NDSolveValue for 2-D heat equation (PDE) coupled with ODE boundary and DiracDelta function

Question

Recently, I am studying on the point source problem (PDE with DiracDelta) for heat equation. The equations are as follow

The problems are arisen from not only the DiracDelta issues but the nonlinear boundary condition at X = 0.

But I still try to using numerical method to solve, and the code is given below

 RegularizedDelta[Gamma_, X_List, Xs_List] := 
     Piecewise[{{Times @@ 
         Thread[1/(4 Gamma) (1 + Cos[Pi/(2 Gamma) (X - Xs)])], 
        And @@ Thread[RealAbs[X - Xs] <= 2 Gamma]}, {0, True}}];
Timing[
    a=2;b=15;c=25;A=1000;
     Monitor[AbsoluteTiming[sys =
        {
    D[rou[X,Y,T],T]==D[rou[X,Y,T],X,X]+D[rou[X,Y,T],Y,Y], 
     + aRegularizedDelta[0.01, {X}, {1}]RegularizedDelta[0.01, {Y}, {0}] -
            NeumannValue[brou[X,Y,T]-phi[T], X == 0],
         rou[X,Y,0] == 0, cD[phi[T],T]== b*rou[0,Y,T]-phi[T],phi[0]=0
         };
   {rou,phi} = 
    NDSolveValue[sys, {rou,phi}, {X, 0, A}, {Y, 0, A}, {T, 0, 1000000}, 
     Method -&gt; {&quot;MethodOfLines&quot;, 
       &quot;SpatialDiscretization&quot; -&gt; {&quot;FiniteElement&quot;, 
         &quot;MeshOptions&quot; -&gt; {&quot;MaxCellMeasure&quot; -&gt; 0.009}}}, 
     EvaluationMonitor :&gt; (currentTime = Row[{&quot;t =. &quot;, CForm[t]}]), 
     InterpolationOrder -&gt; All];], currentTime];]

However, MMA gave a warning message:

NDSolveValue::underdet: There are more dependent variables, {rou[X,Y,T],phi[T]}, than equations, so the system is underdetermined.

This problem bothers me for few months. Does anyone have experience to address this issue? Many thanks.

As mentioned under your previous question, NDSolve cannot handle coupled PDE and ODE. — xzczd, Sep 08 '21 at 11:24
As xzczd correctly points out, currently (V12.3) you can not solve ODEs coupled to PDEs. In the HeatTransfer PDE model tutorial there is an example of a similar case and how it is solved with the finite element method, perhaps useful. — user21, Sep 08 '21 at 12:42
Thank for your kindly replies. Actually I am now trying pdetoode developed by xzczd but facing the memory issue. I will update my code based on this. — LingLong, Sep 08 '21 at 21:39
I’m voting to close this question because the OP has not updated the code code as promised — m_goldberg, Sep 14 '21 at 00:57
I decide to close this issue because I solve it in another way and avoid the ODE coupling with PDE issue. So the issue of that becomes minor. — LingLong, Sep 15 '21 at 23:18

NDSolveValue for 2-D heat equation (PDE) coupled with ODE boundary and DiracDelta function

0 Answers0