Gradient Calculation in Assessing Solutions of Boundary Problems for Higher-Order Linear Parabolic Equations

The initial boundary value problem for generalization of a parabolic equation is considered. The equation of interest contains fourth-order derivatives with respect to the spatial variable, rather than the usual second-order derivatives. The uniqueness of the solution is considered in the case where the boundary conditions correspond to rigid attachment. Integral assessment yields an integral identity that may be used to assess the uniqueness of solution. For the quadratic functional showing the deviation of the solution from the target function, the gradient is calculated and used in classical gradient-based optimization.