scholarly journals Local fractional heat and wave equations with Laguerre type derivatives

2020 ◽  
Vol 24 (4) ◽  
pp. 2575-2580 ◽  
Author(s):  
Chun-Fu Wei
Keyword(s):  
Differential Equation ◽  
Bounded Domain ◽  
Wave Equations ◽  
Separation Of Variables ◽  
Fractional Pde

In this paper, we investigate a local fractional PDE with Laguerre type derivative. The considered equation represents a general extension of the classical heat and wave equations. The method of separation of variables is used to solve the differential equation defined in a bounded domain.

scholarly journals Symmetry determination and nonlinearization of a nonlinear time-fractional partial differential equation

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190564
Author(s):  
Zhi-Yong Zhang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Lie Algebra ◽  
Infinitesimal Generator ◽  
Linear Time ◽  
Lie Symmetry ◽  
Fractional Partial Differential Equation ◽  
Partial Differential ◽  
Fractional Pde ◽  
Leading Position

We first show that the infinitesimal generator of Lie symmetry of a time-fractional partial differential equation (PDE) takes a unified and simple form, and then separate the Lie symmetry condition into two distinct parts, where one is a linear time-fractional PDE and the other is an integer-order PDE that dominates the leading position, even completely determining the symmetry for a particular type of time-fractional PDE. Moreover, we show that a linear time-fractional PDE always admits an infinite-dimensional Lie algebra of an infinitesimal generator, just as the case for a linear PDE and a nonlinear time-fractional PDE admits, at most, finite-dimensional Lie algebra. Thus, there exists no invertible mapping that converts a nonlinear time-fractional PDE to a linear one. We illustrate the results by considering two examples.

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].

ON AN INTEGRO-DIFFERENTIAL EQUATION OF PSEUDOPARABOLIC-PSEUDOHYPERBOLIC TYPE WITH DEGENERATE KERNELS

2018 ◽  
Vol 52 (1 (245)) ◽  
pp. 19-26 ◽  
Author(s):  
T.K. Yuldashev
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Fourier Method ◽  
Integro Differential Equation ◽  
The One

In the article the questions of solvability of boundary value problem for a homogeneous pseudoparabolic-pseudohyperbolic type integro-differential equation with degenerate kernels are considered. The Fourier method based on separation of variables is used. A criterion for the one-valued solvability of the considering problem is found. Under this criterion the one-valued solvability of the problem is proved.

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 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.

On the uniqueness of inverse problems for the reduced wave equation with unknown embedded obstacles

2021 ◽  
Author(s):  
Jiaqing Yang ◽  
Meng Ding ◽  
Keji Liu
Keyword(s):  
Wave Equation ◽  
Inverse Problems ◽  
Bounded Domain ◽  
Open Subset ◽  
Wave Equations ◽  
A Priori ◽  
Singularity Analysis ◽  
Piecewise Constant ◽  
Small Domain ◽  
Dirichlet To Neumann Map

Abstract In this paper, we consider inverse problems associated with the reduced wave equation on a bounded domain Ω belongs to R^N (N >= 2) for the case where unknown obstacles are embedded in the domain Ω. We show that, if both the leading and 0-order coefficients in the equation are a priori known to be piecewise constant functions, then both the coefficients and embedded obstacles can be simultaneously recovered in terms of the local Dirichlet-to-Neumann map defined on an arbitrary small open subset of the boundary \partial Ω. The method depends on a well-defined coupled PDE-system constructed for the reduced wave equations in a sufficiently small domain and the singularity analysis of solutions near the interface for the model.

Effect of Vertical Elastic Baffle on Liquid Sloshing in Rectangular Rigid Container

2021 ◽  
pp. 2150167
Author(s):  
Xun Meng ◽  
Ding Zhou ◽  
Jiadong Wang
Keyword(s):  
Wave Equations ◽  
Velocity Potential ◽  
Separation Of Variables ◽  
Structural Instability ◽  
Dynamic Responses ◽  
Mode Shapes ◽  
System Response ◽  
Coupled Mode ◽  
Response Equation ◽  
Liquid Oscillation

Sloshing may induce adverse loads to cause structural instability and damage. A vertical elastic baffle mounted at the inside bottom of a rectangular container is used as an anti-slosh device to attenuate the liquid oscillation. A semi-analytical model is presented to analyze the hydroelastic problem. The liquid is partitioned into four simple sub-domains with three hypothetical interfaces. The velocity potential of each sub-domain is analytically deduced by the separation of variables. The baffle deflection is expanded into the Fourier series by its dry modals. The eigenvalue equation is formulated by plugging the velocity potentials into the sloshing conditions, interface continuity conditions, as well as the dominant equation and compatibility conditions of the baffle. Then, the velocity potential is expressed by a complete basis of the coupled mode shapes for the system considered under lateral excitation. The system response equation is constituted by inserting the velocity potential into wave equations and baffle equation. The proposed method is verified by comparing the present results with the available data. In addition, numerical analyses are performed to examine the effects of baffle parameters on the natural frequencies, mode shapes and dynamic responses of the container. The sloshing frequency may be altered to a higher value due to the installation of the elastic baffle.

scholarly journals Nonlinear Green’s Functions for Wave Equation with Quadratic and Hyperbolic Potentials

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Asatur Zh. Khurshudyan
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Wave Equations ◽  
Separation Of Variables ◽  
Linear Equations ◽  
Second Order ◽  
Nonlinear Wave Equations ◽  
One Dimensional ◽  
Exponential Nonlinearity ◽  
Second Order Differential Equations

The advantageous Green’s function method that originally has been developed for nonhomogeneous linear equations has been recently extended to nonlinear equations by Frasca. This article is devoted to rigorous and numerical analysis of some second-order differential equations new nonlinearities by means of Frasca’s method. More specifically, we consider one-dimensional wave equation with quadratic and hyperbolic nonlinearities. The case of exponential nonlinearity has been reported earlier. Using the method of generalized separation of variables, it is shown that a hierarchy of nonlinear wave equations can be reduced to second-order nonlinear ordinary differential equations, to which Frasca’s method is applicable. Numerical error analysis in both cases of nonlinearity is carried out for various source functions supporting the advantage of the method.

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