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Recently, it was found that a new set of simple techniques allow one to conveniently express ordinary integrals through differentiation. These techniques add to the general toolbox for integration and integral transforms such as the Fourier…
The paper suggests a Hamiltonian formulation for delay ordinary differential equations (DODEs). Such equations are related to DODEs with a Lagrangian formulation via a delay analog of the Legendre transformation. The Hamiltonian delay…
We show that Toda lattices with the exceptional Cartan matrices are Liouville type systems. For these systems of equations, we obtain explicit formulas for the invariants and generalized Laplace invariants.
In a bounded smooth domain of Rn, we shall establish Kato's inequalities involving measures for p-Laplace operator up to the boundary under suitable assumptions. Keywords: Kato's inequality, $p$-Laplace operator
Darboux transformations are non-group type symmetries of linear differential operators. One can define Darboux transformations algebraically by the intertwining relation $ML=L_1M$ or the intertwining relation $ML=L_1N$ in the cases when the…
We propose a novel numerical algorithm utilizing model reduction for computing solutions to stationary partial differential equations involving the spectral fractional Laplacian. Our approach utilizes a known characterization of the…
We introduce a fractional calculus on time scales using the theory of delta (or nabla) dynamic equations. The basic notions of fractional order integral and fractional order derivative on an arbitrary time scale are proposed, using the…
We derive certain systems of differential equations for matrix elements of products and iterates of logarithmic intertwining operators among strongly graded generalized modules for a strongly graded conformal vertex algebra under suitable…
The classical $S_n$ equations of Carlson and Lee have been a mainstay in multi-dimensional radiation transport calculations. In this paper, an alternative to the $S_n$ equations, the "Lagrange Discrete Ordinate" (LDO) equations are derived.…
The intertwining operator technique is applied to difference Schroedinger equations with operator-valued coefficients. It is shown that these equations appear naturally when a discrete basis is used for solving a multichannel Schroedinger…
Differential constraints compatible with the linearized equations of partial differential equations are examined. Recursion operators are obtained by integrating the differential constraints.
The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations…
We introduce a dimensional splitting method based on the intertwining property of the Radon transform, with a particular focus on its applications related to hyperbolic partial differential equations (PDEs). This dimensional splitting has…
Recently a new technique in the harmonic analysis on symmetric spaces was suggested based on certain remarkable representations of affine and double affine Hecke algebras in terms of Dunkl and Demazure operators instead of Lie groups and…
It is well-known that if a symplectic integrator is applied to a Hamiltonian system, then the modified equation, whose solutions interpolate the numerical solutions, is again Hamiltonian. We investigate this property from the variational…
We investigate the properties of some recently developed variable-order differential operators involving order transition functions of exponential type. Since the characterisation of such operators is performed in the Laplace domain it is…
In this paper, we study the Lagrangian functions for a class of second-order differential systems arising from physics. For such systems, we present necessary and sufficient conditions for the existence of Lagrangian functions. Based on the…
In this paper, we present a new derivative via the Laplace transform. The Laplace transform leads to a natural form of the fractional derivative which is equivalent to a Riemann-Liouville derivative with fixed terminal point. We first…
The paper introduces Laplace-type operators for functions defined on the tangent space of a Finsler Lie algebroid, using a volume form on the prolongation of the algebroid. It also presents the construction of a horizontal Laplace operator…
We consider the generalization of Laplace invariants to linear differential systems of arbitrary rank and dimension. We discuss completeness of certain subsets of invariants.