Integration of NDSolve interpolating function contains overflow, indeterminate or infinity

Question

I am solving the following PDE:

T = 1 + 2/2 (1 + Tanh[(x - 0.5) 8])
Q = -D[T, x]
eqn1 := G[x, v] - D[(1/3)*D[G[x, v], x], x] == (-v^4)*Exp[-v]*D[Q, x]

sol = NDSolve[eqn1, G, {x, 0, 1}, {v, 0, 15}]

and plot the solution

DensityPlot[G[x, v] /. sol, {x, 0, 1}, {v, 0, 15}]

which looks well-behaved. However, when I try to integrate the solution over one of the variables (v), Mathematica says "The integrand has evaluated to Overflow, Indeterminate, or Infinity for all sampling points in the region with boundaries":

intGdv = FunctionInterpolation[NIntegrate[G[x, v] /. sol, {v, 0, 15}], {x, 0, 1}]

Eventually, I want to differentiate this integral wrt x, i.e. to find $\partial_x(intGdv)$. What function can I use to perform this differentiation and why does Mathematica complain about the quality of the integrand above?

Answer 1

Instead of trying to manipulate the output of NDSolve, it is much easier to just augment the NDSolve so that it directly outputs what you need. For example:

{gsol, intGdv} = NDSolveValue[
    {eqn1, D[int[x,v], v] == G[x, v], int[x, 0] == 0},
    {G, int},
    {x, 0, 1},
    {v, 0, 15}
];

NDSolveValue::femibcnd: No DirichletCondition or Robin-type NeumannValue was specified for {G}; the result may not be unique.

Here are some density plots of the integral and its derivative:

DensityPlot[intGdv[x, v], {x, 0, 1}, {v, 0, 15}]
DensityPlot[Derivative[1, 0][intGdv][x, v], {x, 0, 1}, {v, 0, 15}]