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first.sce
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first.sce
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// This is a script to solve the Airy SLE
N = 1000;
M = round(N/3);
NN = 10000;
n = 0:N;
nsub = n(2:N);
nn = 0:N;
// Initial conditions
a = 0;
b = 200;
k = 1;
// Chebyshev grid
x = cos(n'*%pi/N);
xx = cos(nn'*%pi/N);
xsub = x(2:N);
// map Chebyshev extrema grid to integration interval: [a,b]
y = (b-a)/2*x+(a+b)/2;
// y without endpoints a and b
ysub = (b-a)/2*xsub+(a+b)/2;
// Linearly-spaced grid on [-1,1] for later interpolation
xx = linspace(-1,1,NN+1)';
// map interpolation grid to integration interval: [a,b]
yy = (b-a)/2*xx+(a+b)/2;
// More square-shaped matrices
// Chebyshev T(x) matrix for extrema grid
T = cos(acos(x)*n);
// T without x endpoints
Tsub = T(2:N,:);
// T on the xx grid
TT = cos(acos(xx)*n);
// Chebyshev U(x) matrix for extrema grid without endpoints (singularities exist there)
Usub = diag(1./sqrt(1-xsub.^2))*sin(acos(xsub)*n);
// First derivative of the Chebyshev T(x) function, on the extrema grid without endpoints
dTsub = Usub*diag(n);
// dT with endpoints
dT = [-((-1).^n).*n.^2; dTsub; n.^2];
// Second derivative of T(x) on extrema grid without endpoints
d2Tsub = diag(1./(1-xsub.^2))*(diag(xsub)*Usub-Tsub*diag(n))*diag(n);
// Second derivative of T(x) on extrema grid with endpoints
d2T = [((-1).^n).*(n.^2).*(n.^2-1)/3; d2Tsub; (n.^2).*(n.^2-1)/3];
// Second-order differentiation matrix for extrema grid
D2 = d2T/T;
// Second-order differentiation matrix for extrema grid without endpoints
E2 = D2(2:N,2:N);
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// Computation //
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//// H is the matrix version of the Sturm-Liouville operator.
//// 4/(b-a)^2*E2 is the second-order differentiation matrix on the mapped interval
H = - 4/((b-a)^2)*E2 + k*diag(ysub);
//// Solution to eigenvalue problem H Y = Lam Y
[Y, LAM] = spec(H);
//// Convert LAM (diagonal matrix) to vertical vector.
Lam = diag(LAM);
//// Order Lam in ascending order
[Lam, IX] = gsort(Lam, 'r', 'i');
//// Order Y in the same order
Y = Y(:,IX);
//// Add boundary values
Y = [zeros(1,N-1); Y; zeros(1,N-1)];
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// Interpolate //
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//// Determine expansion coefficients for Chebyshev series
aa = T\Y;
//// Interpolate to linear grid
YY = TT*aa;
//// Amplitude
Amp = YY.^2;
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// Plotting //
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
figure(1)
plot(yy,Amp(:,100))