Search Results (8)

Some classical texts on the Sturm–Liouville equation ${(p(x)y^{\prime })}^{\prime }-q(x)y+\lambda \rho (x)y=0$ are revised to highlight further properties of its solutions. Often, in the treatment of the ensuing integral equations, $\rho =const$ is assumed (and, further, $\rho =1$). Instead, here we preserve $\rho (x)$ and make a simple change only of the independent variable that reduces the Sturm–Liouville equation to $y^{″}-q(x)y+\lambda \rho (x)y=0$. We show that many results are identical with those with $\lambda \rho -q=const$. This is true in particular for the mean value of the oscillations and for the analog of the Riemann–Lebesgue Theorem. From a mechanical point of view, what is now the total energy is not a constant of the motion, and nevertheless, the equipartition of the energy is still verified and, at least approximately, it does so also for a class of complex $\lambda$. We provide here many detailed properties of the solutions of the above equation, with $\rho =\rho (x)$. The conclusion, as we may easily infer, is that, for large enough $\lambda$, locally, the solutions are trigonometric functions. We give the proof for the closure of the set of solutions through the Phragmén–Lindelöf Theorem, and show the separate dependence of the solutions from the real and imaginary components of $\lambda$. The particular case of $q(x)=\alpha \rho (x)$ is also considered. A direct proof of the uniform convergence of the Fourier series is given, with a statement identical to the classical theorem. Finally, the proof of J. von Neumann of the completeness of the Laguerre and Hermite polynomials in non-compact sets is revisited, without referring to generating functions and to the Weierstrass Theorem for compact sets. The possibility of the existence of a general integral transform is then investigated. Full article

Investigation of the generalized trigonometric and hyperbolic functions containing two parameters has been a very active research area over the last decade. We believe, however, that their monotonicity and convexity properties with respect to parameters have not been thoroughly studied. In this paper, we make an attempt to fill this gap. Our results are not complete; for some functions, we manage to establish (log)-convexity/concavity in parameters, while for others, we only managed the prove monotonicity, in which case we present necessary and sufficient conditions for convexity/concavity. In the course of the investigation, we found two hypergeometric representations for the generalized cosine and hyperbolic cosine functions which appear to be new. In the last section of the paper, we present four explicit integral evaluations of combinations of generalized trigonometric/hyperbolic functions in terms of hypergeometric functions. Full article

This article introduces $(p,q)$-analogs of the gamma integral operator and discusses their expansion to power functions, $(p,q)$-exponential functions, and $(p,q)$-trigonometric functions. Additionally, it validates other findings concerning $(p,q)$-analogs of the gamma integrals to unit step functions as well as first- and second-order $(p,q)$-differential operators. In addition, it presents a pair of $(p,q)$-convolution products for the specified $(p,q)$-analogs and establishes two $(p,q)$-convolution theorems. Full article

Nonlinear fractional partial differential equations (NLFPDEs) are widely used in simulating a variety of phenomena arisen in several disciplines such as applied mathematics, engineering, physics, and a wide range of other applications. Solitary wave solutions of NLFPDEs have become a significant tool in understanding the long-term dynamics of these events. This article primarily focuses on using the improved modified extended tanh-function algorithm to determine certain traveling wave solutions to the space-time fractional symmetric regularized long wave (SRLW) equation, which is used to discuss space-charge waves, shallow water waves, etc. The Jumarie’s modified Riemann-Liouville derivative is successfully used to deal with the fractional derivatives, which appear in the SRLW problem. We find many traveling wave solutions on the form of trigonometric, hyperbolic, complex, and rational functions. Furthermore, the performance of the employed technique is investigated in comparison to other techniques such as the Oncoming $exp(-\mathsf{\Theta }(q\left)\right)$-expansion method and the extended Jacobi elliptic function expansion strategy. Some obtained results are graphically displayed to show their physical features. The findings of this article demonstrate that the used approach enables us to handle more NLFPDEs that emerge in mathematical physics. Full article

In this study, we present a sensorless, robust, and physiological tracking control method to drive the operational speed of implantable rotary blood pumps (IRBPs) for patients with heart failure (HF). The method used sensorless measurements of the pump flow to track the desired reference flow ($Q_r$). A dynamical estimator model was used to estimate the average pump flow (${\hat{Q}}_{est}$) based on pulse-width modulation (PWM) signals. A proportional-integral (PI) controller integrated with a fuzzy logic control (FLC) system was developed to automatically adapt the pump flow. The $Q_r$ was modeled as a constant and trigonometric function using an elastance function ($E(t$)) to achieve a variation in the metabolic demand. The proposed method was evaluated in silico using a lumped parameter model of the cardiovascular system (CVS) under rest and exercise scenarios. The findings demonstrated that the proposed control system efficiently updated the pump speed of the IRBP to avoid suction or overperfusion. In all scenarios, the numerical results for the left atrium pressure ($P_{la}$), aortic pressure ($P_{ao}$), and left ventricle pressure ($P_{lv}$) were clinically accepted. The ${\hat{Q}}_{est}$ accurately tracked the $Q_r$ within an error of 0.25 L/min. Full article

Utilizing $\left(p,q\right)$-numbers and $\left(p,q\right)$-concepts, in 2016, Duran et al. considered $\left(p,q\right)$-Genocchi numbers and polynomials, $\left(p,q\right)$-Bernoulli numbers and polynomials and $\left(p,q\right)$-Euler polynomials and numbers and provided multifarious formulas and properties for these polynomials. Inspired and motivated by this consideration, many authors have introduced $(p,q)$-special polynomials and numbers and have described some of their properties and applications. In this paper, using the $(p,q)$-cosine polynomials and $(p,q)$-sine polynomials, we consider a novel kinds of $(p,q)$-extensions of geometric polynomials and acquire several properties and identities by making use of some series manipulation methods. Furthermore, we compute the $\left(p,q\right)$-integral representations and $\left(p,q\right)$-derivative operator rules for the new polynomials. Additionally, we determine the movements of the approximate zerosof the two mentioned polynomials in a complex plane, utilizing the Newton method, and we illustrate them using figures. Full article

Evaluation of images of special functions under operators of fractional calculus has become a hot topic with hundreds of recently published papers. These are growing daily and we are able to comment here only on a few of them, including also some of the latest of 2019–2020, just for the purpose of illustrating our unified approach. Many authors are producing a flood of results for various operators of fractional order integration and differentiation and their generalizations of different special (and elementary) functions. This effect is natural because there are great varieties of special functions, respectively, of operators of (classical and generalized) fractional calculus, and thus, their combinations amount to a large number. As examples, we mentioned only two such operators from thousands of results found by a Google search. Most of the mentioned works use the same formal and standard procedures. Furthermore, in such results, often the originals and the images are special functions of different kinds, or the images are not recognized as known special functions, and thus are not easy to use. In this survey we present a unified approach to fulfill the mentioned task at once in a general setting and in a well visible form: for the operators of generalized fractional calculus (including also the classical operators of fractional calculus); and for all generalized hypergeometric functions such as ${}_p{\mathsf{\Psi }}_q$ and ${}_p{F}_q$, Fox H- and Meijer G-functions, thus incorporating wide classes of special functions. In this way, a great part of the results in the mentioned publications are well predicted and appear as very special cases of ours. The proposed general scheme is based on a few basic classical results (from the Bateman Project and works by Askey, Lavoie–Osler–Tremblay, etc.) combined with ideas and developments from more than 30 years of author’s research, and reflected in the cited recent works. The main idea is as follows: From one side, the operators considered by other authors are cases of generalized fractional calculus and so, are shown to be (m-times) compositions of weighted Riemann–Lioville, i.e., Erdélyi–Kober operators. On the other side, from each generalized hypergeometric function ${}_p{\mathsf{\Psi }}_q$ or ${}_p{F}_q$ ($p\le q$ or $p=q+1$) we can reach, from the final number of applications of such operators, one of the simplest cases where the classical results are known, for example: to ${}_0{F}_{q-p}$ (hyper-Bessel functions, in particular trigonometric functions of order $(q-p)$), ${}_0{F}_0$ (exponential function), or ${}_1{F}_0$ (beta-distribution of form ${(1-z)}^{\alpha }{z}^{\beta })$. The final result, written explicitly, is that any GFC operator (of multiplicity $m\ge 1$) transforms a generalized hypergeometric function into the same kind of special function with indices p and q increased by m. Full article

This paper constructs and introduces $(p,q)$ -cosine and $(p,q)$ -sine Bernoulli polynomials using $(p,q)$ -analogues of ${(x+a)}^n$ . Based on these polynomials, we discover basic properties and identities. Moreover, we determine special properties using $(p,q)$ -trigonometric functions and verify various symmetric properties. Finally, we check the symmetric structure of the approximate roots based on symmetric polynomials. Full article