scholarly journals NUMERICAL ANALYSIS OF NON-LINEAR VIBRATIONS OF A FRACTIONALLY DAMPED CYLINDRICAL SHELL UNDER THE ADDITIVE COMBINATIONAL INTERNAL RESONANCE

2018 ◽  
Vol 14 (4) ◽  
pp. 42-58
Author(s):  
Basem Ajarmah ◽  
Marina V. Shitikova
Keyword(s):  
Cylindrical Shell ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Internal Resonance ◽  
Nonlinear Differential Equations ◽  
Surrounding Medium ◽  
Non Linear ◽  
Generalized Displacements ◽  
Linear Vibrations ◽  
Fractional Differential

Non-linear damped vibrations of a cylindrical shell subjected to the additive type combinational internal resonance are investigated numerically using two different numerical methods. The damping features of the surrounding medium are described by the fractional derivative Kelvin-Voigt model involving the Riemann-Liouville fractional derivatives. Within the first method, the generalized displacements of a coupled set of nonlinear ordinary differential are estimated using numerical solution of nonlinear multi-term fractional differential equations by the procedure based on the reduction of the problem to a system of fractional differential equations. According to the second method, the amplitudes and phases of nonlinear vibrations are estimated from the governing nonlinear differential equations describing amplitude-and-phase modulations for the case of the additive combinational internal resonance. A good agreement in results is declared

scholarly journals Numerical analysis of non-linear vibrations of a fractionally damped cylindrical shell under the conditions of combinational internal resonance

2018 ◽  
Vol 148 ◽  
pp. 03006 ◽  
Author(s):  
Yury A. Rossikhin ◽  
Marina V. Shitikova ◽  
Basem Ajarmah
Keyword(s):  
Cylindrical Shell ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Internal Resonance ◽  
Nonlinear Differential Equations ◽  
Surrounding Medium ◽  
Non Linear ◽  
Fractional Differential ◽  
Comparison Of The Results

Non-linear damped vibrations of a cylindrical shell embedded into a fractional derivative medium are investigated for the case of the combinational internal resonance, resulting in modal interaction, using two different numerical methods with further comparison of the results obtained. The damping properties of the surrounding medium are described by the fractional derivative Kelvin-Voigt model utilizing the Riemann-Liouville fractional derivatives. Within the first method, the generalized displacements of a coupled set of nonlinear ordinary differential equations of the second order are estimated using numerical solution of nonlinear multi-term fractional differential equations by the procedure based on the reduction of the problem to a system of fractional differential equations. According to the second method, the amplitudes and phases of nonlinear vibrations are estimated from the governing nonlinear differential equations describing amplitude-and-phase modulations for the case of the combinational internal resonance. A good agreement in results is declared.

scholarly journals Non-Linear Macroeconomic Models of Growth with Memory

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2078 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Economic Growth ◽  
Differential Equations ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Continuous Time ◽  
Non Linear ◽  
Standard Models ◽  
Fractional Differential ◽  
Derivatives Of ◽  
With Memory

In this article, two well-known standard models with continuous time, which are proposed by two Nobel laureates in economics, Robert M. Solow and Robert E. Lucas, are generalized. The continuous time standard models of economic growth do not account for memory effects. Mathematically, this is due to the fact that these models describe equations with derivatives of integer orders. These derivatives are determined by the properties of the function in an infinitely small neighborhood of the considered time. In this article, we proposed two non-linear models of economic growth with memory, for which equations are derived and solutions of these equations are obtained. In the differential equations of these models, instead of the derivative of integer order, fractional derivatives of non-integer order are used, which allow describing long memory with power-law fading. Exact solutions for these non-linear fractional differential equations are obtained. The purpose of this article is to study the influence of memory effects on the rate of economic growth using the proposed simple models with memory as examples. As the methods of this study, exact solutions of fractional differential equations of the proposed models are used. We prove that the effects of memory can significantly (several times) change the growth rate, when other parameters of the model are unchanged.

scholarly journals Ulam-Hyers Stability for Cauchy Fractional Differential Equation in the Unit Disk

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Rabha W. Ibrahim
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Unit Disk ◽  
Fractional Differential Equations ◽  
Fractional Operators ◽  
Owa Operators ◽  
Non Linear ◽  
Fractional Differential

We prove the Ulam-Hyers stability of Cauchy fractional differential equations in the unit disk for the linear and non-linear cases. The fractional operators are taken in sense of Srivastava-Owa operators.

Numerical and analytical solutions of nonlinear differential equations involving fractional operators with power and Mittag-Leffler kernel

2018 ◽  
Vol 13 (1) ◽  
pp. 13 ◽  
Author(s):  
H. Yépez-Martínez ◽  
J.F. Gómez-Aguilar
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Nonlinear Differential Equations ◽  
Fractional Operators ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Fractional Differential ◽  
Derivative Order

Analytical and numerical simulations of nonlinear fractional differential equations are obtained with the application of the homotopy perturbation transform method and the fractional Adams-Bashforth-Moulton method. Fractional derivatives with non singular Mittag-Leffler function in Liouville-Caputo sense and the fractional derivative of Liouville-Caputo type are considered. Some examples have been presented in order to compare the results obtained, classical behaviors are recovered when the derivative order is 1.

A numerical method for solving fractional differential equations

2019 ◽  
Vol 36 (2) ◽  
pp. 551-568
Author(s):  
Zain ul Abdeen ◽  
Mujeeb ur Rehman
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Fractional Differential Equations ◽  
Quadrature Rule ◽  
Benchmark Problems ◽  
Computational Technique ◽  
Algebraic Equations ◽  
Content Type ◽  
Non Linear ◽  
Fractional Differential

Purpose The purpose of this paper is to present a computational technique based on Newton–Cotes quadrature rule for solving fractional order differential equation. Design/methodology/approach The numerical method reduces initial value problem into a system of algebraic equations. The method presented here is also applicable to non-linear differential equations. To deal with non-linear equations, a recursive sequence of approximations is developed using quasi-linearization technique. Findings The method is tested on several benchmark problems from the literature. Comparison shows the supremacy of proposed method in terms of robust accuracy and swift convergence. Method can work on several similar types of problems. Originality/value It has been demonstrated that many physical systems are modelled more accurately by fractional differential equations rather than classical differential equations. Therefore, it is vital to propose some efficient numerical method. The computational technique presented in this paper is based on Newton–Cotes quadrature rule and quasi-linearization. The key feature of the method is that it works efficiently for non-linear problems.

scholarly journals Study of an implicit type coupled system of fractional differential equations by means of topological degree theory

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Muhammad Sarwar ◽  
Anwar Ali ◽  
Mian Bahadur Zada ◽  
Hijaz Ahmad ◽  
Taher A. Nofal
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Topological Degree ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Degree Theory ◽  
Coupled System ◽  
Sufficient Condition ◽  
Topological Degree Theory ◽  
Fractional Differential

AbstractIn this work, a sufficient condition required for the presence of positive solutions to a coupled system of fractional nonlinear differential equations of implicit type is studied. To study sufficient conditions essential for the existence of unique solution degree theory is used. Two examples are given to illustrate the established results.

scholarly journals Generalized exp-function method for non-linear space-time fractional differential equations

2014 ◽  
Vol 18 (5) ◽  
pp. 1573-1576 ◽  
Author(s):  
Li-Mei Yan ◽  
Feng-Sheng Xu
Keyword(s):  
Differential Equations ◽  
Linear Space ◽  
Fractional Differential Equations ◽  
Function Method ◽  
Fractional Derivatives ◽  
Space Time ◽  
Fractional Partial Differential Equation ◽  
Non Linear ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

A generalized exp-function method is proposed to solve non-linear space-time fractional differential equations. The basic idea of the method is to convert a fractional partial differential equation into an ordinary equation with integer order derivatives by fractional complex transform. To illustrate the effectiveness of the method, space-time fractional asymmetrical Nizhnik-Novikor-Veselov equation is considered. The fractional derivatives in the present paper are in Jumarie?s modified Riemann-Liouville sense.

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