scholarly journals Numerical Solution of the Bagley-Torvik Equation Using the Integer-Order Derivatives Expansion

2018 ◽  
Vol 6 (2) ◽  
Author(s):  
Afrah Sadiq Hasan
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Numerical Solution ◽  
Fractional Order ◽  
Infinite Series ◽  
Exact Analytical Solution ◽  
Second Order ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Integer Order

Numerical solution of the well-known Bagley-Torvik equation is considered. The fractional-order derivative in the equation is converted, approximately, to ordinary-order derivatives up to second order. Approximated Bagley-Torvik equation is obtained using finite number of terms from the infinite series of integer-order derivatives expansion for the Riemann–Liouville fractional derivative. The Bagley-Torvik equation is a second-order differential equation with constant coefficients. The derived equation, by considering only the first three terms from the infinite series to become a second-order ordinary differential equation with variable coefficients, is numerically solved after it is transformed into a system of first-order ordinary differential equations. The approximation of fractional-order derivative and the order of the truncated error are illustrated through some examples. Comparison between our result and exact analytical solution are made by considering an example with known analytical solution to show the preciseness of our proposed approach.

Geometric interpretation of fractional-order derivative

2016 ◽  
Vol 19 (5) ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Differential Geometry ◽  
Fractional Order ◽  
Infinite Series ◽  
Fractional Derivatives ◽  
Geometric Interpretation ◽  
Order Derivative ◽  
Jet Bundles ◽  
Fractional Order Derivative ◽  
Integer Order ◽  
Derivatives Of

AbstractA new geometric interpretation of the Riemann-Liouville and Caputo derivatives of non-integer orders is proposed. The suggested geometric interpretation of the fractional derivatives is based on modern differential geometry and the geometry of jet bundles. We formulate a geometric interpretation of the fractional-order derivatives by using the concept of the infinite jets of functions. For this interpretation, we use a representation of the fractional-order derivatives by infinite series with integer-order derivatives. We demonstrate that the derivatives of non-integer orders connected with infinite jets of special type. The suggested infinite jets are considered as a reconstruction from standard jets with respect to order.

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.

Subharmonic Resonance of Duffing Oscillator With Fractional-Order Derivative

2016 ◽  
Vol 11 (5) ◽  
Author(s):  
Nguyen Van Khang ◽  
Truong Quoc Chien
Keyword(s):  
Analytical Solution ◽  
Fractional Order ◽  
Duffing Oscillator ◽  
Existence Condition ◽  
Lyapunov Theory ◽  
Subharmonic Resonance ◽  
System Stability ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Approximately Analytical Solution

In this paper, the subharmonic resonance of Duffing oscillator with fractional-order derivative is investigated using the averaging method. First, the approximately analytical solution and the amplitude–frequency equation are obtained. The existence condition for subharmonic resonance based on the approximately analytical solution is then presented, and the corresponding stability condition based on Lyapunov theory is also obtained. Finally, a comparison between the fractional-order and the traditional integer-order of Duffing oscillators is made using numerical simulation. The influences of the parameters in fractional-order derivative on the steady-state amplitude, the amplitude–frequency curves, and the system stability are also investigated.

Edge enhancement of magnetic data using fractional-order-derivative filters

Geophysics ◽  
2015 ◽  
Vol 80 (1) ◽  
pp. J7-J17 ◽  
Author(s):  
Muzaffer Özgü Arısoy ◽  
Ünal Dikmen
Keyword(s):  
Fractional Order ◽  
Order Derivative ◽  
Magnetic Data ◽  
Frequency Noise ◽  
Edge Enhancement ◽  
Traditional Use ◽  
Data Set ◽  
Fractional Order Derivative ◽  
Detection Techniques ◽  
Integer Order

Edge enhancement and detection techniques are fundamental operations in magnetic data interpretation. Many techniques for edge enhancement have been developed, some based on profile data and others designed for grid-based data sets. Methods that are traditionally applied to magnetic data, such as total horizontal derivative (THD) and analytic signal (AS), require the computation of integer-order horizontal and vertical derivatives of the magnetic data. However, if the data set contains features with a large variation in amplitude, then the features with small amplitudes may be difficult to outline. In addition, because most edge enhancement and detection filters are derivative-based filters, they also amplify high-frequency noise content in the data. As a result, the accuracy of derivative-based filters is restricted to data of high quality. We suggested the modification of the THD and AS filters by combining the amplitude spectra of fractional-order-derivative filters with ad hoc phase spectra, particularly designed for edge detection in magnetic data. We revealed the capability of the proposed algorithm on synthetic magnetic data and on aeromagnetic data from Turkey. Compared with the traditional use of THD and AS (with integer-order derivatives), we developed the method based on fractional-order derivatives that produced more effective results in terms of suppressing noise and delineating the edges of deep sources.

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