# Getting errors NIntegrate::nlim and NIntegrate::slwcon [closed]

Here is the modified question:

First: Define

phi[x_]:=Piecewise[{{1, 0 <= x < 1}}, 0]

psi1[x_] := (phi[2 x] - phi[2 x - 1]);

psijk[x_, j_, k_] := Piecewise[{{(Sqrt[2])^j psi1[2^j x - k],0 <= j}, {(2)^j psi1[2^j (x - k)], j < 0}}];

I am trying to solve the equation in the below picture by finding an approximation solution for u(x) using wavelets. What I did is as follows: I used wavelet functions and substitute the wavelet instead of u(x).

I multiplied both sides of the equation by psijk[x,l,s] and integrate both sides from 0 to 1, this known in wavelets as the Galerkin Method. So we have the following system,

Sum[C [j, k]* PSI[j, k, l, s], {j, -2, 2}, {k, -4, 3}, {l, -2, 2}, {s, -4, 3}] = NIntegrate[f[x]*psijk[x, l, s], {x, 0, 1}]

where

PSI[j_, k_, l_, s_] := NIntegrate[psijk[x, j, k]* psijk[x, l, s], {x, 0, 1}]- NIntegrate[psijk[x, l, s] * NIntegrate[psijk[t, j, k]*(1/(-t + x)^(1/4)), {t, 0, x}], {x, 0, 1}]-NIntegrate[psijk[x, l, s]*NIntegrate[psijk[t, j, k]*(1/(-t + x)^(1/4)), {t, 0, 1}], {x,0, 1}];

and

f[x_]:=16/165 (-1 + x)^(3/4) + 48/385 (-1 + x)^(3/4) x + x^2 + (256 (-1 +x)^(3/4) x^2)/1155 - (256 x^(11/4))/231 - x^3 - (512 (-1 + x)^(3/4) x^3)/1155 + (1024 x^(15/4))/1155, and C[j,k] to be found!

Then to find these coefficients, I defined,

ce = ArrayReshape[Table[PSI[j, k, l, s], {j, -2, 2}, {k, -4, 3}, {l, -2, 2}, {s, -4, 3}], {40, 40}];

coef=NIntegrate[f[x]*psijk[x, l, s], {x, 0, 1}]

and

ae = PseudoInverse[ce].coef;

However, I am getting error and the solution is incorrect as the exact solution is u[x]=x^2(1-x).

The errors came as some comments is from the ranges {x, 0, 1}, {t, 0, 1}, but I dont know how to fix this issue as the method I used is by integrating both sides of the equation from 0 to 1.

I hope the question is clear for everyone. Your help is extremely appreciated!

## closed as unclear what you're asking by Anton Antonov, m_goldberg, MarcoB, Johu, José Antonio Díaz NavasOct 19 '18 at 11:44

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• Try different Methods you can find within NIntegrate. – Αλέξανδρος Ζεγγ Oct 16 '18 at 7:59
• You should interchange the integration parameters NIntegrate[..., {t, 0, x}], {x, 0, 1}]-> NIntegrate[..., {x, 0, 1}, {t, 0, x}]] (first error!) – Ulrich Neumann Oct 16 '18 at 9:24
• What is your reason for including the definition of f in this question? It doesn't seem to called any other function in the question. – m_goldberg Oct 16 '18 at 22:14
• When t == 1 and x == 0, (-t + x)^(1/4) is complex. – Michael E2 Oct 16 '18 at 22:59

If you combine the iterated integrals into multiple integrals, apply PiecewiseExpand to the last integral, and further specify the singularity t == x in the last integral, it seems to work.

PSI[j_, k_, l_, s_] :=
NIntegrate[psijk[x, j, k]*psijk[x, l, s], {x, 0, 1}] -
NIntegrate[
psijk[x, l, s]*psijk[t, j, k]*(1/(-t + x)^(1/4)),
{x, 0, 1}, {t, 0, x}] -
NIntegrate[
psijk[x, l, s]*psijk[t, j, k]*(1/(-t + x)^(1/4)) // PiecewiseExpand,
{x, 0, 1}, {t, 0, 1}, Exclusions -> t == x];

PSI[1, 1, 1, 1]
(*  0.767956 + 0.0606109 I  *)


You get a complex value because near t == 1, x == 0, the factor (-t + x)^(1/4) is complex.

• Might want to set a finite AccuracyGoal, since the integrals may be zero. E.g. AccuracyGoal ->12. – Michael E2 Oct 16 '18 at 23:18
• How we can avoid x-t to be a negative? I need it only positive for any value in the domain of 0<t<x<1. – Mutaz Oct 17 '18 at 6:32
• @Mutaz The integration ranges you give in the 3rd integral are {x, 0, 1} and {t, 0, 1}. Perhaps you meant to put {t, 0, x} for the t range in the 3rd, just as you did in the 2nd integral? (If you do that, the integral is computed without any fuss. No need for PiecewiseExpand either.) – Michael E2 Oct 17 '18 at 10:13
• But it supposed to be as it is – Mutaz Oct 17 '18 at 17:42
• @Mutaz If (-t+x)^(1/4) and {x, 0, 1}, {t, 0, 1} are both correct, then complex numbers cannot be avoided. It is not a numerical issue, but a mathematical one. – Michael E2 Oct 17 '18 at 22:49