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.

The Closed-Form Solutions For The Breakdown Voltages Of 6H-SiC Reachthrough Diodes

1998 ◽  
Vol 512 ◽  
Author(s):  
You-Sang Lee ◽  
D.-S. Byeon ◽  
Y.-I. Choi ◽  
I.-Y. Park ◽  
Min-Koo Han
Keyword(s):  
Closed Form ◽  
Breakdown Voltage ◽  
Epitaxial Layer ◽  
Doping Concentration ◽  
Impact Ionization ◽  
Analytic Solutions ◽  
Ionization Coefficient ◽  
Closed Form Solutions ◽  
The Impact ◽  
Effective Ionization

ABSTRACTThe closed-form analytic solutions for the breakdown voltage of 6H-SiC RTD, reachthrough diode, having the structure of p+-n-n+, are successfully derived by solving the impact ionization integral using effective ionization coefficient in the reachthrough condition. In the region of the lowly doped epitaxial layer, the breakdown voltages of 6H-SiC RTD nearly constant with the increased doping concentration. Also the breakdown voltages of 6H-SiC RTD decrease, in the region of the highly doped epitaxial layer, which coincides with Baliga'seq. [1].

scholarly journals Riccati equations

1987 ◽  
Vol 10 (1) ◽  
pp. 205-207
Author(s):  
Lloyd K. Williams
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Closed Form ◽  
Riccati Equations ◽  
Second Order ◽  
Closed Form Solutions ◽  
The Matrix ◽  
Matrix Case

In this paper we find closed form solutions of some Riccati equations. Attention is restricted to the scalar as opposed to the matrix case. However, the ones considered have important applications to mathematics and the sciences, mostly in the form of the linear second-order ordinary differential equations which are solved herewith.

scholarly journals Degenerate Canonical Forms of Ordinary Second-Order Linear Homogeneous Differential Equations

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 94
Author(s):  
Dimitris M. Christodoulou ◽  
Eric Kehoe ◽  
Qutaibeh D. Katatbeh
Keyword(s):  
Differential Equations ◽  
Fundamental Equation ◽  
Mathematical Physics ◽  
Analytic Solutions ◽  
Second Order ◽  
Canonical Forms ◽  
Order Linear ◽  
Physical Problems ◽  
Radial Schrödinger Equation ◽  
Linear Homogeneous Differential Equation

For each fundamental and widely used ordinary second-order linear homogeneous differential equation of mathematical physics, we derive a family of associated differential equations that share the same “degenerate” canonical form. These equations can be solved easily if the original equation is known to possess analytic solutions, otherwise their properties and the properties of their solutions are de facto known as they are comparable to those already deduced for the fundamental equation. We analyze several particular cases of new families related to some of the famous differential equations applied to physical problems, and the degenerate eigenstates of the radial Schrödinger equation for the hydrogen atom in N dimensions.

scholarly journals F-Operators for the Construction of Closed Form Solutions to Linear Homogenous PDEs with Variable Coefficients

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 918
Author(s):  
Zenonas Navickas ◽  
Tadas Telksnys ◽  
Romas Marcinkevicius ◽  
Maosen Cao ◽  
Minvydas Ragulskis
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Closed Form ◽  
Analytic Solutions ◽  
Variable Coefficients ◽  
Partial Differential ◽  
Computational Framework ◽  
Closed Form Solutions

A computational framework for the construction of solutions to linear homogenous partial differential equations (PDEs) with variable coefficients is developed in this paper. The considered class of PDEs reads: ∂p∂t−∑j=0m∑r=0njajrtxr∂jp∂xj=0 F-operators are introduced and used to transform the original PDE into the image PDE. Factorization of the solution into rational and exponential parts enables us to construct analytic solutions without direct integrations. A number of computational examples are used to demonstrate the efficiency of the proposed scheme.

The Artificial Hamiltonian, First Integrals, and Closed-Form Solutions of Dynamical Systems for Epidemics

2018 ◽  
Vol 73 (4) ◽  
pp. 323-330 ◽  
Author(s):  
Rehana Naz ◽  
Imran Naeem
Keyword(s):  
Dynamical Systems ◽  
Hamiltonian System ◽  
Closed Form ◽  
Hamiltonian Systems ◽  
Physical Structure ◽  
First Integrals ◽  
Second Order ◽  
Hamiltonian Structures ◽  
Closed Form Solutions ◽  
First Order

AbstractThe non-standard Hamiltonian system, also referred to as a partial Hamiltonian system in the literature, of the form ${\dot q^i} = \frac{{\partial H}}{{\partial {p_i}}},{\text{ }}{\dot p^i} = - \frac{{\partial H}}{{\partial {q_i}}} + {\Gamma ^i}(t,{\text{ }}{q^i},{\text{ }}{p_i})$ appears widely in economics, physics, mechanics, and other fields. The non-standard (partial) Hamiltonian systems arise from physical Hamiltonian structures as well as from artificial Hamiltonian structures. We introduce the term ‘artificial Hamiltonian’ for the Hamiltonian of a model having no physical structure. We provide here explicitly the notion of an artificial Hamiltonian for dynamical systems of ordinary differential equations (ODEs). Also, we show that every system of second-order ODEs can be expressed as a non-standard (partial) Hamiltonian system of first-order ODEs by introducing an artificial Hamiltonian. This notion of an artificial Hamiltonian gives a new way to solve dynamical systems of first-order ODEs and systems of second-order ODEs that can be expressed as a non-standard (partial) Hamiltonian system by using the known techniques applicable to the non-standard Hamiltonian systems. We employ the proposed notion to solve dynamical systems of first-order ODEs arising in epidemics.

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