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We present a semi-decision procedure to tackle first order differential equations, with Liouvillian functions in the solution (LFOODEs). As in the case of the Prelle-Singer procedure, this method is based on the knowledge of the integrating…
The Jacobi system on a full-line lattice is considered when it contains additional weight factors. A factorization formula is derived expressing the scattering from such a generalized Jacobi system in terms of the scattering from its…
Given a formal context, an ordinal factor is a subset of its incidence relation that forms a chain in the concept lattice, i.e., a part of the dataset that corresponds to a linear order. To visualize the data in a formal context, Ganter and…
We describe a method of obtaining closed-form complete solutions of certain second-order linear partial differential equations with more than two independent variables. This method generalizes the classical method of Laplace transformations…
We analyze the factorization method (introduced by Kirsch in 1998 to solve inverse scattering problems at fixed frequency from the far field operator) for a general class of boundary conditions that generalizes impedance boundary…
The work is devoted to constructing a wide class of differential-functional dynamical systems, whose rich algebraic structure makes their integrability analytically effective. In particular, there is analyzed in detail the operator Lax type…
A method to calculate the adjoint solution for a large class of partial differential equations is discussed. It differs from the known continuous and discrete adjoint, including automatic differentiation. Thus, it represents an alternative,…
We obtain regularity conditions of a new type of problems of the calculus of variations with second-order derivatives. As a corollary, we get non-occurrence of the Lavrentiev phenomenon. Our main result asserts that autonomous integral…
We study linear difference equations with variable coefficients in a ring using a new nonlinear method. In a ring with identity, if the homogeneous part of the linear equation has a solution in the unit group of the ring (i.e., a unitary…
This paper is concerned with factor left prime factorization problems for multivariate polynomial matrices without full row rank. We propose a necessary and sufficient condition for the existence of factor left prime factorizations of a…
Fractional order differential and difference equations are used to model systems with memory. Variable order fractional equations are proposed to model systems where the memory changes in time. We investigate stability conditions for linear…
We consider high-order splitting schemes for large-scale differential Riccati equations. Such equations arise in many different areas and are especially important within the field of optimal control. In the large-scale case, it is critical…
General solutions of nonlinear ordinary differential equations (ODEs) are in general difficult to find although powerful integrability techniques exist in the literature for this purpose. It has been shown that in some scalar cases…
This work investigates diagonalization-based methods for efficiently solving linear evolution problems, with a particular focus on the heat equation. The plain diagonalization of the differential operator, though effective for elliptic…
Differential constraints compatible with the linearized equations of partial differential equations are examined. Recursion operators are obtained by integrating the differential constraints.
We discuss two optimization problems related to the fractional $p$-Laplacian. First, we prove the existence of at least one minimizer for the principal eigenvalue of the fractional $p$-Laplacian with Dirichlet conditions, with a bounded…
While existing algorithms may be used to solve a linear system over a general field in matrix-multiplication time, the complexity of constructing a symmetric triangular factorization (LDL) has received relatively little formal study. The…
Lie's linearizability criteria for scalar second-order ordinary differential equations had been extended to systems of second-order ordinary differential equations by using geometric methods. These methods not only yield the linearizing…
Based on the continuous time random walk, we derive the Fokker-Planck equations with Caputo-Fabrizio fractional derivative, which can effectively model a variety of physical phenomena, especially, the material heterogeneities and structures…
The article shows how the factorization of an arbitrary Lorentz transformation is performed. That is, representation of an arbitrary Lorentz transformation as a sequence of spatial rotation and boost or boost and spatial rotation. Relations…