An integer order approximation method based on stability boundary locus for fractional order derivative/integrator operators

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.

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

Geophysics ◽

10.1190/geo2013-0473.1 ◽

2015 ◽

Vol 80 (1) ◽

pp. J7-J17 ◽

Cited By ~ 5

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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Design of fractional order sliding mode controller via non-integer order backstepping by fractional order derivative of Lyapnov function

Proceedings of the 2014 International Conference on Advanced Mechatronic Systems ◽

10.1109/icamechs.2014.6911645 ◽

2014 ◽

Cited By ~ 1

Author(s):

Takahiro Takamatsu ◽

Koushirou Kubo ◽

Hiromitsu Ohmori

Keyword(s):

Fractional Order ◽

Sliding Mode ◽

Order Derivative ◽

Sliding Mode Controller ◽

Fractional Order Derivative ◽

Integer Order

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

Science Journal of University of Zakho ◽

10.25271/2018.6.2.437 ◽

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.

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

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.11.86 ◽

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.

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

Advances in Difference Equations ◽

10.1186/s13662-020-03087-w ◽

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.

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