scholarly journals Nonlocal Symmetries for Time-Dependent Order Differential Equations

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 771
Author(s):  
Andrei Ludu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Variable Coefficients ◽  
Time Dependent ◽  
Harmonic Numbers ◽  
Nonlocal Symmetries ◽  
The Riemann Zeta Function ◽  
Symmetry Of The Solution ◽  
New Type ◽  
Riemann Zeta

A new type of ordinary differential equation is introduced and discussed: time-dependent order ordinary differential equations. These equations are solved via fractional calculus by transforming them into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order differential equation represent deformations of the solutions of the classical (integer order) differential equations, mapping them into one-another as limiting cases. This equation can also move, remove or generate singularities without involving variable coefficients. An interesting symmetry of the solution in relation to the Riemann zeta function and Harmonic numbers is observed.

scholarly journals Nonlocal Symmetries for Time-Dependent Order Differential Equations

2018 ◽  
Author(s):  
Andrei Ludu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Time Dependent ◽  
Harmonic Numbers ◽  
Nonlocal Symmetries ◽  
Integrable Kernel ◽  
The Riemann Zeta Function ◽  
Symmetry Of The Solution ◽  
New Type ◽  
Riemann Zeta

A new type of ordinary differential equation is introduced and discussed, namely, the time-dependent order ordinary differential equations. These equations can be solved via fractional calculus and are mapped into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order differential equations smoothly deforms solutions of the classical integer order ordinary differential equations into one-another, and can generate or remove singularities. An interesting symmetry of the solution in relation to the Riemann zeta function and Harmonic numbers was also proved.

scholarly journals Approximate Solutions of Fisher's Type Equations with Variable Coefficients

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
A. H. Bhrawy ◽  
M. A. Alghamdi
Keyword(s):  
Differential Equations ◽  
Legendre Polynomials ◽  
Nonlinear Partial Differential Equations ◽  
Approximate Solutions ◽  
High Accuracy ◽  
Variable Coefficients ◽  
Time Dependent ◽  
First Order ◽  
Spatial Derivatives ◽  
Interpolation Nodes

The spectral collocation approximations based on Legendre polynomials are used to compute the numerical solution of time-dependent Fisher’s type problems. The spatial derivatives are collocated at a Legendre-Gauss-Lobatto interpolation nodes. The proposed method has the advantage of reducing the problem to a system of ordinary differential equations in time. The four-stage A-stable implicit Runge-Kutta scheme is applied to solve the resulted system of first order in time. Numerical results show that the Legendre-Gauss-Lobatto collocation method is of high accuracy and is efficient for solving the Fisher’s type equations. Also the results demonstrate that the proposed method is powerful algorithm for solving the nonlinear partial differential equations.

Generalized Bernoulli Wick differential equation

2021 ◽  
Vol 24 (01) ◽  
pp. 2150008
Author(s):  
Sami H. Altoum ◽  
Aymen Ettaieb ◽  
Hafedh Rguigui
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Exponential Growth ◽  
Present Method ◽  
Dual Space ◽  
Entire Functions ◽  
Wick Product ◽  
Derivation Operator ◽  
New Type ◽  
Minimal Type

Based on the distributions space on [Formula: see text] (denoted by [Formula: see text]) which is the topological dual space of the space of entire functions with exponential growth of order [Formula: see text] and of minimal type, we introduce a new type of differential equations using the Wick derivation operator and the Wick product of elements in [Formula: see text]. These equations are called generalized Bernoulli Wick differential equations which are the analogue of the classical Bernoulli differential equations. We solve these generalized Wick differential equations. The present method is exemplified by several examples.

CONDITIONS OF EXISTENCE THE TRACE OF FUNCTIONS FROM SPACE WITH MULTIWEIGHTED DERIVATIVES IN A SPECIAL POINT

2020 ◽  
pp. 123-128
Author(s):  
А. Kalybay ◽  
◽  
Zh. Keulimzhaeva ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Functional Space ◽  
Variable Coefficients ◽  
Weight Functions ◽  
Weight Sobolev Space ◽  
Strongly Degenerate

When solving differential equations with variable coefficients, especially when the coefficients degenerate at the boundary of a given domain, problems arise in the formulation of boundary value problems. Usually, differential equations with variable coefficients are investigated in a suitable weight functional space. Often in the role of such spaces the weight Sobolev space or various generalizations are considered, which are currently sufficiently studied. However, in some cases, when the coefficients of the considered differential equation are strongly degenerate, the formulation of boundary value problems becomes problematic. In this work, we consider the so-called space with multiweighted derivatives, where after each derivative, the function is multiplied by the weight function and then the next derivative is taken. By controlling the behavior of the weight functions on the boundary, strongly degenerate equations can be investigated. Here we investigate the existence of traces on the boundary of a function from such spaces.

New formula for solution of one class of linear differential equations of the second order with the variable coefficients

2020 ◽  
Vol 8 (3) ◽  
pp. 61-68
Author(s):  
Avyt Asanov ◽  
Kanykei Asanova
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Variable Coefficients ◽  
Second Order ◽  
Linear Differential Equations ◽  
Common Solution ◽  
The Common ◽  
Linear Differential ◽  
New Formula

Exact solutions for linear and nonlinear differential equations play an important rolein theoretical and practical research. In particular many works have been devoted tofinding a formula for solving second order linear differential equations with variablecoefficients. In this paper we obtained the formula for the common solution of thelinear differential equation of the second order with the variable coefficients in themore common case. We also obtained the new formula for the solution of the Cauchyproblem for the linear differential equation of the second order with the variablecoefficients.Examples illustrating the application of the obtained formula for solvingsecond-order linear differential equations are given.Key words: The linear differential equation, the second order, the variablecoefficients,the new formula for the common solution, Cauchy problem, examples.

scholarly journals Some further results on oscillation of neutral differential equations

1992 ◽  
Vol 46 (1) ◽  
pp. 149-157 ◽  
Author(s):  
Jianshe Yu ◽  
Zhicheng Wang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Variable Coefficients ◽  
Neutral Differential Equation ◽  
Neutral Differential Equations ◽  
All Solutions ◽  
Image Position

We obtain new sufficient conditions for the oscillation of all solutions of the neutral differential equation with variable coefficientswhere P, Q, R ∈ C([t0, ∞), R+), r ∈ (0, ∞) and τ, σ ∈ [0, ∞). Our results improve several known results in papers by: Chuanxi and Ladas; Lalli and Zhang; Wei; Ruan.

V. Mechanical integration of the linear differential equations of the second order with variable coefficients

1876 ◽  
Vol 24 (164-170) ◽  
pp. 269-271 ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Mathematical Physics ◽  
Variable Coefficients ◽  
Second Order ◽  
Sectional Area ◽  
Practical Solution ◽  
Hanging Chain ◽  
Linear Differential

Every linear differential equation of the second order may, as is known, be reduced to the form d / dx (1/P du / dx ) = u , . . . . . . (1) where P is any given function of x . On account of the great importance of this equation in mathematical physics (vibrations of a non-uniform stretched cord, of a hanging chain, water in a canal of non-uniform breadth and depth, of air in a pipe of non-uniform sectional area, conduction of heat along a bar of non-uniform fiction or non-uniform conductivity, Laplace’s differential equation of the tides, &c. &c.), I have long endeavoured to obtain a means of faciliiting its practical solution.

scholarly journals On integrals associated with the free particle wave packet

2020 ◽  
Vol 26 (2) ◽  
pp. 257-262
Author(s):  
Alexander E. Patkowski
Keyword(s):  
Wave Packet ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Schrodinger Equation ◽  
Free Particle ◽  
Time Dependent ◽  
Half Integer ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Made In

AbstractWe discuss some properties of integrals associated with the free particle wave packet, {\psi(x,t)}, which are solutions to the time-dependent Schrödinger equation for a free particle. Some noteworthy discussion is made in relation to integrals which have appeared in the literature. We also obtain formulas for half-integer arguments of the Riemann zeta function.

scholarly journals The MHD Newtonian hybrid nanofluid flow and mass transfer analysis due to super-linear stretching sheet embedded in porous medium

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
U. S. Mahabaleshwar ◽  
T. Anusha ◽  
M. Hatami
Keyword(s):  
Differential Equation ◽  
Mass Transfer ◽  
Differential Equations ◽  
Chemical Reaction ◽  
Concentration Field ◽  
Variable Coefficients ◽  
Hybrid Nanofluid ◽  
Nanofluid Flow ◽  
Linear Ordinary Differential Equations ◽  
Non Linear

AbstractThe steady magnetohydrodynamics (MHD) incompressible hybrid nanofluid flow and mass transfer due to porous stretching surface with quadratic velocity is investigated in the presence of mass transpiration and chemical reaction. The basic laminar boundary layer equations for momentum and mass transfer, which are non-linear partial differential equations, are converted into non-linear ordinary differential equations by means of similarity transformation. The mass equation in the presence of chemical reaction is a differential equation with variable coefficients, which is transformed to a confluent hypergeometric differential equation. The mass transfer is analyzed for two different boundary conditions of concentration field that are prescribed surface concentration (PSC) and prescribed mass flux (PMF). The asymptotic solution of concentration filed for large Schmidt number is analyzed using Wentzel-Kramer-Brillouin (WKB) method. The parameters influence the flow are suction/injection, superlinear stretching parameter, porosity, magnetic parameter, hybrid nanofluid terms, Brinkman ratio and the effect of these are analysed using graphs.

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