MATHEMATICAL AND STATISTICAL ANALYSIS OF RL AND RC FRACTIONAL-ORDER CIRCUITS

The RL and RC circuits are analyzed in this research paper. The classical model of these circuits is generalized using the modern concept of fractional derivative with Mittag-Leffler function in its kernel. The fractional differential equations are solved for exact solutions using the Laplace transform technique and the inverse transformation. The obtained solutions are plotted and presented in tables to show the effect of resistance, inductance and fractional parameter on current and voltage. Furthermore, the statistical analysis is presented to predict the seasonal of time and other parameters on the current flowing in the circuit. The statistical analysis shows that the variation in current is insignificant with respect to time and is more significant with respect to other parameters.

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A fractional differential equation with multi-point strip boundary condition involving the Caputo fractional derivative and its Hyers–Ulam stability

Boundary Value Problems ◽

10.1186/s13661-021-01549-y ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Mehboob Alam ◽

Akbar Zada ◽

Ioan-Lucian Popa ◽

Alireza Kheiryan ◽

Shahram Rezapour ◽

...

Keyword(s):

Differential Equation ◽

Fractional Derivative ◽

Fractional Differential Equation ◽

Caputo Fractional Derivative ◽

Integral Boundary Conditions ◽

Ulam Stability ◽

Integral Boundary ◽

Laplace Transform Technique ◽

The Laplace Transform ◽

Fractional Differential

AbstractIn this work, we investigate the existence, uniqueness, and stability of fractional differential equation with multi-point integral boundary conditions involving the Caputo fractional derivative. By utilizing the Laplace transform technique, the existence of solution is accomplished. By applying the Bielecki-norm and the classical fixed point theorem, the Ulam stability results of the studied system are presented. An illustrative example is provided at the last part to validate all our obtained theoretical results.

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Analysis of two colliding fractionally damped spherical shells in modelling blunt human head impacts

Open Physics ◽

10.2478/s11534-013-0194-4 ◽

2013 ◽

Vol 11 (6) ◽

Author(s):

Yury Rossikhin ◽

Marina Shitikova

Keyword(s):

Fractional Derivative ◽

Human Head ◽

Contact Spot ◽

Spherical Shells ◽

Spherical Object ◽

Frontal Impacts ◽

Age Related ◽

Laplace Transform Technique ◽

The Laplace Transform ◽

The Impact

AbstractThe collision of two elastic or viscoelastic spherical shells is investigated as a model for the dynamic response of a human head impacted by another head or by some spherical object. Determination of the impact force that is actually being transmitted to bone will require the model for the shock interaction of the impactor and human head. This model is indended to be used in simulating crash scenarios in frontal impacts, and provide an effective tool to estimate the severity of effect on the human head and to estimate brain injury risks. The model developed here suggests that after the moment of impact quasi-longitudinal and quasi-transverse shock waves are generated, which then propagate along the spherical shells. The solution behind the wave fronts is constructed with the help of the theory of discontinuities. It is assumed that the viscoelastic features of the shells are exhibited only in the contact domain, while the remaining parts retain their elastic properties. In this case, the contact spot is assumed to be a plane disk with constant radius, and the viscoelastic features of the shells are described by the fractional derivative standard linear solid model. In the case under consideration, the governing differential equations are solved analytically by the Laplace transform technique. It is shown that the fractional parameter of the fractional derivative model plays very important role, since its variation allows one to take into account the age-related changes in the mechanical properties of bone.

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On applications of Caputo k-fractional derivatives

Advances in Difference Equations ◽

10.1186/s13662-019-2369-9 ◽

2019 ◽

Vol 2019 (1) ◽

Author(s):

Ghulam Farid ◽

Naveed Latif ◽

Matloob Anwar ◽

Ali Imran ◽

Muhammad Ozair ◽

...

Keyword(s):

Laplace Transform ◽

Fractional Derivative ◽

Fractional Derivatives ◽

Fractional Integral ◽

Integral Inequalities ◽

Chebyshev Inequality ◽

Absolutely Continuous ◽

Laplace Transform Technique ◽

The Laplace Transform ◽

Type Variable

Abstract This research explores Caputo k-fractional integral inequalities for functions whose nth order derivatives are absolutely continuous and possess Grüss type variable bounds. Using Chebyshev inequality (Waheed et al. in IEEE Access 7:32137–32145, 2019) for Caputo k-fractional derivatives, several integral inequalities are derived. Further, Laplace transform of Caputo k-fractional derivative is presented and Caputo k-fractional derivative and Riemann–Liouville k-fractional integral of an extended generalized Mittag-Leffler function are calculated. Moreover, using the extended generalized Mittag-Leffler function, Caputo k-fractional differential equations are presented and their solutions are proposed by applying the Laplace transform technique.

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Characterizations of two different fractional operators without singular kernel

Mathematical Modelling of Natural Phenomena ◽

10.1051/mmnp/2018070 ◽

2019 ◽

Vol 14 (3) ◽

pp. 302 ◽

Cited By ~ 37

Author(s):

Mehmet Yavuz

Keyword(s):

Fractional Derivative ◽

Real Life ◽

Nonlinear Problems ◽

The Other ◽

Singular Kernel ◽

Sinc Function ◽

Life Problems ◽

The Laplace Transform ◽

Fractional Differential ◽

Fractional Derivative Operators

In this paper, we analyze the behaviours of two different fractional derivative operators defined in the last decade. One of them is defined with the normalized sinc function (NSF) and the other one is defined with the Mittag-Leffler function (MLF). Both of them have a non-singular kernel. The fractional derivative operator defined with the MLF is developed by Atangana and Baleanu (ABO) in 2016 and the other operator defined with the normalized sinc function (NSFDO) is created by Yang et al. in 2017. These mentioned operators have some advantages to model the real life problems and to solve them. On the other hand, since the Laplace transform (LT) of the ABO can be calculated more easily, it can be preferred to solve linear/nonlinear problems. In this study, we use the perturbation method with coupled the LTs of these operators to analyze their performance in solving some fractional differential equations. Furthermore, by constructing the error analysis, we test the practicability and usefulness of the method.

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On Hyers–Ulam stability of fractional differential equations with Prabhakar derivatives

Analysis ◽

10.1515/anly-2017-0029 ◽

2018 ◽

Vol 38 (1) ◽

pp. 37-46 ◽

Cited By ~ 3

Author(s):

Mohammad Hossein Derakhshan ◽

Alireza Ansari

Keyword(s):

Differential Equations ◽

Laplace Transform ◽

Fractional Derivative ◽

Fractional Differential Equations ◽

Ulam Stability ◽

Nonlinear Fractional Differential Equations ◽

The Laplace Transform ◽

Fractional Differential ◽

Linear And Nonlinear ◽

The Stability

AbstractIn this article, we study the Hyers–Ulam stability of the linear and nonlinear fractional differential equations with the Prabhakar derivative. By using the Laplace transform, we show that the introduced fractional differential equations with the Prabhakar fractional derivative is Hyers–Ulam stable. The results generalize the stability of ordinary and fractional differential equations in the Riemann–Liouville sense.

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On Solutions of the Initial Value Problem for the Three-Term Fractional Differential Equation with Caputo Derivatives

Symmetry ◽

10.3390/sym12081355 ◽

2020 ◽

Vol 12 (8) ◽

pp. 1355

Author(s):

Stanisław Kukla ◽

Urszula Siedlecka

Keyword(s):

Differential Equation ◽

Fractional Differential Equation ◽

Double Series ◽

Pochhammer Symbol ◽

Generalized Hypergeometric Functions ◽

Caputo Derivatives ◽

Laplace Transform Technique ◽

The Laplace Transform ◽

Fractional Differential ◽

Finite Sum

In this paper, two forms of an exact solution and an analytical–numerical solution of the three-term fractional differential equation with the Caputo derivatives are presented. The Prabhakar function and an asymptotic expansion are utilized to present the double series solution. Using properties of the Pochhammer symbol, a solution is obtained in the form of an infinite series of generalized hypergeometric functions. As an alternative for the series solutions of the fractional commensurate equation, a solution received by an analytical–numerical method based on the Laplace transform technique is proposed. This solution is obtained in the form of a finite sum of the Mittag-Leffler type functions. Numerical examples and a discussion are presented.

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Analytic Solution for theRLElectric Circuit Model in Fractional Order

Abstract and Applied Analysis ◽

10.1155/2014/343814 ◽

2014 ◽

Vol 2014 ◽

pp. 1-5 ◽

Cited By ~ 10

Author(s):

P. V. Shah ◽

A. D. Patel ◽

I. A. Salehbhai ◽

A. K. Shukla

Keyword(s):

Differential Equation ◽

Laplace Transform ◽

Fractional Derivative ◽

Fractional Differential Equation ◽

Fractional Order ◽

Analytic Solution ◽

Electrical Circuit ◽

Special Cases ◽

The Laplace Transform ◽

Fractional Differential

This paper provides an analytic solution ofRLelectrical circuit described by a fractional differential equation of the order0<α≤1. We use the Laplace transform of the fractional derivative in the Caputo sense. Some special cases for the different source terms have also been discussed.

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Exact Solution of Fractional Convective Casson Fluid Through an Accelerated Plate

CFD letters ◽

10.37934/cfdl.13.6.1525 ◽

2021 ◽

Vol 13 (6) ◽

pp. 15-25

Author(s):

Muhammad Nazirul Shahrim ◽

Ahmad Qushairi Mohamad ◽

Lim Yeou Jiann ◽

Muhamad Najib Zakaria ◽

Sharidan Shafie ◽

...

Keyword(s):

Fractional Derivative ◽

Newtonian Fluid ◽

Casson Fluid ◽

Electrical Networks ◽

Moving Plate ◽

Transform Method ◽

The Laplace Transform ◽

Physical Phenomena ◽

The Impact ◽

Fractional Parameter

Fractional derivative has perfectly adopted to model few physical phenomena such as viscoelasticity of coiling polymers, traffic construction, fluid dynamics and electrical networks. However, the application of the fractional derivatives for describing the physical characteristics of non-Newtonian fluid over a moving plate is still rare. In the present study, the effect of the Caputo fractional derivative on the Casson fluid flow which is induced by an accelerated plate is analytically analysed. The governing equations are initially transformed into dimensionless expressions by using suitable dimensionless variables. Then the Laplace transform method is utilized to calculate the exact solutions for the fractional governing partial differential equations. The obtained solutions are validated by comparing the results for specific case with the existing solutions in the literature. The impact of fractional parameter, Prandtl number, and time on the velocity and temperature profiles are graphically showed and discussed. The results depict that the temperature and velocity increase with the increment of fractional parameter and time. Interestingly, the velocity decreases at region near the plate but is enhanced at the area far away from the plate when the Casson fluid parameter is increased. This study is essential in understanding the factional non-Newtonian fluid flows which is more realistic in nature.

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Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

Advances in Difference Equations ◽

10.1186/s13662-021-03472-z ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Raheel Kamal ◽

Kamran ◽

Gul Rahmat ◽

Ali Ahmadian ◽

Noreen Izza Arshad ◽

...

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Laplace Transform ◽

Fractional Derivative ◽

Meshless Method ◽

Inverse Laplace Transform ◽

Contour Integral ◽

Partial Differential ◽

One Dimensional ◽

The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

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Nonexistence Results for Higher Order Fractional Differential Inequalities with Nonlinearities Involving Caputo Fractional Derivative

Mathematics ◽

10.3390/math9161866 ◽

2021 ◽

Vol 9 (16) ◽

pp. 1866

Author(s):

Mohamed Jleli ◽

Bessem Samet ◽

Calogero Vetro

Keyword(s):

Fractional Derivative ◽

Caputo Fractional Derivative ◽

A Priori Estimates ◽

Sufficient Conditions ◽

Global Solutions ◽

Higher Order ◽

Differential Inequalities ◽

Structure Of Solutions ◽

Nonlinear Capacity ◽

Fractional Differential

Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function.

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