I would like to add gaussian noise for each x value of my sample data and then plot

Question

I am having some problems in plotting this. Everything is ok until the plot statement where nothing plots. Can someone please help me so that it can plot something. The following is my code:

j = 10; 
s = 0; r = 0;

B[n_] = Integrate[2*Sin[n*Pi*x]*(x), {x, 0, 1}];
u[x_, psi_] = Sum[B[n]*Sin[n*Pi*x]*Exp[-(n*Pi)^2*psi], {n, 1, j}];

K[x_, psi_] = 
  Sum[Sin[n*Pi*x]*
    Sin[n*Pi*
      psi]*(2*Exp[-(n*Pi)^2*
         Abs[s + r]] - (Exp[-(n*Pi)^2*Abs[s - r]] - 
         Exp[-(n*Pi)^2*(s + r)])/(n*Pi)^2 ), {n, 1, j}];

w = RandomReal[NormalDistribution[0, 1], 101];
d = Round[100*x + 1];
S = Total[Total[u[x, psi]/Length[u[x, psi]]] + w[d]]

T[x_, psi_] = Integrate[K[x - y, psi]*(y)*S, {y, -10, 10}]

Plot3D[T[x, psi], {x, 0, 1}, {psi, 0.01, 1}, 
 AxesLabel -> {"x", "t", "Temperature"}, Boxed -> False, 
 Mesh -> False]

Basically, I have some data from "u" and I want to make it noisy (from "w") for each "x" value and then perform the convolution in "T" and plot.

I will really appreciate anyone's kind help.

Thanks very much!

Answer 1

I'm not sure that I understand the problem you're trying to solve. However, modifying your code as shown below allows it to run - I rephrased several expression to be functions (a good rule of thumb is to use := if the left hand side involves a pattern, like B[n_]) and I removed some code that was apparently trying to treat scalars as vectors.

j = 10; s = 0; r = 0;

ClearAll[B];
B[n_] := B[n] = Integrate[2*Sin[n*Pi*a]*(a), {a, 0, 1}];

ClearAll[u];
u[x_, psi_] := Sum[B[n]*Sin[n*Pi*x]*Exp[-(n*Pi)^2*psi], {n, 1, j}];

K[x_, psi_] := 
  Sum[Sin[n*Pi*x]*
    Sin[n*Pi*
      psi]*(2*Exp[-(n*Pi)^2*Abs[s + r]] - (Exp[-(n*Pi)^2*Abs[s - r]] -
          Exp[-(n*Pi)^2*(s + r)])/(n*Pi)^2), {n, 1, j}];

S[x_, psi_] := u[x, psi] + RandomReal[NormalDistribution[]]

T[x_, psi_] := Integrate[K[x - y, psi]*(y)*S[x, psi], {y, -10, 10}]

Plot3D[T[x, psi], {x, 0, 1}, {psi, 0.01, 1}, 
 AxesLabel -> {"x", "t", "Temperature"}, Boxed -> False, 
 Mesh -> False]

After running for some time (~ 1 hour) it produces the plot below

There is probably a much more efficient way to produce this plot using a more idiomatic approach. If you could provide more detailed information about what you're trying to do with the code you posted, then maybe I or others could give you a more useful answer.