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We provide two methods for computation of continuum backstepping kernels that arise in control of continua (ensembles) of linear hyperbolic PDEs and which can approximate backstepping kernels arising in control of a large-scale, PDE system…
Algorithms for the computation of the real zeros of hypergeometric functions which are solutions of second order ODEs are described. The algorithms are based on global fixed point iterations which apply to families of functions satisfying…
We are interested in the numerical solution of coupled nonlinear partial differential equations (PDEs) in two and three dimensions. Under certain assumptions on the domain, we take advantage of the Kronecker structure arising in standard…
We prove the convergence of hyperbolic approximations for several classes of higher-order PDEs, including the Benjamin-Bona-Mahony, Korteweg-de Vries, Gardner, Kawahara, and Kuramoto-Sivashinsky equations, provided a smooth solution of the…
The present work proposes a well-balanced finite volume-type numerical method for the solution of non-conservative hyperbolic partial differential equations (PDEs) with source terms. The method is characterized, first, by the use of a…
We study an expansion method for high-dimensional parabolic PDEs which constructs accurate approximate solutions by decomposition into solutions to lower-dimensional PDEs, and which is particularly effective if there are a low number of…
The purpose of this paper is twofold. An immediate practical use of the presented algorithm is its applicability to the parametric solution of underdetermined linear ordinary differential equations (ODEs) with coefficients that are…
In the second half of the 19th century Darboux obtained determinant formulae that provide the general solution for a linear hyperbolic second order PDE with finite Laplace series. These formulae played an important role in his study of the…
We establish a consistency result by comparing two independent notions of generalised solutions to a large class of linear hyperbolic first order PDE systems with constant coefficients, showing that they eventually coincide. The first is…
A linear implicit finite difference method is proposed for the approximation of the solution to a periodic, initial value problem for a Schrodinger-Hirota equation. Optimal, second order convergence in the discrete $H^1-$norm is proved,…
In this paper, we develop an ensemble-based time-stepping algorithm to efficiently find numerical solutions to a group of linear, second-order parabolic partial differential equations (PDEs). Particularly, the PDE models in the group could…
We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural…
In this work, we determine the full expression for the global truncation error of hyperbolic partial differential equations (PDEs). In particular, we use theoretical analysis and symbolic algebra to find exact expressions for the…
This paper establishes the equivalence between systems described by a single first-order hyperbolic partial differential equation and systems described by integral delay equations. System-theoretic results are provided for both classes of…
Under consideration are mathematical models of heat and mass transfer. We study inverse problems of recovering lower-order coefficients in a second order parabolic equation. The coefficients are representable in the form of a finite…
In this paper we present a direct formula for the solution of the general second order linear ordinary differential equation as our main result such that the parameters required for the formula are determined using another differential…
We propose an integral transform, called metamorphism, which allow us to reduce the order of a differential equation. For example, the second order Helmholtz equation is transformed into a first order equation, which can be solved by the…
Fast and accurate solution of time-dependent partial differential equations (PDEs) is of key interest in many research fields including physics, engineering, and biology. Generally, implicit schemes are preferred over the explicit ones for…
A multi-scale method for the hyperbolic systems governing sediment transport in subcritical case is developed. The scale separation of this problem is due to the fact that the sediment transport is much slower than flow velocity. We first…
In this paper, using the similarity method, we construct particular solutions with singularities for degenerate high-order equations. The considered equations have singularities of the first and second kind. Particular solutions are…