A Pedagogical Introduction to the Unified Transform Method: The Heat Equation on a Finite Interval

Abstract

This paper presents a detailed application of the Unified Transform Method (Fokas method) to the one-dimensional heat equation on $[0,1]$ with Dirichlet boundary conditions. The analysis formulates the Initial-Boundary Value Problem and derives an integral representation of the solution via a generalised spatial Fourier transform with complex spectral parameter $\lambda \in \mathbb{C}$, yielding the Global Relation -- an algebraic identity coupling the initial datum, prescribed boundary values, and unknown Neumann data. The unknowns are eliminated by exploiting the symmetry $\lambda \mapsto -\lambda$, reducing the solution to a contour integral over $\partial D^+$. An explicit evaluation is carried out for exponential initial datum $u_0(x)=e^{-x}$ and Dirichlet conditions $g_0(t)=\cos(t)$, $h_0(t)=e^{-1}\cos(t)$. The integral representation is analysed in the complex plane, with emphasis on exponential decay and analyticity, providing rigorous justification for contour deformation via Cauchy's Theorem and Jordan's Lemma. Numerical implementation in Maple uses a trapezoidal contour parametrisation ensuring exponential decay along each segment; the solution over $x\in[0,1]$, $t\in[0,2\pi]$ matches prescribed data to machine precision. The results confirm the analytical and numerical efficacy of the Unified Transform for classical parabolic problems and illustrate how rigorous contour analysis yields stable, accurate solutions.