scholarly journals Time Dimensional Consistency Aware Analysis of Voltage Mode and Current Mode Active Fractional Circuits

2019 ◽  
Vol 13 (1) ◽  
pp. 81-93
Author(s):  
Rawid Banchuin ◽  
Roungsan Chaisrichaoren
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Fractional Derivatives ◽  
Current Mode ◽  
Equation Approach ◽  
Time Dimension ◽  
Current Mode Circuits ◽  
Time Component ◽  
Fractional Differential

In this research, the analysis of the active fractional circuits has been performed by using the fractional differential equation approach. Both voltage and current mode circuits have been taken into account.  The fractional time component parameters have been included in the derivative terms within the fractional differential equations. This is because the consistency in time dimension between the fractional derivative and the conventional one which is also related to the physical measurability, is concerned. The fractional derivatives have been interpreted in Caputo sense. The resulting analytical solutions of the time dimensional consistency aware fractional differential equations have been determined. We have found that the dimensional consistency between both sides of the equations of the solutions which cannot be achieved in the previous works, can be obtained. By applying different source terms to the obtained analytical solutions, the response of both voltage and current mode circuits have been determined and the behaviours of the circuits have been analysed. The fractional time constant and pole locations in the F-plane of these circuits have been determined. Their dynamic behaviours, stabilities have been analysed. Moreover, the discussion on circuit realizations with fractional capacitor has also been made.

scholarly journals A generalization of truncated M-fractional derivative and applications to fractional differential equations

2020 ◽  
Vol 5 (1) ◽  
pp. 171-188 ◽  
Author(s):  
Esin İlhan ◽  
İ. Onur Kıymaz
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Integer Order ◽  
Derivative Type ◽  
Fractional Differential

AbstractIn this paper, our aim is to generalize the truncated M-fractional derivative which was recently introduced [Sousa and de Oliveira, A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties, Inter. of Jour. Analy. and Appl., 16 (1), 83–96, 2018]. To do that, we used generalized M-series, which has a more general form than Mittag-Leffler and hypergeometric functions. We called this generalization as truncated ℳ-series fractional derivative. This new derivative generalizes several fractional derivatives and satisfies important properties of the integer-order derivatives. Finally, we obtain the analytical solutions of some ℳ-series fractional differential equations.

scholarly journals Analysis of Fractal Wave Equations by Local Fractional Fourier Series Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yong-Ju Yang ◽  
Dumitru Baleanu ◽  
Xiao-Jun Yang
Keyword(s):  
Fourier Series ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Present Method ◽  
Fractional Derivatives ◽  
Wave Equations ◽  
Series Method ◽  
Fourier Series Method ◽  
Fractional Differential

The fractal wave equations with local fractional derivatives are investigated in this paper. The analytical solutions are obtained by using local fractional Fourier series method. The present method is very efficient and accurate to process a class of local fractional differential equations.

scholarly journals The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Xianzhen Zhang ◽  
Zuohua Liu ◽  
Hui Peng ◽  
Xianmin Zhang ◽  
Shiyong Yang
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
General Solution ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Order System ◽  
Impulsive Systems ◽  
Fractional Differential ◽  
Noninstantaneous Impulses

Based on some recent works about the general solution of fractional differential equations with instantaneous impulses, a Caputo-Hadamard fractional differential equation with noninstantaneous impulses is studied in this paper. An equivalent integral equation with some undetermined constants is obtained for this fractional order system with noninstantaneous impulses, which means that there is general solution for the impulsive systems. Next, an example is given to illustrate the obtained result.

scholarly journals Positive solutions for a system of fractional differential equations with p-Laplacian operator and multi-point boundary conditions

2018 ◽  
Vol 23 (5) ◽  
pp. 771-801 ◽  
Author(s):  
Rodica Luca
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Existence Results ◽  
Laplacian Operator ◽  
Point Boundary ◽  
Existence And Nonexistence ◽  
Fractional Differential

>We investigate the existence and nonexistence of positive solutions for a system of nonlinear Riemann–Liouville fractional differential equations with parameters and p-Laplacian operator subject to multi-point boundary conditions, which contain fractional derivatives. The proof of our main existence results is based on the Guo–Krasnosel'skii fixed-point theorem.

scholarly journals EXISTENCE AND STABILITY RESULTS FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH TWO CAPUTO FRACTIONAL DERIVATIVES

2019 ◽  
pp. 341
Author(s):  
Mohamed Houas ◽  
Mohamed Bezziou
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Contraction Mapping ◽  
Nonlocal Boundary ◽  
Stability Of Solutions ◽  
Caputo Fractional Derivatives ◽  
Existence And Stability ◽  
Fractional Differential ◽  
Stability Results

In this paper, we discuss the existence, uniqueness and stability of solutions for a nonlocal boundary value problem of nonlinear fractional differential equations with two Caputo fractional derivatives. By applying the contraction mapping and O’Regan fixed point theorem, the existence results are obtained. We also derive the Ulam-Hyers stability of solutions. Finally, some examples are given to illustrate our results.

scholarly journals Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Khalid Hattaf
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Direct Method ◽  
Lyapunov Direct Method ◽  
Science And Engineering ◽  
Fractional Differential ◽  
The Stability

This paper aims to study the stability of fractional differential equations involving the new generalized Hattaf fractional derivative which includes the most types of fractional derivatives with nonsingular kernels. The stability analysis is obtained by means of the Lyapunov direct method. First, some fundamental results and lemmas are established in order to achieve the goal of this study. Furthermore, the results related to exponential and Mittag–Leffler stability existing in recent studies are extended and generalized. Finally, illustrative examples are presented to show the applicability of our main results in some areas of science and engineering.

scholarly journals Solving Linear and Nonlinear Fractional Differential Equations Using Spline Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Adel Al-Rabtah ◽  
Shaher Momani ◽  
Mohamed A. Ramadan
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Spline Functions ◽  
Exact Analytical Solutions ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential ◽  
Linear And Nonlinear ◽  
Good Agreement

Suitable spline functions of polynomial form are derived and used to solve linear and nonlinear fractional differential equations. The proposed method is applicable for0<α≤1andα≥1, whereαdenotes the order of the fractional derivative in the Caputo sense. The results obtained are in good agreement with the exact analytical solutions and the numerical results presented elsewhere. Results also show that the technique introduced here is robust and easy to apply.

On the Application of Fractional Derivatives to the Study of Memristor Dynamics

2020 ◽  
pp. 342-363
Author(s):  
Rawid Banchuin
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Phase Portraits ◽  
Fractional Differential

In this chapter, the authors report their work on the application of fractional derivative to the study of the memristor dynamic where the effects of the parasitic fractional elements of the memristor have been studied. The fractional differential equations of the memristor and the memristor-based circuits under the effects of the parasitic fractional elements have been formulated and solved both analytically and numerically. Such effects of the parasitic fractional elements have been studied via the simulations based on the obtained solutions where many interesting results have been proposed in the work. For example, it has been found that the parasitic fractional elements cause both charge and flux decay of the memristor and the impasse point breaking of the phase portraits between flux and charge of the memristor-based circuits similarly to the conventional parasitic elements. The effects of the order and the nonlinearity of the parasitic fractional elements have also been reported.

scholarly journals A multiple-step adaptive pseudospectral method for solving multi-order fractional differential equations

2019 ◽  
Vol 8 (1) ◽  
pp. 702-718
Author(s):  
Mahmoud Mashali-Firouzi ◽  
Mohammad Maleki
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Pseudospectral Method ◽  
Algebraic Equations ◽  
Computational Accuracy ◽  
Step Size ◽  
Collocation Points ◽  
Fractional Differential ◽  
Multiple Step

Abstract The nonlocal nature of the fractional derivative makes the numerical treatment of fractional differential equations expensive in terms of computational accuracy in large domains. This paper presents a new multiple-step adaptive pseudospectral method for solving nonlinear multi-order fractional initial value problems (FIVPs), based on piecewise Legendre–Gauss interpolation. The fractional derivatives are described in the Caputo sense. We derive an adaptive pseudospectral scheme for approximating the fractional derivatives at the shifted Legendre–Gauss collocation points. By choosing a step-size, the original FIVP is replaced with a sequence of FIVPs in subintervals. Then the obtained FIVPs are consecutively reduced to systems of algebraic equations using collocation. Some error estimates are investigated. It is shown that in the present multiple-step pseudospectral method the accuracy of the solution can be improved either by decreasing the step-size or by increasing the number of collocation points within subintervals. The main advantage of the present method is its superior accuracy and suitability for large-domain calculations. Numerical examples are given to demonstrate the validity and high accuracy of the proposed technique.

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