Basics of fractional calculus and fractional order differential equations

Traditional continuous-time filters are of integer order which the power loss of passive power filter is general very much. However, using fractional calculus, filters may also be represented by the more general fractional-order differential equations. In this work, firstly, the passive elements were described with fractional-order differential equations depending on the introduction of fractional calculus application research. Secondly, the mathematical model of fractional-order filters was derived and discussed which includes high impedance at a certain frequency and low impedance at others, and the integer-order filters are only a tight subset of fractional-order filters that are testified. At last, the filter design idea to the fractional-order domain is developed and the better filter performance of the fractional-order passive power filter is validated by the mathematical model analysis and simulation results.

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Legendre Seudospectral Method For The Approximate Solutions Some Of The Fractional-Order Differential Equations

i-manager’s Journal on Mathematics ◽

10.26634/jmat.2.2.2314 ◽

2013 ◽

Vol 2 (2) ◽

pp. 22-27

Author(s):

Zeliha SARIATES K�RPINAR ◽

M�nevver TUZ

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Approximate Solutions ◽

Fractional Order Differential Equations

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A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2019-0281 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Khalid K. Ali ◽

Mohamed A. Abd El salam ◽

Emad M. H. Mohamed

Keyword(s):

Differential Equations ◽

Collocation Method ◽

Fractional Order ◽

Linear Functional ◽

Numerical Technique ◽

Operational Matrix ◽

Fractional Order Differential Equations ◽

The Given ◽

Functional Argument ◽

Special Case

AbstractIn this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient.

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Extension of natural transform method with Daftardar-Jafari polynomials for fractional order differential equations

Alexandria Engineering Journal ◽

10.1016/j.aej.2021.01.051 ◽

2021 ◽

Vol 60 (3) ◽

pp. 3205-3217

Author(s):

Rashid Nawaz ◽

Nasir Ali ◽

Laiq Zada ◽

Kottakkkaran Sooppy Nisar ◽

M.R. Alharthi ◽

...

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Transform Method ◽

Fractional Order Differential Equations

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Analysis of Homotopy Perturbation Method for Solving Fractional Order Differential Equations

Mathematics ◽

10.3390/math7010040 ◽

2019 ◽

Vol 7 (1) ◽

pp. 40 ◽

Cited By ~ 15

Author(s):

Shumaila Javeed ◽

Dumitru Baleanu ◽

Asif Waheed ◽

Mansoor Shaukat Khan ◽

Hira Affan

Keyword(s):

Differential Equations ◽

Perturbation Method ◽

Fractional Order ◽

Homotopy Perturbation Method ◽

Fractional Partial Differential Equations ◽

Partial Differential ◽

Fractional Order Differential Equations ◽

Homotopy Perturbation ◽

The Given ◽

Ordinary Derivative

The analysis of Homotopy Perturbation Method (HPM) for the solution of fractional partial differential equations (FPDEs) is presented. A unified convergence theorem is given. In order to validate the theory, the solution of fractional-order Burger-Poisson (FBP) equation is obtained. Furthermore, this work presents the method to find the solution of FPDEs, while the same partial differential equation (PDE) with ordinary derivative i.e., for α = 1 , is not defined in the given domain. Moreover, HPM is applied to a complicated obstacle boundary value problem (BVP) of fractional order.

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