scholarly journals Exact analytic solutions of unsolvable classes of first and second order nonlinear ODEs (Part I: Abel’s equations)

2005 ◽  
Vol 18 (2) ◽  
pp. 155-162 ◽  
Author(s):  
Dimitrios E. Panayotounakos
Keyword(s):  
Analytic Solutions ◽  
Second Order ◽  
Nonlinear Odes ◽  
Exact Analytic Solutions

scholarly journals Closed-Form Solutions of the Thomas-Fermi in Heavy Atoms and the Langmuir-Blodgett in Current Flow ODEs in Mathematical Physics

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Efstathios E. Theotokoglou ◽  
Theodoros I. Zarmpoutis ◽  
Ioannis H. Stampouloglou
Keyword(s):  
Closed Form ◽  
Current Flow ◽  
Mathematical Physics ◽  
Analytic Solutions ◽  
Second Order ◽  
Closed Form Solutions ◽  
Langmuir Blodgett ◽  
Thomas Fermi ◽  
Heavy Atoms ◽  
Exact Analytic Solutions

Two kinds of second-order nonlinear, ordinary differential equations (ODEs) appearing in mathematical physics are analyzed in this paper. The first one concerns the Thomas-Fermi (TF) equation, while the second concerns the Langmuir-Blodgett (LB) equation in current flow. According to a mathematical methodology recently developed, the exact analytic solutions of both TF and LB ODEs are proposed. Both of these are nonlinear of the second order and by a series of admissible functional transformations are reduced to Abel’s equations of the second kind of the normal form. The closed form solutions of the TF and LB equations in the phase and physical plane are given. Finally a new interesting result has been obtained related to the derivative of the TF function at the limit.

CRYSTALLINE GRAVITY

2009 ◽  
Vol 24 (18n19) ◽  
pp. 3243-3255 ◽  
Author(s):  
GERARD 't HOOFT
Keyword(s):  
Finite Number ◽  
Lorentz Group ◽  
Degrees Of Freedom ◽  
Holonomy Group ◽  
Discrete Subgroup ◽  
Analytic Solutions ◽  
Space Time ◽  
Exact Analytic Solutions ◽  
Finite Domains ◽  
Dimensional Crystal

Matter interacting classically with gravity in 3+1 dimensions usually gives rise to a continuum of degrees of freedom, so that, in any attempt to quantize the theory, ultraviolet divergences are nearly inevitable. Here, we investigate a theory that only displays a finite number of degrees of freedom in compact sections of space-time. In finite domains, one has only exact, analytic solutions. This is achieved by limiting ourselves to straight pieces of string, surrounded by locally flat sections of space-time. Next, we suggest replacing in the string holonomy group, the Lorentz group by a discrete subgroup, which turns space-time into a 4-dimensional crystal with defects.

scholarly journals Exact analytical solutions of continuously graded models of flat lenses based on transformation optics

2017 ◽  
Vol 30 (4) ◽  
pp. 639-646 ◽  
Author(s):  
Mariana Dalarsson ◽  
Raj Mittra
Keyword(s):  
Magnetic Fields ◽  
Longitudinal Direction ◽  
Middle Layer ◽  
Analytic Solutions ◽  
Transformation Optics ◽  
Confluent Hypergeometric Functions ◽  
Graded Index ◽  
Exact Analytic Solutions ◽  
Electric And Magnetic Fields ◽  
Physical Insight

We present a study of exact analytic solutions for electric and magnetic fields in continuously graded flat lenses designed utilizing transformation optics. The lenses typically consist of a number of layers of graded index dielectrics in both the radial and longitudinal directions, where the central layer in the longitudinal direction primarily contributes to a bulk of the phase transformation, while other layers act as matching layers and reduce the reflections at the interfaces of the middle layer. Such lenses can be modeled as compact composites with continuous permittivity (and if needed) permeability functions which asymptotically approach unity at the boundaries of the composite cylinder. We illustrate the proposed procedures by obtaining the exact analytic solutions for the electric and magnetic fields for one simple special class of composite designs with radially graded parameters. To this purpose we utilize the equivalence between the Helmholtz equation of our graded flat lens and the quantum-mechanical radial Schr?dinger equation with Coulomb potential, furnishing the results in the form of Kummer confluent hypergeometric functions. Our approach allows for a better physical insight into the operation of our transformation optics-based graded lenses and opens a path toward novel designs and approaches.

scholarly journals New Soliton Solutions for the Higher-Dimensional Non-Local Ito Equation

2021 ◽  
Vol 10 (1) ◽  
pp. 374-384
Author(s):  
Mustafa Inc ◽  
E. A. Az-Zo’bi ◽  
Adil Jhangeer ◽  
Hadi Rezazadeh ◽  
Muhammad Nasir Ali ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Solution Process ◽  
Soliton Solutions ◽  
Analytic Solutions ◽  
Bright Solitons ◽  
Exact Analytic Solutions ◽  
Higher Dimensional ◽  
Non Local ◽  
Free Parameters ◽  
Ito Equation

Abstract In this article, (2+1)-dimensional Ito equation that models waves motion on shallow water surfaces is analyzed for exact analytic solutions. Two reliable techniques involving the simplest equation and modified simplest equation algorithms are utilized to find exact solutions of the considered equation involving bright solitons, singular periodic solitons, and singular bright solitons. These solutions are also described graphically while taking suitable values of free parameters. The applied algorithms are effective and convenient in handling the solution process for Ito equation that appears in many phenomena.

Computerized Symbolic Computation on a Sixth-order Model for Liquid Waves in the Presence of Surface Tension or a Floating Ice

2004 ◽  
Vol 59 (12) ◽  
pp. 997-1003 ◽  
Author(s):  
Bo Tian ◽  
Yi-Tian Gao
Keyword(s):  
Surface Tension ◽  
Symbolic Computation ◽  
Analytic Solutions ◽  
Rapid Expansion ◽  
Liquid Propellant ◽  
Science And Engineering ◽  
Exact Analytic Solutions ◽  
Surface Liquid ◽  
Rational Expressions ◽  
Similarity Reductions

Computerized symbolic computation reflects the rapid expansion of computer sciences in various fields of science and engineering, while the studies on the liquid surfaces for rivers, oceans, aviation kerosene, liquid propellant for rockets, etc., are of current interest. In the presence of surface tension or sea ice, and with symbolic computation, the Hărăgus-Courcelle-Il’ichev model for surface liquid waves is hereby investigated. Several similarity reductions are presented, some of which are explicitly written out as exact analytic solutions having their rational expressions with respect to the dimensionless spatial variables of the model.

scholarly journals Internal Gravity Waves in a Saturated Moist Neutral Atmosphere

2013 ◽  
Vol 70 (12) ◽  
pp. 3693-3709 ◽  
Author(s):  
David J. Muraki ◽  
Richard Rotunno
Keyword(s):  
Numerical Solutions ◽  
Internal Gravity Waves ◽  
Analytic Solutions ◽  
Neutral Atmosphere ◽  
Atmospheric Flow ◽  
Start Up ◽  
Exact Analytic Solutions ◽  
Downward Displacement ◽  
Neutral Flow ◽  
Saturated Atmosphere

Abstract This work is motivated by an unusual feature associated with the start-up of a moist nearly neutral atmospheric flow over a mountain ridge that was previously observed in a full-physics numerical model. In that study, the upstream propagation of a wave of subsidence precluded the establishment of upward-displaced and saturated flow that might be expected upstream of the topography. This phenomenon was hypothesized to be a consequence of the peculiar property of saturated moist neutral flow: an upward air parcel displacement produces zero buoyancy, while a downward displacement desaturates the air parcel and produces a positive buoyancy anomaly. In the present study, this hypothesis is confirmed within numerical solutions to a reduced system of equations that incorporates the saturated-atmosphere property in a particularly simple manner. The relatively uncomplicated nature of these solutions motivates the numerical solution of a further simplified initial-value problem for both nonhydrostatic and hydrostatic flow. Exact analytic solutions are developed for the latter hydrostatic case, which explains the upstream-propagating wave of subsidence as a shock phenomenon.

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