Legendre polynomials for solving 2nd order ODE

How can i employ legendre polynomials for numerical examples? how to map the boundary conditions?

[Shamanta Sharmin]

The code i wrote is this:

clear all;

close all;

clc

warning off;

%initialization

syms x;

int_initial=0; % x=0 mapped [Mapping has been done! 0<x<1, VVI]

int_final=pi/2; % x=1 mapped

N=5; % number of coeffienct= N+1

% Bernstein Polynomial

a=int_initial;

b=int_final;

for i=1:N+1

leg_poly(1,i)=collect(1/factorial(i)*diff((x^2-x)^i,i)-(-1)^i)*(x-1);

end

leg_poly

% least square function

fi(x)=1-4*x/pi; % initial function

u=[fi(x) leg_poly]% polynomial function fi(x)+ fi_N(x)

funct(x)=[cos(2*x) zeros(1,N+1)] % given function

deriv_u=(diff(u,x,2))

% % funct2(x)=[(x^2) zeros(1,N+1)]

% %diff(u, x,2)

% l= x*diff(u, x,2)

%

% m=diff(u, x,1) %*transpose(funct1(x))

% %m=transpose(funct(x))*diff(u,x,1)

% deriv_u=l+m

%

% %H=(-deriv_u+funct(x))

% % double derivative

R=simplify(-deriv_u-9*u+funct(x)) % residual

size(R)

for n=1:N+1 %N+1

% WR(n,:)=vpa(int(R(n+1)*R,x,int_initial,int_final))

WR(n,:)=vpa(int(R(n+1)*R,x,int_initial,int_final));

end

WR; % coefficient matrix with equation

% solving system of equations

B=-1*WR(:,1)

A=WR(:,2:end)

% A\B

%

% %adjustment of matrix A and B due to Berstein polynomial

A(:,[1 end])=0;

A([1 end],:)=0

A;

B([1 end],:)=0

B;

% solving for systems of equations

C=linsolve(A,B) % coefficient matrix

leg_poly.*C'

%plugging the values of coefficient in least squares expression

u_least(x)=expand(sum([fi(x) leg_poly.*C']))

% evaluating polynomials

u_least_val=@(x)u_least(x)

format long

value= 0:0.01:pi/2;

least_square_method_sol=double((u_least_val(value))')

exact_fun=(4/5)*cos(3*x)+(4/5)*sin(3*x)+(1/5)*cos(2*x)

exact_sol_exp=@(x)((4/5)*cos(3*x)+(4/5)*sin(3*x)+(1/5)*cos(2*x)); % exact solution taken for 0:0.1:2

exact_sol=exact_sol_exp(value)'

value_list=value'

error=least_square_method_sol-exact_sol

% comparison with exact solution

T=table(value_list,least_square_method_sol,exact_sol,error)