scholarly journals Asymptotics solutions of a singularly perturbed integro-differential fractional order derivative equation with rapidly oscillating coefficients

2021 ◽  
Vol 104 (4) ◽  
pp. 56-67
Author(s):  
M.A. Bobodzhanova ◽  
◽  
B.T. Kalimbetov ◽  
G.M. Bekmakhanbet ◽  
◽  
...  
Keyword(s):  
Fractional Order ◽  
Regularization Method ◽  
Careful Analysis ◽  
Singularly Perturbed ◽  
Order Derivative ◽  
Time Interval ◽  
Resonance Case ◽  
Fractional Order Derivative ◽  
Oscillating Coefficients ◽  
Complex Cases

In this paper, the regularization method of S.A.Lomov is generalized to the singularly perturbed integrodifferential fractional-order derivative equation with rapidly oscillating coefficients. The main goal of the work is to reveal the influence of the oscillating components on the structure of the asymptotics of the solution to this problem. The case of the absence of resonance is considered, i.e. the case when an integer linear combination of a rapidly oscillating inhomogeneity does not coincide with a point in the spectrum of the limiting operator at all points of the considered time interval. The case of coincidence of the frequency of a rapidly oscillating inhomogeneity with a point in the spectrum of the limiting operator is called the resonance case. This case is supposed to be studied in our subsequent works. More complex cases of resonance (for example, point resonance) require more careful analysis and are not considered in this work.

scholarly journals Singularly Perturbed Integro-Differential Equations With Rapidly Oscillating Coefficients and With Rapidly Changing Kernel in the Case of a Multiple Spectrum

2021 ◽  
Vol 20 ◽  
pp. 84-96
Author(s):  
Burkhan Kalimbetov ◽  
Valery Safonov
Keyword(s):  
Integral Operator ◽  
Careful Analysis ◽  
Singularly Perturbed ◽  
Time Interval ◽  
Simple Spectrum ◽  
Integral Term ◽  
System A ◽  
Oscillating Coefficients ◽  
Oscillating Coefficient ◽  
Complex Cases

The paper investigates a system with rapidly oscillating coefficients and with a rapidly decreasing kernel of the integral operator. Previously, only differential problems of this type were studied in which the integral term was absent. The presence of an integral operator significantly affects the development of an algorithm for asymptotic solutions, for the implementation of which it is necessary to take into account essentially singularities generated by the rapidly decreasing spectral value of the kernel of the integral operator. In addition, resonances can arise in the problem under consideration (i.e., the case can be realized when an integer linear combination of the eigenvalues of the rapidly oscillating coefficient coincides with the points of the spectrum of the limiting operator over the entire considered time interval), as well as the case where the eigenvalue of the rapidly oscillating coefficient coincides with the points spectrum of the limiting operator. This case generates a multiple spectrum of the original singularly perturbed integro-differential system. A similar problem was previously considered in the case of a simple spectrum. More complex cases of resonance (for example, point resonance) require more careful analysis and are not considered in this article.

GENERALIZED THEORY OF MAGNETO-THERMO-VISCOELASTIC SPHERICAL CAVITY PROBLEM UNDER FRACTIONAL ORDER DERIVATIVE: STATE SPACE APPROACH

2020 ◽  
Vol 9 (11) ◽  
pp. 9769-9780
Author(s):  
S.G. Khavale ◽  
K.R. Gaikwad
Keyword(s):  
State Space ◽  
Fractional Order ◽  
Order Derivative ◽  
Laplace Inversion ◽  
State Space Approach ◽  
Spherical Region ◽  
Fractional Order Derivative ◽  
Numerical Laplace Inversion ◽  
Mathcad Software ◽  
Space Approach

This paper is dealing the modified Ohm's law with the temperature gradient of generalized theory of magneto-thermo-viscoelastic for a thermally, isotropic and electrically infinite material with a spherical region using fractional order derivative. The general solution obtained from Laplace transform, numerical Laplace inversion and state space approach. The temperature, displacement and stresses are obtained and represented graphically with the help of Mathcad software.

scholarly journals Novel hyperbolic and exponential ansatz methods to the fractional fifth-order Korteweg–de Vries equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Choonkil Park ◽  
R. I. Nuruddeen ◽  
Khalid K. Ali ◽  
Lawal Muhammad ◽  
M. S. Osman ◽  
...  
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Mathematical Physics ◽  
Order Derivative ◽  
Conformable Fractional Derivative ◽  
Fractional Order Derivative ◽  
De Vries ◽  
Fifth Order ◽  
2D And 3D

Abstract This paper aims to investigate the class of fifth-order Korteweg–de Vries equations by devising suitable novel hyperbolic and exponential ansatze. The class under consideration is endowed with a time-fractional order derivative defined in the conformable fractional derivative sense. We realize various solitons and solutions of these equations. The fractional behavior of the solutions is studied comprehensively by using 2D and 3D graphs. The results demonstrate that the methods mentioned here are more effective in solving problems in mathematical physics and other branches of science.

scholarly journals Iterative analysis of non-linear Swift-Hohenberg equations under nonsingular fractional order derivative

2021 ◽  
pp. 104080
Author(s):  
Israr Ahmad ◽  
Thabet Abdeljawad ◽  
Ibrahim Mahariq ◽  
Kamal Shah ◽  
Nabil Mlaiki ◽  
...  
Keyword(s):  
Fractional Order ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Iterative Analysis ◽  
Non Linear

scholarly journals Multiplicity Result of Positive Solutions for Nonlinear Differential Equation of Fractional Order

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Yang Liu ◽  
Zhang Weiguo
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Nonlinear Differential Equation ◽  
Positive Solutions ◽  
Fractional Order ◽  
Point Theorem ◽  
Boundary Value ◽  
Order Derivative ◽  
Fractional Order Derivative

We investigate the existence of multiple positive solutions for a class of boundary value problems of nonlinear differential equation with Caputo’s fractional order derivative. The existence results are obtained by means of the Avery-Peterson fixed point theorem. It should be point out that this is the first time that this fixed point theorem is used to deal with the boundary value problem of differential equations with fractional order derivative.

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