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This is my code, i want to solve system by 14 equations and 14 unknowns(rr list).But the code doesn't run...

Clear["Global`*"] T[0, t_] = 1; T[1, t_] = t; T[n_, t_] := 2 *t*T[n - 1, t] - T[n - 2, t]; For[n = 0, n <= 7, n++, Print["T[", n, ",t]= ", T[n, t], "\n"]]; tableoft = tt /.NSolve[T[7, tt], tt]; tableoft[[1]]; Subscript[z, 1][t_] = Sum[Simplify[Subscript[a, j]*T[j, t]], {j, 0, 6}]; Subscript[z, 2][t_] = Sum[Simplify[Subscript[b, l]*T[l, t]], {l, 0, 6}]; f[t_] = (t^4/6 - t^3/3 + t) /. t -> 1/2*(τ + 1); Subscript[k, 1][t_, s_] = s^3 /. s -> 1/4*(τ + 1)*(r + 1); Subscript[k, 1][t, s]*Subscript[z, 1][r]; p[τ_] = Integrate[Subscript[k, 1][t, s]*Subscript[z, 1][r], {r, -1, 1}] Subscript[k, 2][t_, s_] = -2 (t - s) /. t -> 1/2*(τ + 1) /. s -> 1/4*(τ + 1)*(r + 1); Subscript[k, 2][t, s]*Subscript[z, 2][r]; pp[τ_] = Integrate[Subscript[k, 2][t, s]*Subscript[z, 2][r], {r, -1, 1}]; Subscript[g, 2][τ_] = Expand[(f[t] + p[τ] + pp[τ])^2] // N; Subscript[δ, 1][τ_] = ((Subscript[z, 1][τ])*((f[t] + p[τ] + pp[τ])) - 1) // N; Subscript[δ, 2][τ_] = (Subscript[z, 2][τ] -Subscript[g, 2][τ]) // N; Table[N[Subscript[δ, 1][tableoft[[i]]]], {i, 1, 7}]; Table[N[Subscript[δ, 2][tableoft[[i]]]], {i, 1, 7}]; r = Flatten[Table[N[{Subscript[δ, 1][tableoft[[i]]], Subscript[δ, 2][tableoft[[i]]]}], {i, 1, 7}]]; rr =Table[Simplify[r[[i]]] == 0, {i, 1, 14}] list =Flatten[Table[{Subscript[a, i], Subscript[b, i]}, {i, 0, 6}]]; Solve[rr, list]

How to get solution fast?

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You could use FindRoot like so:

sol = FindRoot[rr, Thread[{list, ConstantArray[0, Length[list]]}]] (* {Subscript[a, 0] -> 18.9824, Subscript[b, 0] -> 0.43251, Subscript[a, 1] -> -33.3436, Subscript[b, 1] -> 0.65013, Subscript[a, 2] -> 27.5785, Subscript[b, 2] -> 0.279464, Subscript[a, 3] -> -22.0304, Subscript[b, 3] -> 0.0771869, Subscript[a, 4] -> 16.5283, Subscript[b, 4] -> 0.0177385, Subscript[a, 5] -> -11.0201, Subscript[b, 5] -> 0.00270039, Subscript[a, 6] -> 5.51, Subscript[b, 6] -> 0.000329395} *)

Another way is to remove the Solve and instead do a minimization. Transform the equations lhs == rhs into lhs - rhs == 0 if they aren't like this already, then minimize the total square error from zero:

{err, sol} = NMinimize[Total[(rr /. Equal -> Subtract)^2], list] (* {1.00027*10^-26, {Subscript[a, 0] -> 18.9824, Subscript[b, 0] -> 0.43251, Subscript[a, 1] -> -33.3436, Subscript[b, 1] -> 0.65013, Subscript[a, 2] -> 27.5785, Subscript[b, 2] -> 0.279464, Subscript[a, 3] -> -22.0304, Subscript[b, 3] -> 0.0771869, Subscript[a, 4] -> 16.5283, Subscript[b, 4] -> 0.0177385, Subscript[a, 5] -> -11.0201, Subscript[b, 5] -> 0.00270039, Subscript[a, 6] -> 5.51, Subscript[b, 6] -> 0.000329395}} *) rr[[All, 1]] /. sol (* {0., -3.46945*10^-18, 2.22045*10^-16, -2.34188*10^-16, 2.22045*10^-15, 5.55112*10^-17, -1.66533*10^-15, 1.77636*10^-15, -4.32987*10^-15, 3.55271*10^-15, -1.9984*10^-15, 0., -5.88418*10^-15, -8.52651*10^-14} *)

As you can see, substituting these values satisfies all equations at an error under 10^-13.

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