Problem: In $L^2(0,\pi)$, $\langle a,b \rangle = \int_0^\pi a(x).b(x) dx$. Finding a function $u(x,t)$ satisfy \begin{align} u_{t} - u_{xx} = F(x,t), \qquad \Omega \times [0,T]. \end{align} Dirichlet condition \begin{align} u(x,t) = 0, \qquad (x,t) \in \partial\Omega \times [0,T]. \end{align} Final value condition \begin{align} u(x,T) = f(x), \qquad x \in \Omega. \end{align} Let $(x,t) \in \Omega \times [0,T] = [0,\pi] \times [0,1].$ And \begin{align} F(x,t) &= \dfrac{1}{6}\big(2t^3+9t^2+7t+1\big)\sin(x),\\ f(x) &= t \sin(x). \end{align} The exact solution \begin{align} u(x,t) = \left(\dfrac{1}{3}t^3+\dfrac{1}{2}t^2+\dfrac{1}{6}t\right)\sin(x). \end{align} The solution as a series in Fourier type \begin{align} u = \sum_{n=1}^{\infty} \langle u, \varphi_n \rangle \varphi_n, \end{align} where $\varphi_n$ is eigenfunctions, $\lambda_n$ is eigenvalues and \begin{align} &\varphi_n = \sqrt{\dfrac{2}{\pi}} \sin(n\pi),\\ &\lambda_n = n^2,\\ &\langle u, \varphi_n \rangle = e^{(T-t)\lambda_n}\langle f, \varphi_n \rangle - \int_t^T e^{(s-t)\lambda_n} \langle F(s), \varphi_n \rangle ds,\\ &~~~~~~~~~~~~= \sqrt{\dfrac{2}{\pi}} e^{(1-t)n^2}\int_0^\pi f(x).sin(nx)dx - \sqrt{\dfrac{2}{\pi}}\int_t^1 e^{(s-t)n^2} \int_0^\pi F(x,s).\sin(nx)dx ds. \end{align} Matlab's numerical solution does not converge to the exact solution?
clf
clear
Nx = 100;
X = linspace(0,pi,Nx+1);
dx = pi/Nx;
Nt = 100;
T0 = 1;
T = linspace(0,T0,Nt+1);
dt = T0/Nt;
t_ob = 0.2;
u_ex = @(x,t) ((t.^3)/3+(t.^2)/2+t/6).*sin(x);
plotu_ex = u_ex(X,t_ob);
F = @(x,t) (1/6)*sin(x).*(2*(t.^3)+9*(t.^2)+7*t+1);
f = u_ex(X,T0);