The classical half-line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $$bq(0,t) + {q_x}(0,t) = 0$$ is replaced with a dynamic Robin condition; $$b = b(t)$$ is allowed to vary in time. Applications include convective heating by a corrosive liquid. We present a solution representation and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half-line, with arbitrary linear dynamic boundary conditions.

热方程的经典半线 Robin 问题可以通过空间傅里叶变换方法求解。在这项工作中，我们研究了将静态 Robin 条件$$bq(0,t) + {q_x}(0,t) = 0$$替换为动态 Robin 条件的问题；$$b = b(t)$$允许随时间变化。应用包括腐蚀性液体的对流加热。我们通过对 Fokas 变换方法的扩展提出了一种解决方案表示并证明其有效性。我们展示了如何将问题简化为 Dirichlet 边界值的变系数分数线性常微分方程。我们实施分数 Frobenius 方法来求解这个方程，并证明原问题的近似解中的误差适当地收敛。我们还证明了热方程的原始动态 Robin 问题的解决方案的存在性和唯一性的论点。最后，我们将这些结果扩展到半线上任意空间阶的线性演化方程，具有任意线性动态边界条件。