Reduction of the two‐dimensional master equation to a Smoluchowsky type differential equation with application to CH4→CH3+H

1993 ◽  
Vol 98 (11) ◽  
pp. 8673-8679 ◽  
Author(s):  
S. H. Robertson ◽  
A. I. Shushin ◽  
D. M. Wardlaw
Keyword(s):  
Differential Equation ◽  
Master Equation ◽  
Two Dimensional ◽  
Type Differential Equation

scholarly journals BOUNDARY-VALUE PROBLEM FOR A TWO-DIMENSIONAL SECOND ORDER-TYPE EQUATION WITH DISCRETE ADDITIVE AND MULTIPLICATIVE DERIVATIVES

2021 ◽  
Vol 1 (4(68)) ◽  
pp. 61-63
Author(s):  
V. Sultanova
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
General Solution ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Type Equation ◽  
Order Type ◽  
Second Order ◽  
Two Dimensional ◽  
Type Differential Equation

The present paper is concerned with the study of solutions to the boundary-value problem for a two-dimensional second order-type differential equation with a discrete additive derivative for one argument and a discrete multiplicative derivative for another argument. We will determine the general solution of the considered equation, containing some derived sequences. Further, these unknown sequences are determined using an assigned boundary condition.

scholarly journals Exact Solutions and Conserved Vectors of the Two-Dimensional Generalized Shallow Water Wave Equation

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1439
Author(s):  
Chaudry Masood Khalique ◽  
Karabo Plaatjie
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Elliptic Integral ◽  
Momentum Conservation ◽  
Two Dimensional ◽  
Order Ordinary Differential Equation ◽  
Closed Form Solutions ◽  
Shallow Water Wave

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing the three translation symmetries of the equation, a fourth-order ordinary differential equation is obtained and solved in terms of an incomplete elliptic integral. Moreover, with the aid of Kudryashov’s approach, more closed-form solutions are constructed. In addition, energy and linear momentum conservation laws for the underlying equation are computed by engaging the multiplier approach as well as Noether’s theorem.

scholarly journals DIFFERENTIAL EQUATION APPROACH FOR ONE- AND TWO-DIMENSIONAL LATTICE GREEN'S FUNCTION

2007 ◽  
Vol 21 (02n03) ◽  
pp. 139-154 ◽  
Author(s):  
J. H. ASAD
Keyword(s):  
Differential Equation ◽  
Green’S Function ◽  
Recurrence Relation ◽  
Green's Function ◽  
Two Dimensional ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
The One ◽  
Lattice Green's Function ◽  
Lattice Green’S Function

A first-order differential equation of Green's function, at the origin G(0), for the one-dimensional lattice is derived by simple recurrence relation. Green's function at site (m) is then calculated in terms of G(0). A simple recurrence relation connecting the lattice Green's function at the site (m, n) and the first derivative of the lattice Green's function at the site (m ± 1, n) is presented for the two-dimensional lattice, a differential equation of second order in G(0, 0) is obtained. By making use of the latter recurrence relation, lattice Green's function at an arbitrary site is obtained in closed form. Finally, the phase shift and scattering cross-section are evaluated analytically and numerically for one- and two-impurities.

Additional spectral properties of the fourth-order Bessel-type differential equation

2005 ◽  
Vol 278 (12-13) ◽  
pp. 1538-1549 ◽  
Author(s):  
W. N. Everitt ◽  
H. Kalf ◽  
L. L. Littlejohn ◽  
C. Markett
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Spectral Properties ◽  
Type Differential Equation

scholarly journals Regime of small number of photons in the cavity for a singleemitter laser

2021 ◽  
Vol 2103 (1) ◽  
pp. 012158
Author(s):  
N V Larionov
Keyword(s):  
Differential Equation ◽  
Exact Solutions ◽  
Master Equation ◽  
Cavity Mode ◽  
Photon Number ◽  
System Of Equations ◽  
Equation Approach ◽  
Master Equation Approach ◽  
Single Emitter ◽  
Analytical Expressions

Abstract The model of a single-emitter laser generating in the regime of small number of photons in the cavity mode is theoretically investigated. Based on a system of equations for different moments of the field operators the analytical expressions for mean photon number and photon number variance are obtained. Using the master equation approach the differential equation for the phase-averaged quasi-probability Q is derived. For some limiting cases the exact solutions of this equation are found.

scholarly journals Solutions de similitude d'un jeu différentiel stochastique

2006 ◽  
Vol Volume 5, Special Issue TAM... ◽  
Author(s):  
Mario Lefebvre
Keyword(s):  
Differential Equation ◽  
Stochastic Process ◽  
Partial Differential Equation ◽  
Differential Equations ◽  
Similarity Solutions ◽  
Explicit Solutions ◽  
Two Dimensional ◽  
Controlled Processes ◽  
International Audience ◽  
First Time

International audience A two-dimensional controlled stochastic process defined by a set of stochastic differential equations is considered. Contrary to the most frequent formulation, the control variables appear only in the infinitesimal variances of the process, rather than in the infinitesimal means. The differential game ends the first time the two controlled processes are equal or their difference is equal to a given constant. Explicit solutions to particular problems are obtained by making use of the method of similarity solutions to solve the appropriate partial differential equation. On considère un processus stochastique commandé bidimensionnel défini par un ensemble d'équations différentielles stochastiques. Contrairement à la formulation la plus fréquente, les variables de commande apparaissent dans les variances infinitésimales du processus, plutôt que dans les moyennes infinitésimales. Le jeu différentiel prend fin lorsque les deux processus sont égaux ou que leur différence est égale à une constante donnée. Des solutions explicites à des problèmes particuliers sont obtenues en utilisant la méthode des similitudes pour résoudre l'équation aux dérivées partielles appropriée.

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