${\text{PDL}}_g $ Splines Defined by Partial Differential Operators with Initial and Boundary Value Conditions

Abstract

As a natural generalization of $L_g $ splines and thin-plate splines, ${\text{PDL}}_g $ splines are introduced in this paper. A ${\text{PDL}}_g $ spline is defined as a solution of the optimal scattered-data interpolation problem described by a general higher-order separable linear partial differential operator (which may or may not incorporate time), with (initial-terminal and) boundary value conditions over (the time domain and) an arbitrary bounded spatial domain with a continuous and piecewise smooth boundary. A closed-form expression for the ${\text{PDL}}_g $ spline is obtained by means of the reproducing kernel of a related Hilbert space. A feasible constructive method for finding the reproducing kernel via a fundamental solution (or Green’s function) of an induced linear and self-adjoint partial differential operator is also included. This method provides a way to take care of the initial-terminal and boundary value conditions. Moreover, an explicit formulation for finding the best approximation, i...