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Introduction to Special Functions for Applied Mathematics
No full textIntroduction to Special Functions for Applied Mathematics introduces readers to the topic of special functions, with a particular focus on applications. Designed to build swiftly from the more basic special functions towards more advanced material, the book is ideally suited for an intensive one semester course. Complemented with various solved examples and exercises to support students and instructors, the book can be used for both self-study and directed learning
A Forgotten Differential Equation Studied by Jacopo Riccati Revisited in Terms of Lie Symmetries
Full text linkIn this paper we present a two parameter family of differential equations treated by Jacopo Riccati, which does not appear in any modern repertoires and we extend the original solution method to a four parameter family of equations, translating the Riccati approach in terms of Lie symmetries. To get the complete solution, hypergeometric functions come into play, which, of course, were unknown in Riccati’s time. Re-discovering the method introduced by Riccati, called by himself dimidiata separazione (splitted separation), we arrive at the closed form integration of a differential equation, more general to the one treated in Riccati’s contribution, and which also does not appear in the known repertoires
Dynamical systems in analysing competitiveness and co-existence among technologies
No full textThe problem of competitiveness among different technologies is analysed using ordinary differential equations and dynamic systems techniques. Specifically, we have made large use of Lyapunov linearized stability criterion. We have dealt with different kinds of conditions in the market: the static, the dynamic and unbalanced. In all cases we have considered the problem of penetrability of the market at the very beginning of the introduction of a new technology. © 1997 Taylor & Francis Group, LLC
A nonlinear oscillator modeling the transverse vibrations of a rod
No full textThis paper has been originated by the motion analysis of a flexible rod whose extreme (pivoted) sections cannot undergo any displacement, but only rotate. First an historical outline of the rod tranverse vibration analysis is provided. The dynamical elastica of such a system can be easy expressed through a linear fourth order PDE solved by a Fourier series: the rod resulting movement is then known by armonical superposition, but the overall vibration rod's period is analytically unexpressed. The authors propose an alternative analytic treatment designing a discrete model, made of a single degree of freedom oscillator, ruled by a nonlinear second order ODE, here solved through the elliptic integrals of first and third kind. This greater complexity is balanced by having found a closed formula for computing the overall vibration period by means of a suitable punctual oscillator
The Solow model improved through the logistic manpower growth law
No full textIn this paper the neo-classical economic Solow-Swan model (1956) has been improved replacing its Malthusian manpower law with the Verhulst (logistic) one. The relevant ordinary differential equation for the ratio capital/work has been then integrated in closed form via the Hypergeometric function 2F1. The logistic growth injection for the manpower is detected to induce a more slow dynamics onto the Solow-Swan system, which keeps its stability. Increasing developments are displayed as the technologic progress rises. Further sceneries are tested and the congruence of the new solution with the classical one is shown switching to zero the selflimitation coefficent in the logistic law. © 2003 Università degli Studi di Ferrara
Hypergeometric Identities Related to Roberts Reductions of Hyperelliptic Integrals
Full text linkIn this article starting from some reductions of hyperelliptic integrals of genus 3 into elliptic integrals, due to Michael Roberts (A Tract on the addition of Elliptic and hyperelliptic integrals, Hodger, Foster and Co, 1871) we obtain several identities which, to the best of our knowledge, are all new. The strategy used at this purpose is to evaluate Roberts integrals, in two different ways, on one side by means of elliptic integrals, obtained from the Roberts method of reduction and, on the other side, using multivariate hypergeometric functions
A new method for the explicit integration of Lotka-Volterra equations
No full textIn this work we study some first order nonlinear ordinary differential equations describing the time evolution (or "motion") of those hamiltonian systems provided with a first integral linking implicitly both variables to a motion constant. An application has been performed on the Lotka-Volterra predator-prey system, turning to a strongly nonlinear- differential equation in the phase variables. Our method grasps all the capabilities of modern computer algebra in order to solve (algebraic approximation) some equations of third and fourth degree with intricate forcing terms, obtaining symbolic explicit expressions osculating the solution in a neighborhood of the initial conditions. Another approach is also developed managing a Taylor truncated series and inverting it (asymptotic approximation). After having evaluated how both approximations differ from the traditional numerical techniques, finally we accomplish the much more probatory control of the approximants' accuracy referred, through the motion constant, to the first integral of the equation itself
A historical outline of the theorem of implicit functions
No full textIn this article a historical outline of the implicit functions theory is presented starting from the wiewpoint of Descartes algebraic geometry (1637) and Leibniz (1676 or 1677), Johann Bernoulli (1695) and Euler (1748) infinitesimal calculus. The critical contribution is highlighted due to the Italian mathematician Ulisse Dini who settled the matter in modern form inside the real functions theory. The paper supplies the documented proof of Dini's priority in the so called implicit functions theorem. In the meanwhile the historical lack in attributing the theorem to Dini can be ascribed to the fact that he published his proof only in his Lezioni [3], written for supporting his teaching
A Structural Approach to Gudermannian Functions
Full text linkThe Gudermannian function relates the circular angle to the hyperbolic one when their cosines are reciprocal. Whereas both such angles are halved areas of circular and hyperbolic sectors, it is natural to develop similar considerations within the study of a class of curves images of maps with constant areal speed. After a brief exposition of some use of the Gudermannian in applied sciences, we proceed to illustrate the class of curves, called Keplerian curves, which can be parametrised by a map m= (cos m, sin m) whose areal speed is 1. In the next Sections, after a detailed study of p-circular and hyperbolic Fermat curves Fp and Fp∗ , we define the p-Gudermannian as the primitive of the derivative of the p-hyperbolic sine divided by the square of the p-hyperbolic cosine: all the analogues of the classical identities are proven. Having realised that such curves correspond to each other by means a homology, we extend our study to a wide class of Keplerian curves and their homologues; once again, defined the Gudermannian in an identical manner, all the analogues of classical identities subsist. Below, three examples are detailed. The last paragraph further extends this consideration, eliminating the hypothesis that the curves are parametrised by maps with areal speed 1. The Appendix illustrates integrating techniques for systems defining the Fermat curves and determining the inverse of their tangent function
The Wallis Products for Fermat Curves
Full text linkAfter revisiting the properties of generalized trigonometric functions,
i.e. the trigonometric function linked to the planar (Fermat) curve ,
using the tool of Keplerian trigonometry
we present the extension to this class of functions of the Wallis product,
discovering connections with the representations of ordinary trigonometric functions by means of infinite products
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