Frontiers in Applied Mathematics and Statistics | Mathematical Physics section | New and Recent Articles
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RSS Feed for Mathematical Physics section in the Frontiers in Applied Mathematics and Statistics journal | New and Recent Articlesen-usFrontiers Feed Generator,version:12020-02-28T19:05:58.2812692+00:0060https://www.frontiersin.org/articles/10.3389/fams.2019.00011
https://www.frontiersin.org/articles/10.3389/fams.2019.00011
The Fractional Laguerre Equation: Series Solutions and Fractional Laguerre Functions2019-02-20T00:00:00ZRasha ShatSafa AlrefaiIslam AlhamaydaAlaa SarhanMohammed Al-RefaiIn this paper, we propose a fractional generalization of the well-known Laguerre differential equation. We replace the integer derivative by the conformable derivative of order 0 < α < 1. We then apply the Frobenius method with the fractional power series expansion to obtain two linearly independent solutions of the problem. For certain eigenvalues, the infinite series solution truncate to obtain the singular and non-singular fractional Laguerre functions. We obtain the fractional Laguerre functions in closed forms, and establish their orthogonality result. The applicability of the new fractional Laguerre functions is illustrated.]]>https://www.frontiersin.org/articles/10.3389/fams.2018.00030
https://www.frontiersin.org/articles/10.3389/fams.2018.00030
The Modeling of Shock-Wave Pressures, Energies, and Temperatures Within the Human Brain Due to Improvised Explosive Devices (IEDs) Using the Transport and Burgers' Equations2018-08-02T00:00:00ZStephen W. MasonThis second paper adopts a more rigorous, in-depth approach to modeling the resulting dynamic-pressures in the human brain, following a transitory improvised explosive device (IED) shock-wave entering the head. Determining more complicated boundary conditions, a set of particular-solutions for both Burgers' and the Transport equations has been obtained to describe the highly damped neurological pressures, complete with respective graphical plots. Many of these two-dimensional solution-curves closely resemble the Friedlander curve [1–4], not only illustrating enormous over-pressures that result almost immediately after the initial impact, but under-pressures experimentally depicted in all cases, due to oscillatory motion. It appears, given experimental evidence, that most—if not all—of these models can be aptly described by damped sinusoidal functions, these facts being further corroborated by existing literature, referencing models expounded by Friedlander's seminal work [1–4]. Using other advanced mathematical techniques, such as the Hopf-Cole Transformation, application of the Dirac-delta function and the Heat-Diffusion equation, expressions have been determined to model and predict the associated energies and temperatures within this paper.]]>https://www.frontiersin.org/articles/10.3389/fams.2018.00029
https://www.frontiersin.org/articles/10.3389/fams.2018.00029
Adoption of the Transport and Burgers' Equations in Modeling Neurological Shock-Waves in the Human Brain Due to Improvised Explosive Devices (IEDs)2018-08-02T00:00:00ZStephen W. MasonThis paper considers the propagation of neurological shock-waves in the human head due to improvised explosive devices (IEDs). The models adopted here use various mathematical techniques, including adoption and application of the two most important partial differential equations (PDEs) in this area, such as the Burgers' and Transport equations—together with a discussion of the inherent mechanics witnessed during experiments. In particular, only a one-dimensional model of the propagation of an intense acoustic compression wave—known as a shock-wave—traveling from air into the human head, is analyzed, using experimental data taken from existing literature. Computer simulations of these models also reproduce published experimental measurements of these acoustic dynamic pressures within the human brain, with graphs describing shock-wave motion in both two- and three- dimensions. There follows analysis and explanations of this phenomena, developed more thoroughly, to explain in detail features of experimentally-observed dynamic-pressures resulting in the brain, after a transitory shock-wave rapidly passes through the human head. The final part of this paper leads to further mathematical exposition—intended to be discussed within future publications—of dynamic-pressures, the latter being explained more comprehensively, and in greater detail, for clarity, especially in terms of the inherent physics and mechanical properties of the ensuing dynamics.]]>https://www.frontiersin.org/articles/10.3389/fams.2016.00005
https://www.frontiersin.org/articles/10.3389/fams.2016.00005
Self-Similar Symmetry Model and Cosmic Microwave Background2016-05-11T00:00:00ZTomohide SonodaIn this paper, we present the self-similar symmetry (SSS) model that describes the hierarchical structure of the universe. The model is based on the concept of self-similarity, which explains the symmetry of the cosmic microwave background (CMB). The approximate length and time scales of the six hierarchies of the universe—grand unification, electroweak unification, the atom, the pulsar, the solar system, and the galactic system—are derived from the SSS model. In addition, the model implies that the electron mass and gravitational constant could vary with the CMB radiation temperature.]]>