In a unidirectional solidification design problem, the solidification velocity and the liquid-side interfacial temperature gradient are of principle interest due to their effect on the morphology of the cast structure. The design challenge is prediction of the temporal conditions at the boundaries, such that the solidification velocity and the liquid-side temperature gradient at the solid-liquid interface follow a predetermined design scenario. The stated problem requires the resolution of two inverse problems: one, in an expanding solid domain and the second, in a shrinking liquid domain. An innovative solution technique is proposed and demonstrated for design of the liquid-side temperature gradient during unidirectional solidification. During the early transient, the control of the interfacial temperature gradient presents a challenge due to the diffusion time between the boundary and the interface. This challenge is met using a combination of initial condition design and time structuring, which allows independent control of the interfacial temperature gradient for the extent of the solidification process. The solution is developed in the context of a classic weighted-residual method, where the temporal variable is treated in an elliptic fashion.