Some effective solutions for Prandtl's type integro-differential equation

Prandtl’s type integro-differential equations with a different coefficient are investigated. Using the methods of the theory of analytic functions and integral transformations, the singular integro-differential equation is reduced to boundary value problems of the theory of analytic functions. Effective solutions of this equation and the corresponding asymptotic estimates are obtained

Asymptotics for Bailey-type mock theta functions

We compute asymptotic estimates for the Fourier coefficients of two mock theta functions, which come from Bailey pairs derived by Lovejoy and Osburn. To do so, we employ the circle method due to Wright and a modified Tauberian theorem. We encounter cancelation in our estimates for one of the mock theta functions due to the auxiliary function $$\theta _{n,p}$$ θ n , p arising from the splitting of Hickerson and Mortenson. We deal with this by using higher-order asymptotic expansions for the Jacobi theta functions.

On the Convergence of the Quasi-Periodic Approximations on a Finite Interval

We investigate the convergence of the quasi-periodic approximations in different frameworks and reveal exact asymptotic estimates of the corresponding errors. The estimates facilitate a fair comparison of the quasi-periodic approximations to other classical well-known approaches. We consider a special realization of the approximations by the inverse of the Vandermonde matrix, which makes it possible to prove the existence of the corresponding implementations, derive explicit formulas and explore convergence properties. We also show the application of polynomial corrections for the convergence acceleration of the quasi-periodic approximations. Numerical experiments reveal the auto-correction phenomenon related to the polynomial corrections so that utilization of approximate derivatives surprisingly results in better convergence compared to the expansions with the exact ones.

Multiplicative Dirac System

We define a Dirac system in multiplicative calculus by some algebraic structures. Asymptotic estimates for eigenfunctions of the multiplicative Dirac system are obtained. Eventually, some fundamental properties of the multiplicative Dirac system are examined in detail.

On rational Abel – Poisson means on a segment and approximations of Markov functions

Approximations on the segment [−1, 1] of Markov functions by Abel – Poisson sums of a rational integral operator of Fourier type associated with the Chebyshev – Markov system of algebraic fractions in the case of a fixed number of geometrically different poles are investigated. An integral representation of approximations and an estimate of uniform approximations are found. Approximations of Markov functions in the case when the measure µ satisfies the conditions suppµ = [1, a], a > 1, dµ(t) = φ(t)dt and φ(t) ≍ (t − 1)α on [1, a], a are studied and estimates of pointwise and uniform approximations and the asymptotic expression of the majorant of uniform approximations are obtained. The optimal values of the parameters at which the majorant has the highest rate of decrease are found. As a corollary, asymptotic estimates of approximations on the segment [−1, 1] are given by the method of rational approximation of some elementary Markov functions under study.

A New Type of Sturm-Liouville Equation in the Non-Newtonian Calculus

In mathematical physics (such as the one-dimensional time-independent Schrödinger equation), Sturm-Liouville problems occur very frequently. We construct, with a different perspective, a Sturm-Liouville problem in multiplicative calculus by some algebraic structures. Then, some asymptotic estimates for eigenfunctions of the multiplicative Sturm-Liouville problem are obtained by some techniques. Finally, some basic spectral properties of this multiplicative problem are examined in detail.

On approximation, in the strong sense, of functions of two variables by trigonometric polynomials

In the paper, we have found order asymptotic estimates of approximations, in the strong sense, relative to given matrix of classes of continuous periodic functions of two variables by some trigonometric polynomials.

Closed geodesics on surfaces without conjugate points

We obtain Margulis-type asymptotic estimates for the number of free homotopy classes of closed geodesics on certain manifolds without conjugate points. Our results cover all compact surfaces of genus at least 2 without conjugate points.