scholarly journals Quantization of Damped Systems Using Fractional WKB Approximation

2018 ◽  
Vol 10 (5) ◽  
pp. 34
Author(s):  
Ola A. Jarabah
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Function ◽  
Fractional Derivatives ◽  
Separation Of Variables ◽  
Classical Mechanics ◽  
Wkb Approximation ◽  
Jacobi Function ◽  
Exact Agreement ◽  
Damped Systems

The Hamilton Jacobi theory is used to obtain the fractional Hamilton-Jacobi function for fractional damped systems. The technique of separation of variables is applied here to solve the Hamilton Jacobi partial differential equation for fractional damped systems. The fractional Hamilton-Jacobi function is used to construct the wave function and then to quantize these systems using fractional WKB approximation. The solution of the illustrative example is found to be in exact agreement with the usual classical mechanics for regular Lagrangian when fractional derivatives are replaced with the integer order derivatives and r-0 .

scholarly journals On the Solvability of a Mixed Problem for a High-Order Partial Differential Equation with Fractional Derivatives with Respect to Time, with Laplace Operators with Spatial Variables and Nonlocal Boundary Conditions in Sobolev Classes

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 235 ◽  
Author(s):  
Onur İlhan ◽  
Shakirbay Kasimov ◽  
Shonazar Otaev ◽  
Haci Baskonus
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Mixed Problem ◽  
Fractional Derivatives ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Order Partial Differential Equation ◽  
Sobolev Classes ◽  
Spatial Variables

In this paper, we study the solvability of a mixed problem for a high-order partial differential equation with fractional derivatives with respect to time, and with Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes.

scholarly journals Multiparameter spectral theory of singular differential operators

1988 ◽  
Vol 31 (1) ◽  
pp. 49-66 ◽  
Author(s):  
B. P. Rynne
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Spectral Theory ◽  
Unbounded Domain ◽  
Differential Operators ◽  
Separation Of Variables ◽  
Singular Case ◽  
Ordinary Differential Operators ◽  
Adjoint Operators ◽  
Unbounded Intervals

In this paper we investigate certain aspects of the multiparameter spectral theory of systems of singular ordinary differential operators. Such systems arise in various contexts. For instance, separation of variables for a partial differential equation on an unbounded domain leads to a multiparameter system of ordinary differential equations, some of which are defined on unbounded intervals. The spectral theory of systems of regular differential operators has been studied in many recent papers, e.g. [1, 3, 6, 9, 19, 21], but the singular case has not received so much attention. Some references for the singular case are [7, 8, 10, 13, 14, 18, 20], in addition general multiparameter spectral theory for self adjoint operators is discussed in [3, 9, 19].

scholarly journals Analytic transient solutions of a cylindrical heat equation

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2617-2628
Author(s):  
K.Y. Kung ◽  
Man-Feng Gong ◽  
H.M. Srivastava ◽  
Shy-Der Lin
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Heat Equation ◽  
Heat Conduction Equation ◽  
Separation Of Variables ◽  
Transient Heat Conduction ◽  
Transient Temperature ◽  
Transient Solutions

The principles of superposition and separation of variables are used here in order to investigate the analytical solutions of a certain transient heat conduction equation. The structure of the transient temperature appropriations and the heat-transfer distributions are summed up for a straight mix of the results by means of the Fourier-Bessel arrangement of the exponential type for the investigated partial differential equation.

scholarly journals Quasigeostrophic Equations for Fractional Powers of Infinitesimal Generators

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Luciano Abadias ◽  
Pedro J. Miana
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Infinitesimal Generator ◽  
Fractional Derivatives ◽  
Integral Inequalities ◽  
Fractional Powers ◽  
Partial Differential ◽  
Infinitesimal Generators ◽  
Positive Kernel ◽  
Quasigeostrophic Equations

In this paper we treat the following partial differential equation, the quasigeostrophic equation: ∂/∂t+u·∇f=-σ-Aαf,  0≤α≤1, where (A,D(A)) is the infinitesimal generator of a convolution C0-semigroup of positive kernel on Lp(Rn), with 1≤p<∞. Firstly, we give remarkable pointwise and integral inequalities involving the fractional powers (-A)α for 0≤α≤1. We use these estimates to obtain Lp-decayment of solutions of the above quasigeostrophic equation. These results extend the case of fractional derivatives (taking A=Δ, the Laplacian), which has been studied in the literature.

scholarly journals Explicit Solution for Cylindrical Heat Conduction

2016 ◽  
Vol 13 (2) ◽  
Author(s):  
Kaitlyn Parsons ◽  
Tyler Reichanadter ◽  
Andi Vicksman ◽  
Harvey Segur
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Analytical Method ◽  
Explicit Solution ◽  
Separation Of Variables ◽  
Partial Differential ◽  
Forcing Function

The heat equation is a partial differential equation that elegantly describes heat conduction or other diffusive processes. Primary methods for solving this equation require time-independent boundary conditions. In reality this assumption rarely has any validity. Therefore it is necessary to construct an analytical method by which to handle the heat equation with time-variant boundary conditions. This paper analyzes a physical system in which a solid brass cylinder experiences heat flow from the central axis to a heat sink along its outer rim. In particular, the partial differential equation is transformed such that its boundary conditions are zero which creates a forcing function in the transform PDE. This transformation constructs a Green’s function, which admits the use of variation of parameters to find the explicit solution. Experimental results verify the success of this analytical method. KEYWORDS: Heat Equation; Bessel-Fourier Decomposition; Cylindrical; Time-dependent Boundary Conditions; Orthogonality; Partial Differential Equation; Separation of Variables; Green’s Functions

ON A FAMILY OF CONVEX SOLUTIONS TO A HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION

2013 ◽  
Vol 10 (04) ◽  
pp. 637-658
Author(s):  
KRZYSZTOF RÓZGA
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Asymptotic Behavior ◽  
Additional Assumption ◽  
Separation Of Variables ◽  
P System ◽  
Hyperbolic Partial Differential Equation ◽  
Satisfactory Description ◽  
Nonlinear Part ◽  
Convexity Conditions

A family of convex solutions of Φxx - f(x)Φyy = 0, for x > 0 and y ∈ ℝ, where f is positive and continuously differentiable in (0, ∞), is discussed. It consists of all convex solutions of that equation which are of the form Φ(x, y) = p(x)q(y). The separation of variables is an easy task to perform. In particular, it results in an explicit form of q(y). Imposing convexity conditions requires however more insight. It is observed that a nonlinear part of those conditions, in case of f′ ≤ 0, is related to an asymptotic behavior of p(x) and p′(x) as x → ∞. Then, under an additional assumption that lim x→∞ f(x) > 0, a satisfactory description of the set of all the functions p(x), which determines convex Φ(x, y) via the formula Φ(x, y) = p(x)q(y), is obtained. So constructed functions Φ(x, y) are convex entropies for the corresponding p-system. Finally two nontrivial examples, involving a modified Bessel and hypergeometric equation are provided.

Mass transfer in droplets determined by analog computation

SIMULATION ◽  
1970 ◽  
Vol 15 (6) ◽  
pp. 241-248 ◽  
Author(s):  
R.M. Wellek ◽  
J.T. Kuo ◽  
R.C. Waggoner
Keyword(s):  
Differential Equation ◽  
Mass Transfer ◽  
Partial Differential Equation ◽  
Separation Of Variables ◽  
Analog Computer ◽  
Internal Circulation ◽  
Eigenvalues And Eigenfunctions ◽  
Partial Differential ◽  
Sturm Liouville ◽  
Single Droplets

A mechanism describing the rate of mass transfer to single droplets with a special type of internal circulation is described by a model consisting of a partial differential equation with two independent variables. A Sturm-Liouville system is obtained when the partial differential equation is transformed into a set of ordinary differential equations by the separation-of-variables technique. The eigenvalues and eigenfunctions which determine the solution to this system are obtained by a two variable search procedure on an iterative-analog computer.

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