This paper presents a semi-analytical method for solving fractional differential equations with strong terms like (exp, sin, cos,Â…). An auxiliary parameter is introduced into the well-known Picard’s method and so called controlled Picard’s method. The proposed approach is based on a combination of controlled Picard’s method with Simpson rule. This approach can cover a wider range of integer and fractional orders differential equations due to the extra auxiliary parameter which enhances the convergence and is suitable for higher order differential equations. The proposed approach can be effectively applied to Bratu’s problem in fractional order domain to predict and calculate all branches of problem solutions simultaneously. Also, it is tested on other fractional differential equations like nonlinear fractional order Sine-Gordon equation. The results demonstrate reliability, simplicity and efficiency of the approach developed. © 2017 University of Bahrain
Modified methods for solving two classes of distributed order linear fractional differential equations
This paper introduces two methods for the numerical solution of distributed order linear fractional differential equations. The first method focuses on initial value problems (IVPs) and based on the ?th Caputo fractional definition with the shifted Chebyshev operational matrix of fractional integration. By applying this method, the IVPs are converted into simple linear differential equations which can be easily handled. The other method focuses on boundary value problems (BVPs) based on Picard’s method frame. This method is based on iterative formula contains an auxiliary parameter which provides a simple way to control the convergence region of solution series. Several numerical examples are used to illustrate the accuracy of the proposed methods compared to the existing methods. Also, the response of mechanical system described by such equations is studied. © 2017 Elsevier Inc.
First-order filters generalized to the fractional domain
Traditional continuous-time filters are of integer order. However, using fractional calculus, filters may also be represented by the more general fractional-order differential equations in which case integer-order filters are only a tight subset of fractional-order filters. In this work, we show that low-pass, high-pass, band-pass, and all-pass filters can be realized with circuits incorporating a single fractance device. We derive expressions for the pole frequencies, the quality factor, the right-phase frequencies, and the half-power frequencies. Examples of fractional passive filters supported by numerical and PSpice simulations are given. © 2008 World Scientific Publishing Company.
On the generalization of second-order filters to the fractional-order domain
This work is aimed at generalizing the design of continuous-time second-order filters to the non-integer-order (fractional-order) domain. In particular, we consider here the case where a filter is constructed using two fractional-order capacitors both of the same order ?. A fractional-order capacitor is one whose impedance is Zc = 1/C(j?) ?, C is the capacitance and ? (0 < ? ? 1) is its order. We generalize the design equations for low-pass, high-pass, band-pass, all-pass and notch filters with stability constraints considered. Several practical active filter design examples are then illustrated supported with numerical and PSpice simulations. Further, we show for the first time experimental results using the fractional capacitive probe described in Ref. 1. © 2009 World Scientific Publishing Company.
Fractional-order RC and RL circuits
This paper is a step forward to generalize the fundamentals of the conventional RC and RL circuits in fractional-order sense. The effect of fractional orders is the key factor for extra freedom, more flexibility, and novelty. The conditions for RC and RL circuits to act as pure imaginary impedances are derived, which are unrealizable in the conventional case. In addition, the sensitivity analyses of the magnitude and phase response with respect to all parameters showing the locations of these critical values are discussed. A qualitative revision for the fractional RC and RL circuits in the frequency domain is provided. Numerical and PSpice simulations are included to validate this study. © Springer Science+Business Media, LLC 2012.
Fractional order filter with two fractional elements of dependant orders
This work is aimed at generalizing the design of continuous-time filters in the non-integer-order (fractional-order) domain. In particular, we consider here the case where a filter is constructed using two fractional-order elements of different orders ? and ?. The design equations for the filter are generalized taking into consideration stability constraints. Also, the relations for the critical frequency points like maximum and minimum frequency points, the half power frequency and the right phase frequency are derived. The design technique presented here is related to a fractional order filter with dependent orders ? and ? related by a ratio k. Frequency transformations from the fractional low-pass filter to both fractional high-pass and band-pass filters are discussed. Finally, case studies of KHN active filter design examples are illustrated and supported with numerical and ADS simulations. © 2012 Elsevier Ltd.
Fractional order oscillators based on operational transresistance amplifiers
In this paper, a general analysis of the fractional order operational transresistance amplifiers (OTRA) based oscillator is presented and validated through eight different circuits which represent two classifications according to the number of OTRAs. The general analytical formulas of the oscillation frequency, condition as well as the phase difference are illustrated for each case and summarized in tables. One of the advantages of the fractional-order circuit is the extra degrees of freedom added from the fractional-order parameters. Moreover different special cases {? = ? ? 1, ? ? ? = 1, ? ? ? = 1} are investigated where the conventional case ? = ? = 1 is included in all of them. Also, the effect of the fractional order parameter on the phase difference between the two oscillator outputs is presented which increases the design flexibility and controllability. The effect of the non-ideal characteristics associated with OTRA on the presented oscillator is also studied. A comparison between the fractional order oscillators with their integer order counterpart is also presented to verify the advantages of the added fractional order parameters. Numerical and spice simulations are given to validate the presented analysis. © 2015 Elsevier GmbH.
Fractional Order Sallen–Key and KHN Filters: Stability and Poles Allocation
This paper presents the analysis for allocating the system poles and hence controlling the system stability for KHN and Sallen–Key fractional order filters. The stability analysis and stability contours for two different fractional order transfer functions with two different fractional order elements are presented. The effect of the transfer function parameters on the singularities of the system is demonstrated where the number of poles becomes dependent on the transfer function parameters as well as the fractional orders. Numerical, circuit simulation, and experimental work are used in the design to test the proposed stability contours. © 2014, Springer Science+Business Media New York.
Introduction
This chapter summarizes the basic linear circuit elements (resistor, capacitor, inductor, and fractional-order elements) with their basic fundamentals and characteristic graphs. Each element was defined by a relation between the state variables of the network: current I, voltage V, charge Q, and flux ?. It also investigates the basic fundamentals of the memristor, its historical background, and its advantages over the last few decades. Moreover, the organization of the book is also discussed. © 2015, Springer International Publishing Switzerland.
Fractional Order Oscillator Design Based on Two-Port Network
In this paper, a general analysis of the generation for all possible fractional order oscillators based on two-port network is presented. Three different two-port network classifications are used with three external single impedances, where two are fractional order capacitors and a resistor. Three possible impedance combinations for each classification are investigated, which give nine possible oscillators. The characteristic equation, oscillation frequency and condition for each presented topology are derived in terms of the transmission matrix elements and the fractional order parameters ? and ?. Mapping between some cases is also illustrated based on similarity in the characteristic equation. The use of fractional order elements ? and ? adds extra degrees of freedom, which increases the design flexibility and frequency band, and provides extra constraints on the phase difference. Study of four different active elements, such as voltage-controlled current source, gyrator, op-amp-based network, and second-generation current-conveyor-based network, serve as a two-port network is presented. The general analytical formulas of the oscillation frequency and condition as well as the phase difference between the two oscillatory outputs are derived and summarized in tables for each designed oscillator network. A comparison between fractional order oscillators with their integer order counterparts is also illustrated where some designs cannot work in the integer case. Numerical Spice simulations and experimental results are given to validate the presented analysis. © 2015, Springer Science+Business Media New York.