Can anyone suggest intial guess(the value of A0) for the system of homogeneous nonlinear equations or any other method to solve the system.
D = 0.1;
L = 0.1;
B = 0.1;
xi = 0.1;
sigma1 = B^3 * D^2 * L^3;
sigma2 = D^2 * L^3;
Ra_values = (10:100:100000).';
solutions = zeros(length(Ra_values), 7);
%Nu = zeros(length(Ra_values),1);
for i = 1:length(Ra_values)
Ra = Ra_values(i);
R = Ra * xi;
f1 = @(A) -((5*A(2)*A(6)*pi^5*B^2*D^2*L^4)/8 + (A(2)*A(6)*pi^5*D^4*L^4)/4)/sigma1 - (B * ((A(4)*A(7)*D^4*pi^5)/4 - (A(1)*L^4*pi^4)/2 - (32*A(5)*L^4*Ra)/9 + (A(1)*L^4*R*pi^2)/2 + (5*A(4)*A(7)*D^2*L^2*pi^5)/8))/sigma2;
f2 = @(A) -(B^2 * (A(2)*A(5)*D^2*L^4*pi^5 - (A(2)*D^2*L^4*pi^4)/2 - (A(1)*A(6)*D^2*L^4*pi^5)/8 - (4*A(6)*D^2*L^4*Ra)/9 + (A(4)*A(7)*D^2*L^4*pi^5)/2 + (3*A(4)*A(7)*D^4*L^2*pi^5)/16 + (A(2)*D^2*L^4*R*pi^2)/4) - (A(2)*D^4*L^4*pi^4)/4)/sigma1 - (B * ((A(4)*A(7)*D^4*pi^5)/8 - (A(2)*L^4*pi^4)/4 - (16*A(6)*L^4*Ra)/9 + (A(2)*L^4*R*pi^2)/4 + (5*A(4)*A(7)*D^2*L^2*pi^5)/16))/sigma2;
f3 = @(A) (B * ((16*A(7)*L^4*Ra)/9 + (A(3)*D^4*pi^4)/4 + (A(3)*L^4*pi^4)/4 - (A(3)*L^4*R*pi^2)/4 + (A(3)*D^2*L^2*pi^4)/2 + (A(1)*A(7)*D^2*L^2*pi^5)/8 - A(3)*A(5)*D^2*L^2*pi^5 - (A(4)*A(6)*D^2*L^2*pi^5)/2))/sigma2 - (B^2 * ((3*A(4)*A(6)*pi^5*D^4*L^2)/16 + (5*A(4)*A(6)*pi^5*D^2*L^4)/16) + (A(4)*A(6)*D^4*L^4*pi^5)/8)/sigma1;
f4 = @(A) (B * pi^2 * (2*A(4)*D^4*pi^2 - 2*A(4)*L^4*R + 2*A(4)*L^4*pi^2 + 4*A(4)*D^2*L^2*pi^2 + A(2)*A(7)*D^2*L^2*pi^3 - 8*A(3)*A(6)*D^2*L^2*pi^3 - 8*A(4)*A(5)*D^2*L^2*pi^3))/(16*D^2*L^3) - ((B^2 * pi^2 * (2*A(4)*D^2*L^4*R - 4*A(4)*D^2*L^4*pi^2 - 4*A(4)*D^4*L^2*pi^2 + 8*A(2)*A(7)*D^2*L^4*pi^3 + A(2)*A(7)*D^4*L^2*pi^3 - A(3)*A(6)*D^2*L^4*pi^3 + A(3)*A(6)*D^4*L^2*pi^3 + 8*A(4)*A(5)*D^2*L^4*pi^3))/16 - (A(4)*D^4*L^4*pi^4)/8)/sigma1;
f5 = @(A) (B^2 * ((pi^5*A(2)^2*D^2*L^4)/4 + (pi^5*A(4)^2*D^4*L^2)/4 + (pi^5*A(4)^2*D^2*L^4)/8) + (A(2)^2*D^4*L^4*pi^5)/4 + (A(4)^2*D^4*L^4*pi^5)/8)/sigma1 + (B * ((A(3)^2*D^4*pi^5)/4 + (A(4)^2*D^4*pi^5)/8 + (8*A(1)*L^4*Ra)/9 + 8*A(5)*L^4*pi^4 - 2*A(5)*L^4*R*pi^2 + (A(3)^2*D^2*L^2*pi^5)/4 + (A(4)^2*D^2*L^2*pi^5)/8))/sigma2;
f6 = @(A) (B^2 * ((4*A(2)*D^2*L^4*Ra)/9 + 2*A(6)*D^2*L^4*pi^4 + (A(1)*A(2)*D^2*L^4*pi^5)/8 + (A(3)*A(4)*D^2*L^4*pi^5)/16 + (3*A(3)*A(4)*D^4*L^2*pi^5)/16 - (A(6)*D^2*L^4*R*pi^2)/4) + (A(6)*D^4*L^4*pi^4)/4)/sigma1 + (B * (4*A(2)*L^4*Ra)/9 + 4*A(6)*L^4*pi^4 + (A(3)*A(4)*D^4*pi^5)/4 - A(6)*L^4*R*pi^2 + (A(3)*A(4)*D^2*L^2*pi^5)/4)/sigma2;
f7 = @(A) (B * ((4*A(3)*L^4*Ra)/9 + (A(7)*D^4*pi^4)/4 + 4*A(7)*L^4*pi^4 - A(7)*L^4*R*pi^2 + 2*A(7)*D^2*L^2*pi^4 + (A(1)*A(3)*D^2*L^2*pi^5)/8 + (A(2)*A(4)*D^2*L^2*pi^5)/16))/sigma2 + (B^2 * ((3*A(2)*A(4)*pi^5*D^4*L^2)/16 + (A(2)*A(4)*pi^5*D^2*L^4)/4) + (A(2)*A(4)*D^4*L^4*pi^5)/4)/sigma1;
F = @(A) [f1(A); f2(A); f3(A); f4(A); f5(A); f6(A); f7(A)];
A0 = [];
A = fsolve(F, A0);
solutions(i, :) = A;
end
A1_1_1 = solutions(:, 1);
A2_1_1 = solutions(:, 5);
Nu = xi - 1./2 - (pi^3 ./ Ra_values) .* A1_1_1 - (8 * pi^3 ./ Ra_values) .* A2_1_1;
