Summary: We consider coupled semi-infinite diffusion problems of the form \(u_t(x, t)- A^2 u_{xx}(x, t)= 0\), \(x> 0\), \(t> 0\), subject to \(u(0, t)= B\) and \(u(x, 0)= 0\), where \(A\) is a matrix \(\mathbb{C}^{r\times r}\), and \(u(x, t)\) and \(B\) are vectors in \(\mathbb{C}^r\). Using the Fourier sine transform, an explicit exact solution of the problem is proposed. Given an admissible error \(\varepsilon\) and a domain \(D(x_0, t_0)= \{(x, t); 0\leq x\leq x_0, t\geq t_0> 0\}\), an analytic approximation solution is constructed so that the error with respect to the exact solution is uniformly upper bounded by \(\varepsilon\) in \(D(x_0, t_0)\).