Linear evolution equations on the half-line with dynamic boundary conditions

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