Derivative at the point

Question

How can I get a derivate at the point? I've specifyed a boundary condition for my PDE:

  lb = - c D[u[x, t], x][a, t] + d u[a, t] == gamma

When I present the function as

a = 0;
c = 1; d= 3;
u[x_, t_] = f[x, t] w[x]
f[x_, t_] = E^(-k t) + 1; 
    k = 5;
w[x_] = x^2 Cos[x];

and try to express the right part:

Solve[lb, gamma]

I get

{{gamma -> -(2 (1 + E^(-5 t)) x Cos[x] - (1 + E^(-5 t)) x^2 Sin[
     x])[0, t]}}

So, variables x and t are still free. But I want Mathematica to substitute x=0 , t=t and get {gamma ->0} above.

Answer 1

a = 0;
c = 1;
d = 3;
k = 5;
f[x_, t_] = E^(-k t) + 1;
w[x_] = x^2 Cos[x];
u[x_, t_] = f[x, t] w[x];

I don't understand what you mean by the notation D[u[x, t], x][a, t] in the definition of lb. Since D[u[x, t], x] is not a pure function, it does not take arguments. Assuming that the definition of lb should read

lb = -c D[u[x, t], x] + d u[a, t] == gamma

(*  -2 (1 + E^(-5 t)) x Cos[x] + (1 + E^(-5 t)) x^2 Sin[x] == gamma  *)

soln = Solve[lb, gamma][[1]] // Simplify

(*  {gamma -> E^(-5 t) (1 + E^(5 t)) x (-2 Cos[x] + x Sin[x])}  *)

lb /. soln // Simplify

(*  True  *)

If instead you mean

lb = -c (D[u[x, t], x] /. x -> a) + d u[a, t] == gamma

(*  0 == gamma  *)

Alternatively, you can write the derivative as

lb = -c Derivative[1, 0][u][a, t] + d u[a, t] == gamma

(*  0 == gamma  *)