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numerical integration - Help For Solving Complicated Function


I have a piece-wise defined function:


a1[x_] = -0.30744074406928307` Cos[1.0065881856430914` x] + 
   0.30744074406928307` Cosh[1.0065881856430914` x] + 
   Sin[1.0065881856430914` x] - Sinh[1.0065881856430914` x];


a2[x_] = 0.0384058298228355` Cos[1.7899950449175461` x] + 
 0.008258500530147013` Cosh[1.7899950449175461` x];

param = {u -> 6/11, y -> 5/11};

\[Phi][x] = 
 Piecewise[{{a1[x], 0 <= x <= u}, {a2[1 - x], u < x <= 1}}, {x, 0, 
    1}] /. param


I can plot it:


Plot[\[Phi][x], {x, 0, 1}]

enter image description here


Now I would like to calculate $n_{PI}$. See here for more details: How to solve this equation numerically or analytically


So... I define some constants:


b = 1;
g = 1;
h = 1;


And try to compute $n_{PI}$:


nom[n_?NumericQ] := 
 NIntegrate[(b \[Phi][x])/(g - n \[Phi][x])^2 + 
   53/100*(h^(1/2) \[Phi][x])/(g - n \[Phi][x])^(3/2) + 
   53/200*(b^(1/4) \[Phi][x])/(g - n \[Phi][x])^(5/4), {x, 0, 1}, 
  Method -> "LocalAdaptive"]

denom[n_?NumericQ] := 
 NIntegrate[(2 b \[Phi][x]^2)/(g - n \[Phi][x])^3 + 
   159/200*(h^(1/2) \[Phi][x]^2)/(g - n \[Phi][x])^(5/2) + 

   53/160*(b^(1/4) \[Phi][x]^2)/(g - n \[Phi][x])^(9/4), {x, 0, 1}, 
  Method -> "LocalAdaptive"]

nPI = FindRoot[n - nom[n]/denom[n] == 0, {n, 0.01}, 
  Method -> {"Newton", "UpdateJacobian" -> 3}]

So far, it seems to work. But then I also want to change the constant b with respect to the x-position so I introduce b as:


b=Piecewise[{{5*10^-6, 0 <= x <= u}, {50*10^-6, u < x <= 1}}, {x, 0, 
    1}] /. param;


But I get the error:



NIntegrate::inumr: The integrand (2 ([Piecewise] Times[<<2>>]+Times[<<2>>]+Sin[<<1>>]+Times[<<2>>] 0<=x<=6/11 1[Plus[<<2>>]] 6/11


)^2)/(1-0.01 Piecewise[{{<<2>>},{<<2>>}},{x,0,1}])^3+(159 ([Piecewise] <<1>>)^2)/(200 (1-0.01 Piecewise[{{<<2>>},{<<2>>}},{x,0,1}])^(5/2))+(53 ([Piecewise] Times[<<2>>]+Times[<<2>>]+Sin[<<1>>]+Times[<<2>>] 0<=x<=6/11 1[Plus[<<2>>]] 6/11


)^2)/(160 (1-0.01 Piecewise[{{<<2>>},{<<2>>}},{x,0,1}])^(9/4)) has evaluated to non-numerical values for all sampling points in the region with boundaries {{6/11,1}}.



Any help would be highly appreciated !!


For easier copy and paste:


a[x_] = -0.30744074406928307` Cos[1.0065881856430914` x] + 
       0.30744074406928307` Cosh[1.0065881856430914` x] + 

       Sin[1.0065881856430914` x] - Sinh[1.0065881856430914` x];

b[x_] = 0.0384058298228355` Cos[1.7899950449175461` x] + 
0.008258500530147013` Cosh[1.7899950449175461` x];
    param = {u -> 6/11, y -> 5/11};

\[Phi][x] = 
     Piecewise[{{a[x], 0 <= x <= u}, {b[1 - x], u < x <= 1}}, {x, 0, 
        1}] /. param


b = Piecewise[{{5*10^-6, 0 <= x <= u}, {50*10^-6, u < x <= 1}}, {x, 0, 
1}] /. param;
g = 1;
h = 1;

nom[n_?NumericQ] := 
     NIntegrate[(b \[Phi][x])/(g - n \[Phi][x])^2 + 
       53/100*(h^(1/2) \[Phi][x])/(g - n \[Phi][x])^(3/2) + 
       53/200*(b^(1/4) \[Phi][x])/(g - n \[Phi][x])^(5/4), {x, 0, 1}, 
      Method -> "LocalAdaptive"]


denom[n_?NumericQ] := 
  NIntegrate[(2 b \[Phi][x]^2)/(g - n \[Phi][x])^3 + 
    159/200*(h^(1/2) \[Phi][x]^2)/(g - n \[Phi][x])^(5/2) + 
    53/160*(b^(1/4) \[Phi][x]^2)/(g - n \[Phi][x])^(9/4), {x, 0, 1}, 
    Method -> "LocalAdaptive"]

 nPI = FindRoot[n - nom[n]/denom[n] == 0, {n, 0.01}, 
    Method -> {"Newton", "UpdateJacobian" -> 3}]

Answer




a1[x_] = -0.30744074406928307` Cos[1.0065881856430914` x] + 
0.30744074406928307` Cosh[1.0065881856430914` x] + 
Sin[1.0065881856430914` x] - Sinh[1.0065881856430914` x];

a2[x_] = 0.0384058298228355` Cos[1.7899950449175461` x] + 
0.008258500530147013` Cosh[1.7899950449175461` x];

param = u -> 6/11;

ϕ[x_] := Piecewise[{{a1[x], 0 <= x <= u}, {a2[1 - x], u < x <= 1}}, {x, 0, 1}] /. param;


b[x_] := Piecewise[{{5*10^-6, 0 <= x <= u}, {50*10^-6, u < x <= 1}}, {x, 0, 1}] /. param;
g = 1;
h = 1;

nom[n_?NumericQ] := 
NIntegrate[(b[x] ϕ[x])/(g - n ϕ[x])^2 + 
53/100*(h^(1/2) ϕ[x])/(g - n ϕ[x])^(3/2) + 
53/200*(b[x]^(1/4) ϕ[x])/(g - n ϕ[x])^(5/4), {x, 0, 1}, 
Method -> "LocalAdaptive"]


denom[n_?NumericQ] := 
NIntegrate[(2 b[x] ϕ[x]^2)/(g - n ϕ[x])^3 + 
159/200*(h^(1/2) ϕ[x]^2)/(g - n ϕ[x])^(5/2) + 
53/160*(b[x]^(1/4) ϕ[x]^2)/(g - n ϕ[x])^(9/4), {x, 0, 1},
Method -> "LocalAdaptive"]

nPI = FindRoot[n - nom[n]/denom[n] == 0, {n, 10}, 
Method -> {"Newton", "UpdateJacobian" -> 3}]


(* {n -> 9.81491} *)

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