scholarly journals Unified Treatment of a Class of Spherically Symmetric Potentials: Quasi-Exact Solution

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
H. Panahi ◽  
M. Baradaran
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Representation Theory ◽  
Exact Solutions ◽  
Lie Algebra ◽  
Algebraic Approach ◽  
Spherically Symmetric ◽  
Exact Solvability ◽  
Enveloping Algebra ◽  
Basic Differential Equation

We investigate the Schrödinger equation for a class of spherically symmetric potentials in a simple and unified manner using the Lie algebraic approach within the framework of quasi-exact solvability. We illustrate that all models give rise to the same basic differential equation, which is expressible as an element of the universal enveloping algebra ofsl(2). Then, we obtain the general exact solutions of the problem by employing the representation theory ofsl(2)Lie algebra.

New exact solutions for the Wick-type stochastic Zakharov–Kuznetsov equation for modelling waves on shallow water surfaces

2017 ◽  
Vol 25 (2) ◽  
Author(s):  
S. Saha Ray ◽  
S. Singh
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solution ◽  
Exact Solutions ◽  
White Noise ◽  
Shallow Water ◽  
Hermite Transform ◽  
New Exact Solutions ◽  
Kudryashov Method ◽  
Stochastic Solution

AbstractIn this article, an exact solution of the Wick-type stochastic Zakharov–Kuznetsov equation has been obtained by using the Kudryashov method. We have used the Hermite transform for transforming the Wick-type stochastic Zakharov–Kuznetsov equation into a deterministic partial differential equation. Also we have applied the inverse Hermite transform for obtaining a set of stochastic solution in the white noise space.

scholarly journals Affinoid Dixmier Modules and the Deformed Dixmier-Moeglin Equivalence

2021 ◽  
Author(s):  
Adam Jones
Keyword(s):  
Representation Theory ◽  
Lie Groups ◽  
Lie Algebra ◽  
Algebraic Structure ◽  
Verma Module ◽  
Finitely Generated ◽  
Enveloping Algebra ◽  
Primitive Ideals

AbstractThe affinoid enveloping algebra $\widehat {U({\mathscr{L}})}_{K}$ U ( L ) ̂ K of a free, finitely generated $\mathbb {Z}_{p}$ ℤ p -Lie algebra ${\mathscr{L}}$ L has proven to be useful within the representation theory of compact p-adic Lie groups, and we aim to further understand its algebraic structure. To this end, we define the notion of a Dixmier module over $\widehat {U({\mathscr{L}})}_{K}$ U ( L ) ̂ K , a generalisation of the Verma module, and we prove that when ${\mathscr{L}}$ L is nilpotent, all primitive ideals of $\widehat {U({\mathscr{L}})}_{K}$ U ( L ) ̂ K can be described in terms of annihilator ideals of Dixmier modules. Using this, we take steps towards proving that this algebra satisfies a version of the classical Dixmier-Moeglin equivalence.

scholarly journals On Ulam Stability of a Generalized Delayed Differential Equation of Fractional Order

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Janusz Brzdęk ◽  
Nasrin Eghbali ◽  
Vida Kalvandi
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Exact Solutions ◽  
Fractional Order ◽  
Nonlinear Operator ◽  
Main Tool ◽  
Ulam Stability ◽  
Speed Of Convergence ◽  
Delayed Differential Equation ◽  
Pointwise Limit

AbstractWe investigate Ulam stability of a general delayed differential equation of a fractional order. We provide formulas showing how to generate the exact solutions of the equation using functions that satisfy it only approximately. Namely, the approximate solution $$\phi $$ ϕ generates the exact solution as a pointwise limit of the sequence $$\varLambda ^n\phi $$ Λ n ϕ with some integral (possibly, nonlinear) operator $$\varLambda $$ Λ . We estimate the speed of convergence and the distance between those approximate and exact solutions. Moreover, we provide some exemplary calculations, involving the Chebyshev and Bielecki norms and some semigauges, that could help to obtain reasonable outcomes for such estimations in some particular cases. The main tool is the Diaz–Margolis fixed point alternative.

scholarly journals Separation of Variables for

2005 ◽  
Vol 48 (4) ◽  
pp. 587-600 ◽  
Author(s):  
Samuel A. Lopes
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Analogous Result ◽  
Separation Of Variables ◽  
Positive Part ◽  
Semisimple Lie Algebra ◽  
Infinite Dimensional ◽  
Enveloping Algebra ◽  
Complex Semisimple ◽  
Quantized Enveloping Algebra

AbstractLet be the positive part of the quantized enveloping algebra . Using results of Alev–Dumas and Caldero related to the center of , we show that this algebra is free over its center. This is reminiscent of Kostant's separation of variables for the enveloping algebra U(g) of a complex semisimple Lie algebra g, and also of an analogous result of Joseph–Letzter for the quantum algebra Ŭq(g). Of greater importance to its representation theory is the fact that is free over a larger polynomial subalgebra N in n variables. Induction from N to provides infinite-dimensional modules with good properties, including a grading that is inherited by submodules.

Modified Auxiliary Differential Equation Method and Exact Solutions Generalized Schrödinger

2014 ◽  
Vol 513-517 ◽  
pp. 4470-4473 ◽  
Author(s):  
Lin Tian ◽  
Yu Ping Qin
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Exact Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Equation Method ◽  
Hyperbolic Function ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Solution ◽  
Auxiliary Differential Equation Method ◽  
Differential Equation Method

This paper describes a method on which modify auxiliary differential equation method by using this method for solving nonlinear partial differential equations and with aid of Maple Software ,we get the exact solution of the generalized schrödinger, including hyperbolic function solutions, trigonometric solution.

scholarly journals SEARCHING EXACT SOLUTIONS FOR COMPACT STARS IN BRANEWORLD: A CONJECTURE

2008 ◽  
Vol 23 (38) ◽  
pp. 3247-3263 ◽  
Author(s):  
J. OVALLE
Keyword(s):  
Exact Solution ◽  
General Relativity ◽  
Exact Solutions ◽  
Degrees Of Freedom ◽  
Compact Stars ◽  
Necessary Condition ◽  
Spherically Symmetric ◽  
Field Equations ◽  
Internal Solution ◽  
Particular Solution

In the context of the braneworld, a method to find consistent solutions to Einstein's field equations in the interior of a spherically symmetric, static and non-uniform stellar distribution with Weyl stresses is developed. This method, based on the fact that any braneworld stellar solution must have the general relativity solution as a limit, produces a constraint which reduces the degrees of freedom on the brane. Hence the nonlocality and non-closure of the braneworld equations can be overcome. The constraint found is physically interpreted as a necessary condition to regain general relativity, and a particular solution for it is used to find an exact and physically acceptable analytical internal solution to no-uniform stellar distributions on the brane. It is shown that such an exact solution is possible due to the fact that bulk corrections to pressure, density and a metric component are a null source of anisotropic effects on the brane. A conjecture is proposed regarding the possibility of finding physically relevant exact solutions to non-uniform stellar distributions on the brane.

ALGEBRAIC APPROACH TO QUASI-EXACT SOLUTIONS OF THE KLEIN–GORDON–COULOMB PROBLEM

2012 ◽  
Vol 27 (30) ◽  
pp. 1250176 ◽  
Author(s):  
H. PANAHI ◽  
M. BARADARAN
Keyword(s):  
Exact Solutions ◽  
Invariant Subspace ◽  
Coulomb Potential ◽  
Gordon Equation ◽  
Algebraic Approach ◽  
Finite Part ◽  
Coulomb Problem ◽  
Exact Solvability ◽  
Klein Gordon ◽  
Klein Gordon Equation

The Klein–Gordon equation in the presence of generalized Coulomb potential is solved and the quasi-exact solutions are obtained via the sl(2) algebraization. The condition of quasi-exact solvability is derived by matching the condition of invariant subspace on the problem. The Lie-algebraic approach of quasi-exact solvability is applied to the problem and the (n+1)×(n+1) matrix for finite values of n is obtained in quite a detailed manner and thereby the finite part of the spectrum is obtained.

scholarly journals On Fuzzy differential equation

2019 ◽  
Vol 11 (2) ◽  
pp. 1-9
Author(s):  
Eman Ali Hussain ◽  
Yahya Mourad Abdul – Abbass
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Exact Solutions ◽  
Turing Machine ◽  
Hybrid Method ◽  
Hybrid Methods ◽  
Fitness Function ◽  
Approximate Solutions ◽  
Fuzzy Differential Equation

In this paper, we introduce a hybrid method to use fuzzy differential equation, and Genetic Turing Machine developed for solving nth order fuzzy differential equation under Seikkala differentiability concept [14]. The Errors between the exact solutions and the approximate solutions were computed by fitness function and the Genetic Turing Machine results are obtained. After comparing the approximate solution obtained by the GTM method with approximate to the exact solution, the approximate results by Genetic Turing Machine demonstrate the efficiency of hybrid methods for solving fuzzy differential equations (FDE).

Difference Schemes for Nonlinear BVPS on the Semiaxis

2007 ◽  
Vol 7 (1) ◽  
pp. 25-47 ◽  
Author(s):  
I.P. Gavrilyuk ◽  
M. Hermann ◽  
M.V. Kutniv ◽  
V.L. Makarov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Adaptive Algorithm ◽  
Order Differential Equation ◽  
Constructive Method ◽  
Numerical Examples ◽  
Exact Difference ◽  
The Given

Abstract The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

scholarly journals CONSTRUCTIVE NONCOMMMUTATIVE INVARIANT THEORY

2021 ◽  
Author(s):  
MÁTYÁS DOMOKOS ◽  
VESSELIN DRENSKY
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Invariant Theory ◽  
Reductive Group ◽  
Symmetric Tensor ◽  
Free Associative Algebra ◽  
Tensor Algebra ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Degree Bounds

AbstractThe problem of finding generators of the subalgebra of invariants under the action of a group of automorphisms of a finite-dimensional Lie algebra on its universal enveloping algebra is reduced to finding homogeneous generators of the same group acting on the symmetric tensor algebra of the Lie algebra. This process is applied to prove a constructive Hilbert–Nagata Theorem (including degree bounds) for the algebra of invariants in a Lie nilpotent relatively free associative algebra endowed with an action induced by a representation of a reductive group.

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