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The efficient solution of discretisations of coupled systems of partial differential equations (PDEs) is at the core of much of numerical simulation. Significant effort has been expended on scalable algorithms to precondition Krylov…
Consider a classical elliptic pseudodifferential operator $P$ on ${\Bbb R}^n$ of order $2a$ ($0<a<1)$ with even symbol. For example, $P=A(x,D)^a$ where $A(x,D)$ is a second-order strongly elliptic differential operator; the fractional…
Learning underlying dynamics from data is important and challenging in many real-world scenarios. Incorporating differential equations (DEs) to design continuous networks has drawn much attention recently, however, most prior works make…
The importance of the theory of pseudo-differential operators in the study of non linear integrable systems is point out. Principally, the algebra $\Xi $ of nonlinear (local and nonlocal) differential operators, acting on the ring of…
Wave-like partial differential equations occur in many engineering applications. Here the engineering setup is embedded into the Hilbert space framework of functional analysis of modern mathematical physics. The notion wave-like is a…
We propose an operational method for the solution of differential equations involving vector products. The technique we propose is based on the use of the evolution operator, defined in such a way that the wealth of techniques developed…
The standard text book theory of ODEs lacks a general method to solve linear equations having variable coefficients, providing instead a collection of special techniques for particular classes of equations. The present article addresses…
Separable coordinate systems are introduced in the complex and real four-dimensional flat spaces. We use maximal Abelian subgroups to generate coordinate systems with a maximal number of ignorable variables. The results are presented (also…
The reduction operators, i.e., the operators of nonclassical (conditional) symmetry, of (1+1)-dimensional second order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary…
In this paper we present an $L^p$-theory for the stochastic partial differential equations (SPDEs in abbreciation) driven by L\'e{}vy processes. Existence and uniqueness of solutions in Sobolev spaces are obtained. The coefficients of SPDEs…
We consider a general discrete Sobolev inner product involving the Hahn difference operator, so this includes the well--known difference operators $\mathscr{D}_{q}$ and $\Delta$ and, as a limit case, the derivative operator. The objective…
We consider a two-dimensional integrable Hamiltonian system with a vector and scalar potential in quantum mechanics. Contrary to the case of a pure scalar potential, the existence of a second order integral of motion does not guarantee the…
Much recent work has addressed the solution of a family of partial differential equations by computing the inverse operator map between the input and solution space. Toward this end, we incorporate function-valued reproducing kernel Hilbert…
Computer algebra procedures to manipulate pseudo-differential operators are implemented to perform calculations with integrable models. We use lazy evaluation and streams to represent and operate with pseudo-differential operators. No order…
We formulate the functional Bethe ansatz for bosonic (infinite dimensional) representations of the Yang-Baxter algebra. The main deviation from the standard approach consists in a half infinite 'Sklyanin lattice' made of the eigenvalues of…
A signature independent formalism is created and utilized to determine the general second-order symmetry operators for Dirac's equation on two-dimensional Lorentzian spin manifolds. The formalism is used to characterize the orthonormal…
An equation containing a fractional power of an elliptic operator of second order is studied for Dirichlet boundary conditions. Finite difference approximations in space are employed. The proposed numerical algorithm is based on solving an…
A formal definition of the graded algebra $\mathcal{R}$ of modular linear differential operators is given and its properties are studied. An algebraic structure of the solutions to modular linear differential equations (MLDEs) is shown. It…
The modified biharmonic equation is encountered in a variety of application areas, including streamfunction formulations of the Navier-Stokes equations. We develop a separation of variables representation for this equation in polar…
Solving partial differential equations (PDEs) by learning the solution operators has emerged as an attractive alternative to traditional numerical methods. However, implementing such architectures presents two main challenges: flexibility…