicitSolver" -> {"GMRES"}}}] What BCs are further need ? I tried many time but can't get any idea, any help please. $\endgroup$
    – FDMEM
    Aug 9, 2023 at 4:49

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