diff options
| -rw-r--r-- | new_schofield.nb | 19414 | ||||
| -rw-r--r-- | schofield.wl | 79 |
2 files changed, 14169 insertions, 5324 deletions
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"Output",ExpressionUUID->"b7f39439-0c1d-40f8-af40-5fed6fd931de"] +Cell[2760211, 54061, 448, 10, 24, "Input",ExpressionUUID->"fd8d9fbd-6567-43cc-a4dc-4c986b02e466"], +Cell[2760662, 54073, 26854, 485, 178, "Output",ExpressionUUID->"b7f39439-0c1d-40f8-af40-5fed6fd931de"] }, Open ]] }, Open ]] } diff --git a/schofield.wl b/schofield.wl index 646881d..8ac7ffc 100644 --- a/schofield.wl +++ b/schofield.wl @@ -1,7 +1,7 @@ BeginPackage["Schofield`"] -$Assumptions = {θc > 0, θc > 1, gC[_] ∈ Reals, B > 0, γ > 0} +$Assumptions = {θc > 0, θc > 1, gC[_] ∈ Reals, B > 0, γ > 0, ξ0 > 0} β[D_:2] := Piecewise[ { @@ -31,28 +31,43 @@ $Assumptions = {θc > 0, θc > 1, gC[_] ∈ Reals, B > 0, γ > 0} ] Δ[D_:2] := β[D] δ[D] -t[θ_] := (θ)^2 - 1 -h[n_][θ_] := (1 - (θ/θc)^2) Sum[gC[i] θ^(2i+1), {i, 0, n}] - -RFf[y_] := π^-1 y Exp[1/y] ExpIntegralEi[-1/y] -RFnonsingular[n_][θ_] := A RFf[B(θc+θ)] + Sum[FC[i] θ^(2i) , {i, 0, n}] -RF[n_][θ_] := A RFf[B(θc-θ)] + RFnonsingular[n][θ] +t[θ_] := ((θ)^2 - 1) +(* +hBasis = Orthogonalize[x^# & /@ Range[0, 20], Function[{f, g}, Integrate[f g (1 - x^2)^2, {x, -1, 1}]]] +h[n_][θ_] := (1 - (θ/θc)^2) Sum[gC[i] hBasis[[2 * i + 2]], {i, 0, n}] /. x -> θ / θc +*) +h[n_][θ_] := (1 - (θ/θc)^2) Sum[gC[i]θ^(2 * i + 1), {i, 0, n}] + +RFLow[B_, θc_][θ_] := (2 E^(1/( + B \[Theta]c)) \[Theta]c ExpIntegralEi[-(1/(B \[Theta]c))] + + E^(1/(B (-\[Theta] + \[Theta]c))) (\[Theta] - \[Theta]c) \ +ExpIntegralEi[1/(B \[Theta] - B \[Theta]c)] - + E^(1/(B \[Theta] + + B \[Theta]c)) (\[Theta] + \[Theta]c) ExpIntegralEi[-(1/( + B \[Theta] + B \[Theta]c))]) +RFHigh[ξ0_][ξ_] := (1+(ξ/ξ0)^2)^(5/6) - 1 + +RF[θ_] := AL RFLow[B, θc][θ] + AH RFHigh[θ0][θ] + AM θ^2 + +ruleB[g_] := B - π / (2 1.3578383417066) (- g'[θc] / t[θc]^Δ[2]) +ruleθ0[g_] := Simplify[g[I θ0]/(-t[I θ0])^Δ[2]/I] - 0.18930 + +eqLowRHS[F_][m_] := SeriesCoefficient[F[θ], {θ, θc, m}, Assumptions -> Join[{θ < θc, θ > 1}, $Assumptions]] + +eqLowLHS[h_][m_] :=D[ + t[θ]^2 (Gl[h[θ] t[θ]^-Δ[2]] + Log[t[θ]^2]/(8 π)), + {θ, m} ] / m! /. θ -> θc -RFfCoeff[m_] := 0 /; m < 2 -RFfCoeff[m_] := Gamma[m - 1] / π /; m > 1 +eqLow[F_, h_][m_] := eqLowRHS[F][m] - eqLowLHS[h][m] -ruleB[g_] := π / (2 1.3578383417066) (- g'[θc] / t[θc]^Δ[2]) +eqHighRHS[F_][m_] := D[F[θ], {θ, m} ] /. θ -> 0 -eqLow[n_, h_][m_] := -A B^m RFfCoeff[m] + D[ - RFnonsingular[n][θ] - t[θ]^2 (Gl[h[θ] t[θ]^-Δ[2]] + Log[t[θ]^2]/(8 π)), - {θ, m} ] / m! /. θ -> θc +eqHighLHS[h_][m_] := D[(-t[θ])^2 (Gh[h[θ] (-t[θ])^-Δ[2]] + Log[(-t[θ])^2]/(8 π)), {θ, m} ] /. θ -> 0 -eqHigh[n_, h_][m_] := D[ - RF[n][θ] - (-t[θ])^2 (Gh[h[θ] (-t[θ])^-Δ[2]] + Log[(-t[θ])^2]/(8 π)), - {θ, m} ] /. θ -> 0 +eqHigh[F_, h_][m_] := eqHighRHS[F][m] - eqHighLHS[h][m] -eqMid[n_, h_][m_] := D[ - RF[n][θ] - t[θ]^2 Log[h[θ]^2]/(8 Δ[2]π) - h[θ]^((2-α[2])/Δ[2]) Φ[η] +eqMid[F_, h_][m_] := D[ + F[θ] - t[θ]^2 Log[h[θ]^2]/(8 Δ[2]π) - h[θ]^((2-α[2])/Δ[2]) Φ[η] /. η -> t[θ] / h[θ]^(1 / Δ[2]), {θ, m} ] /. θ -> 1 @@ -76,7 +91,10 @@ Gls = { 0.18388370, -0.6591714, 2.937665, - -15.61 + -15.61, + 96.76, + -679, + 5340 } Ghs = { @@ -90,7 +108,9 @@ Ghs = { 0, 1457.62, 0, - -25891 + -25891, + 0, + 502000 } dRule[sym_][f_, i_] := Derivative[i[[1]] - 1][sym][0] -> f (i[[1]] - 1)! @@ -99,9 +119,22 @@ dRule[sym_][f_, i_] := Derivative[i[[1]] - 1][sym][0] -> f (i[[1]] - 1)! GlRules = MapIndexed[dRule[Gl], Gls]; GhRules = MapIndexed[dRule[Gh], Ghs]; -rules[g_] := Join[{B -> ruleB[g]}, ΦRules, GlRules, GhRules] - -eq[F_, g_][m_] := Flatten[Join[{eqLow[F, g][#], eqMid[F, g][#]} & /@ Range[0, m], eqHigh[F, g] /@ Range[0, m, 2]]] /. rules[g] +rules[g_] := Join[ΦRules, GlRules, GhRules] +(*ξ0 := 0.18930*) +gC[0] := 1 +tC[0] := 1 + +eq[F_, g_][m_] := Flatten[Join[{ruleB[g], ruleθ0[g]},{eqLow[F, g][#](*, eqMid[F, g][#]*)} & /@ Range[0, m], eqHigh[F, g] /@ Range[2, m, 2]]] //. rules[g] + +newSol[eqs_, oldSol_, newVars_, δ_:0, opts___] := FindRoot[ + eqs, + Join[ + oldSol /. Rule -> List, + Thread[{newVars, δ * RandomVariate[NormalDistribution[], Length[newVars]]}] + ], + MaxIterations -> 10000, + opts +] EndPackage[] |