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Introduction to Linear Algebra with Mathematica

Glossary

The Fokas Method or Unified Transform Method

We consider one-dimensional initial boundary value problem (where α > 0, γ ≥ 0).
\begin{align*} &\mbox{Heat conduction equation:} \qquad &u_t &= \alpha\,u_{xx} - \gamma\,u(x,t) + f(x,t)\qquad\mbox{for } 0 < x < \infty \mbox{ and } t > 0, \\ &\mbox{Boundary condition:} \qquad &u(0,t) &= g(t) , \qquad t > 0, \\ &\mbox{Initial condition:} \qquad &u(x,0) &= u_0 (x) , \qquad 0 < x < \infty . \\ &\mbox{Corner condition:} \qquad &u(x,t) &\sim (0, 0) , \qquad \mbox{ as } (x,t) \to (0,0), \\ &\mbox{Regularity condition:} \qquad & \lim_{x,t \to +\infty}\,|u(x,t)| &= 0 , \quad\qquad u, u_t , u_{xx} \in 𝔏²(R), \\ \end{align*}