Return to Answer

Finally we can test how periodic interpolation is good for this problem. We calculate 160 points in the beginning and 60 random points in the end of interval {NN,0,160}, and compare points with fp. We can check that only 3 points from 220 not follow to fp. Therefore it is good approximation.

we use filtered function fN based on list interpolation

lst = Table[{n, NIntegrate[ Exp[-(xp^2/(2*(30/10^6)^2))]* Evaluate[FNz /. {x -> 0, xp -> xp, nn -> n}], {xp, -300/10^6, 300/10^6}, PrecisionGoal -> 5] // Quiet}, {n, 0, 140}]; fN = Interpolation[lst]; 

Let plot lst to understand why we used filtration

ListLinePlot[lst1] 

With this function we integrate equation as

sol00 = NDSolve[{X''[n] - fN[n] == 0, X[0] == 0, X'[0] == 0}, X, {n, 0, 140}] Plot[X[nn] /. {sol00}, {nn, 0, 140},Frame -> True, FrameLabel -> {"N", "X"}] 

we use filtered function fp based on list interpolation. First we recognize that function fp is periodic with a period of 1

lst1 = Table[{n, NIntegrate[ Exp[-(xp^2/(2*(30/10^6)^2))]* Evaluate[FNz /. {x -> 0, xp -> xp, nn -> n}], {xp, -300/10^6, 300/10^6}, PrecisionGoal -> 5] // Quiet}, {n, 0, 1,.005}]; lst2 = Table[{n, NIntegrate[ Exp[-(xp^2/(2*(30/10^6)^2))]*  Evaluate[FNz /. {x -> 0, xp -> xp, nn -> n}], {xp, -300/10^6, 300/10^6}, PrecisionGoal -> 5] // Quiet}, {n, 1,3,.02}]; ListPlot[{lst1,lst2}] 

So we can make periodic interpolation as follows

fp = Interpolation[Join[lst1, {{1, lst1[[1, 2]]}}],   PeriodicInterpolation -> True] 

With this function we integrate equation as

sol00 = NDSolve[{X''[n] - fp[n] == 0, X[0] == 0, X'[0] == 0}, X, {n, 0, 140}] Plot[X[nn] /. {sol00}, {nn, 0, 140},Frame -> True, FrameLabel -> {"N", "X"}] 

(*constants*)e = -1.60217733*10^-19; m = 9.109389699999999*10^-31; epsilon = 8.854187817620391*10^-12; lw = 0.026; kk = 1; beta = Sqrt[1 - 1/(4000/0.511)^2]; gamma = 4000/0.511; c = 3*10^8; kw = (2 Pi)/0.026; sigma = 10^(-8); coeff = ((e*lw^2)/(gamma*m*beta^2*c^2))* (1/(10^10*e)/((2*Pi*(30/10^6))/10^5))* Exp[-(sigma^2/(2*(10^(-5))^2))]; (*basic equations*) rs2 = {zp, xp + kk/(gamma*kw) Sin[kw*zp], 0}; ro2 = {(nn + 10000)*lw, x + kk/(gamma*kw) Sin[kw*(nn + 10000)*lw], 0}; betas = {beta - kk^2/(4 gamma^2) Cos[2 kw*zp], kk/gamma Cos[kw*zp], 0}; betao = {beta - kk^2/(4 gamma^2) Cos[2 kw*(nn + 10000)*lw], kk/gamma Cos[kw*(nn + 10000)*lw], 0}; betaDot = {(c*kk^2*kw)/(2 gamma^2) Sin[ 2 kw*zp], -((c*kk*kw)/gamma) Sin[kw*zp], 0}; deltar2 = ro2 - rs2; Rgam2 = Sqrt[deltar2[[1]]^2 + deltar2[[2]]^2]; Ec2 = (e/(4 Pi*epsilon)) (deltar2/Rgam2 - betas)/(gamma^2 Rgam2^2 (1 - (deltar2/Rgam2).betas)^3); Erad2 = (e/(4 Pi*epsilon)) Cross[deltar2/Rgam2, Cross[deltar2/Rgam2 - betas, betaDot]]/(c* Rgam2*(1 - (deltar2/Rgam2).betas)^3); Bc2 = Cross[deltar2/Rgam2, Ec2]; Brad2 = Cross[deltar2/Rgam2, Erad2]; Fbc2 = Cross[betao, Bc2]; Fbrad2 = Cross[betao, Brad2]; sumEtran = (Ec2[[2]] + Erad2[[2]]); sumFBtran = Fbc2[[2]] + Fbrad2[[2]]; ZPRIME[nn_?NumericQ, x_?NumericQ] := zp /. FindRoot[sigma == (1/(gamma*kw))*Sqrt[gamma^2 + kk^2]* (EllipticE[kw*(nn + 10000)*lw, kk^2/(gamma^2 + kk^2)] - EllipticE[kw*zp, kk^2/(gamma^2 + kk^2)]) - beta*Sqrt[((nn + 10000)*lw - zp)^2 + (x + (kk*Sin[kw*(nn + 10000)*lw])/(gamma*kw) - (kk*Sin[kw*zp])/(gamma*kw))^2], {zp, 0}]; FNz = coeff*(sumEtran + sumFBtran) /. {zp -> ZPRIME[nn, x]}; 

With this function we integrate equation as

(*constants*)e = -1.60217733*10^-19; m = 9.109389699999999*10^-31; epsilon = 8.854187817620391*10^-12; lw = 0.026; kk = 1; beta = Sqrt[1 - 1/(4000/0.511)^2]; gamma = 4000/0.511; c = 3*10^8; kw = (2 Pi)/0.026; sigma = 10^(-8); coeff = ((e*lw^2)/(gamma*m*beta^2*c^2))* (1/(10^10*e)/((2*Pi*(30/10^6))/10^5))* Exp[-(sigma^2/(2*(10^(-5))^2))]; (*basic equations*) rs2 = {zp, xp + kk/(gamma*kw) Sin[kw*zp], 0}; ro2 = {(nn + 10000)*lw, x + kk/(gamma*kw) Sin[kw*(nn + 10000)*lw], 0}; betas = {beta - kk^2/(4 gamma^2) Cos[2 kw*zp], kk/gamma Cos[kw*zp], 0}; betao = {beta - kk^2/(4 gamma^2) Cos[2 kw*(nn + 10000)*lw], kk/gamma Cos[kw*(nn + 10000)*lw], 0}; betaDot = {(c*kk^2*kw)/(2 gamma^2) Sin[ 2 kw*zp], -((c*kk*kw)/gamma) Sin[kw*zp], 0}; deltar2 = ro2 - rs2; Rgam2 = Sqrt[deltar2[[1]]^2 + deltar2[[2]]^2]; Ec2 = (e/(4 Pi*epsilon)) (deltar2/Rgam2 - betas)/(gamma^2 Rgam2^2 (1 - (deltar2/Rgam2).betas)^3); Erad2 = (e/(4 Pi*epsilon)) Cross[deltar2/Rgam2, Cross[deltar2/Rgam2 - betas, betaDot]]/(c* Rgam2*(1 - (deltar2/Rgam2).betas)^3); Bc2 = Cross[deltar2/Rgam2, Ec2]; Brad2 = Cross[deltar2/Rgam2, Erad2]; Fbc2 = Cross[betao, Bc2]; Fbrad2 = Cross[betao, Brad2]; sumEtran = (Ec2[[2]] + Erad2[[2]]); sumFBtran = Fbc2[[2]] + Fbrad2[[2]]; ZPRIME[nn_?NumericQ, x_?NumericQ] := zp /. FindRoot[sigma == (1/(gamma*kw))*Sqrt[gamma^2 + kk^2]* (EllipticE[kw*(nn + 10000)*lw, kk^2/(gamma^2 + kk^2)] - EllipticE[kw*zp, kk^2/(gamma^2 + kk^2)]) - beta*Sqrt[((nn + 10000)*lw - zp)^2 + (x + (kk*Sin[kw*(nn + 10000)*lw])/(gamma*kw) - (kk*Sin[kw*zp])/(gamma*kw))^2], {zp, 0}]; FNz = coeff*(sumEtran + sumFBtran) /. {zp -> ZPRIME[nn, x-xp]}; 

With this function we integrate equation as