Each component of any solution of a Fuchsian differential system satisfies a Fuchsian differential equation. The set of Fuchsian systems is fibered into equivalence classes. Each class consists of systems with similar sets of matrix residues, the conjugation matrix being the same for all elements of the set. We investigate the corresponding classes of scalar equations.
The confluent Heun equation and the confluent hypergeometric equation are studied in scalar and vector forms with particular emphasis on the role of apparent singularities. A relation to the Painlevé equation is established.
We consider the simplest Fuchsian second-order equations with particular attention to the role of apparent singularities. We show the relation to the Painlevé equation and follow the matrix formulation of the problem.
A symbolic generation of Painlevé equations is developed on the basis of the antiquantization of deformed Heun-class equations. The corresponding CAS Maple package is presented, along with examples of its use. The particular cases of reduced confluent Heun equations are discussed.
We discuss several examples of generating apparent singular points as a result of differentiating particular homogeneous linear ordinary differential equations with polynomial coefficients and formulate two general conjectures on the generation and removal of apparent singularities in arbitrary Fuchsian differential equations with polynomial coefficients. We consider a model problem in polymer physics.
We consider deformed Heun-class equations, i.e., equations of the Heun class with an added apparent singularity. We prove that each deformed Heun-class equation under antiquantization realizes a transfer from the Heun-class equation to the corresponding Painlevé equation, and we completely list such transfers.
We present a new multifactorial global search algorithm (MGSA) and check the operability of the algorithm on the Michalewicz and Rastrigin functions. We discuss the choice of an objective function and additional search criteria in the context of the problem of reactive force field (ReaxFF) optimization and study the ranking of the ReaxFF parameters together with their impact on the objective function...
A Fuchsian 2 × 2 system generating the Painlevé equation P6 is acted on by a polynomial transformation similar to rotation in order to reduce the polynomial degree of matrices in the left- and the right-hand sides of the system. This clarifies the derivation of the Painlevé equation and the study of its symmetries.
A first-order 2 × 2 system equivalent to the Heun equation is obtained. A deformed Heun equation in symmetric form is presented. Series solutions of this equation are presented. A four-parameter subfamily of deformed confluent Heun equations whose solutions have integral representations is found.
Euler integral symmetries relate solutions of ordinary linear differential equations and generate integral representations of the solutions in several cases or relations between solutions of constrained equations. These relations lead to the corresponding symmetries of the monodromy matrices for the differential equations. We discuss Euler symmetries in the case of the deformed confluent Heun equation,...
Euler integral transformations relate solutions of ordinary linear differential equations and generate integral representations of the solutions in a number of cases or relations between solutions of constrained equations (Euler symmetries) in some other cases. These relations lead to the corresponding symmetries of the monodromy matrices. We discuss Euler symmetries in the case of the simplest Fuchsian...
We consider three different models of linear differential equations and their isomonodromic deformations. We show that each of the models has its own specificity, although all of them lead to the same final result. It turns out that isomonodromic deformations are closely related to the Hamiltonian structure of both classical mechanics and quantum mechanics.
Against the background of one of the authors’ (S. Yu. Slavyanov’s) reminiscences of A. A. Bolibrukh, the asymptotic behavior of the spectral curves generated by the boundary-value problem for the confluent Heun equation (that is, an equation with two regular and one irregular singularity) is considered. The spectral curves are constructed for small values of one parameter (the distance between the...
The relationship between the eigenfunctions of a Fredholm-type integral equation with rapidly oscillating kernel and the dynamic mapping is analyzed. Differential operators commuting with the Fourier operator are constructed. These operators are closely related to nontrivial solutions of the unperturbed nonlinear functional equation related to the dynamic mapping. Bibliography: 6 titles.
Under the assumption that potentials in two Schrödinger equations differ by a polynomial of degree k, we derive a (k+4)th-order equation for a function that is a product of solutions of these equations. Several examples of applications in physics are considered.
We propose a method for reconstructing the original profile function in the one-dimensional Fourier transformation from the module of the Fourier transform function analytically. The major concept of the method consists in representing the modeling profile function as a sum of local peak functions. The latter are chosen as eigenfunctions generated by linear differential equations with polynomial coefficients...
An approach to developing a mathematical knowledge base on special functions is described and an example of such a knowledge base is presented. The problem consists of creating a uniform structural approach to classification of special functions of different types. Such an approach was realized in a series of papers by the authors. Two variants of an experimental version of the SFTools knowledge base...
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