Abstract:We construct a class of quadratic irrationals having continued fractions of period \(n\geq2\) with `small’ partial quotients for which specific integer multiples have periodic continued fractions with the length of the period being \(1\), \(2\) or \(4\), and with ‘large’ partial quotients. We then show that numbers in the period of the new continued fraction are functions of the numbers in the periods of the original continued fraction. We also show how polynomials arising from generalizations of these continued fractions are related to Chebyshev and Fibonacci polynomials and, in some cases, have hyperbolic root distribution.

Abstract:In this study, we focus on the slip effects on the peristaltic unsteady flow of magnatohydromagnetic Jeffrey fluid in a flow passage with non-conducting and flexible boundary walls. The effect of the magnetic field with varying thermal conductivity is taken under the influence of heat transfer analysis. The dimensionless system of PDEs is solved analytically, and the obtained results are computed for the temperature, pressure drop, the axial pressure gradient, axial velocity, and then these results are discussed for different values of the physical parameters of our interest. For the stream functions, the contour plots are also obtained which indicates the exact flow behavior within the flow channel, and the effects of the physical parameters on Jeffery fluid within the flow channel are discussed briefly. Our results indicate that the heat transfer coefficient decreases with an increase in thermal slip and velocity slip parameters. Furthermore, it shows that the size of the trapped bolus is greater for the inclined magnetic field as compared to the transverse magnetic field.

Abstract:This paper solves implicit differential equations involving Hilfer-Katugampola fractional derivatives with nonlocal, boundary, and impulsive conditions. In addition, some sufficient conditions are formulated for the existence and uniqueness of solutions to the given problem, and Hyers-Ulam stability results are also presented.

Abstract:In this research paper, the authors studied some problems related to harmonic summability of double Fourier series on Nörlund summability method. These results constitute substantial extension and generalization of related work of Moricz [1] and Rhodes et al., [2]. We also constructed a new result on \((N,p^{(1)}_b,p^{(2)}_a)\) by regular N\”orlund method of summability.

Abstract:The aim of this paper is to present an optimal control problem to reduce the MDR-TB (multidrug-resistant tuberculosis) and XDR-TB (extensively drug-resistant TB) cases, using controls in these compartments and controlling reinfection/reactivation of the bacteria. The model used studies the efficacy of the tuberculosis treatment taking into account the influence of HIV/AIDS and diabetes, and we prove the global stability of the disease-free equilibrium point based on the behavior of the basic reproduction number. Various control strategies are proposed with the combinations of controls. We show the existence of optimal control using Pontryagin’s maximum principle. We solve the optimality system numerically with an algorithm based on forward/backward Runge-Kutta method of the fourth-order. The numerical results indicate that the implementation of the strategy that activates all controls and of type I (starting with the highest controls) is the most cost-effective of the strategies studied. This strategy reduces significantly the number of MDR-TB and XDR-TB cases in all sub-populations, which is an important factor in combating tuberculosis and its resistant strains.

Abstract:In particular, we study repeated integrals and recurrent integrals. For each of these integrals, we develop reduction formulae for both the definite as well as indefinite form. These reduction formulae express these repetitive integrals in terms of single integrals. We also derive a generalization of the fundamental theorem of calculus that expresses a definite integral in terms of an indefinite integral for repeated and recurrent integrals. From the recurrent integral formulae, we derive some partition identities. Then we provide an explicit formula for the \(n\)-th integral of \(x^m(\ln x)^{m’}\) in terms of a shifted multiple harmonic star sum. Additionally, we use this integral to derive new expressions for the harmonic sum and repeated harmonic sum.