Linear PDE with constant coefficients

We discuss practical methods for computing the space of solutions to an arbitrary homogeneous linear system of partial differential equations with constant coefficients. These rest on the Fundamental Principle of Ehrenpreis–Palamodov from the 1960s. We develop this further using recent advances in computational commutative algebra.

Differential invariant method for seeking nonlocally related systems and nonlocal symmetries. II: Connections with the conservation law method

In this paper, we show direct connections between the conservation law (CL)-based method and the differential invariant (DI)-based method for obtaining nonlocally related systems and nonlocal symmetries for a given partial differential equation (PDE) system. For a PDE system with two independent variables, we show that the CL method is a special case for the DI method. For a PDE system with at least three independent variables, we show that the CL method, for a curl-type CL, is a special case for the DI method. We also consider the situation for a self-adjoint, i.e. variational, linear PDE system. Here, a solution of the linear PDE system yields a nonlocally related system for both approaches. In particular, the resulting nonlocally related systems need not be invertibly equivalent. Through an example, we show that three distinct nonlocally related systems can be obtained from an admitted point symmetry.

A solution framework for linear PDE-constrained mixed-integer problems

We present a general numerical solution method for control problems with state variables defined by a linear PDE over a finite set of binary or continuous control variables. We show empirically that a naive approach that applies a numerical discretization scheme to the PDEs to derive constraints for a mixed-integer linear program (MILP) leads to systems that are too large to be solved with state-of-the-art solvers for MILPs, especially if we desire an accurate approximation of the state variables. Our framework comprises two techniques to mitigate the rise of computation times with increasing discretization level: First, the linear system is solved for a basis of the control space in a preprocessing step. Second, certain constraints are just imposed on demand via the IBM ILOG CPLEX feature of a lazy constraint callback. These techniques are compared with an approach where the relations obtained by the discretization of the continuous constraints are directly included in the MILP. We demonstrate our approach on two examples: modeling of the spread of wildfire and the mitigation of water contamination. In both examples the computational results demonstrate that the solution time is significantly reduced by our methods. In particular, the dependence of the computation time on the size of the spatial discretization of the PDE is significantly reduced.

Almost-Periodic Response Solutions for a Forced Quasi-Linear Airy Equation

We prove the existence of almost-periodic solutions for quasi-linear perturbations of the Airy equation. This is the first result about the existence of this type of solutions for a quasi-linear PDE. The solutions turn out to be analytic in time and space. To prove our result we use a Craig–Wayne approach combined with a KAM reducibility scheme and pseudo-differential calculus on $$\mathbb {T}^\infty $$ T ∞ .

Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints

We show general lower semicontinuity and relaxation theorems for linear-growth integral functionals defined on vector measures that satisfy linear PDE side constraints (of arbitrary order). These results generalize several known lower semicontinuity and relaxation theorems for BV, BD, and for more general first-order linear PDE side constrains. Our proofs are based on recent progress in the understanding of singularities of measure solutions to linear PDEs and of the generalized convexity notions corresponding to these PDE constraints.