Riemann-Liouville Definition Research Articles

EFFECT OF FRACTIONAL DERIVATIVE PROPERTIES ON THE PERIODIC SOLUTION OF THE NONLINEAR OSCILLATIONS

A periodic solution of the time-fractional nonlinear oscillator is derived based on the Riemann–Liouville definition of the fractional derivative. In this approach, the particular integral to the fractional perturbed equation is found out. An enhanced perturbation method is developed to study the forced nonlinear Duffing oscillator. The modified homotopy equation with two expanded parameters and an additional auxiliary parameter is applied in this proposal. The basic idea of the enhanced method is to apply the annihilator operator to construct a simplified equation freeness of the periodic force. This method makes the solution process for the forced problem much simpler. The resulting equation is valid for studying all types of possible resonance states. The outcome shows that this alteration method overcomes all shortcomings of the perturbation method and leads to obtain a periodic solution.

An iteration algorithm for the time-independent fractional Schrödinger equation with Coulomb potential

A numerical formula is derived which gives solutions of the fractional Schrodinger equation in time-independent form in the case of Coulomb potential using Riemann–Liouville definition of the fractional derivative and the quadrature methods. The formula is applied for electron in the nucleus field for multiple values of fractional parameter of the space-dependent fractional Schrodinger equation and for each value of the space-dependent fractional parameter, multiple values of energies are applied. Distances are found at which the probability takes its maximum value. Values of energy obtained in this study corresponding to the maximum value of probability are compared with the energy values resulted from the fractional Bohr’s atom formula in the fractional quantum mechanics.

Numerical Simulation of the Fractional Dispersion Advection Equations Based on the Lattice Boltzmann Model

The fractional dispersion advection equations (FDAEs) have recently attracted considerable attention due to their extensive application in the fields of science and engineering. For example, it has been shown that the anomalous solute transport behaviour that exists in hydrology can be well explained by introducing FDAEs. Therefore, the study of FDAEs has profound significance for understanding real transport phenomena in nature. Nevertheless, the existing algorithms for the FDAEs are generally intricate and costly. Therefore, exploiting an efficient solution technique has been a concern for scientists. In an effort to overcome this challenge, a promising lattice Boltzmann (LB) model for the FDAEs is presented in this paper. The Riemann–Liouville definition and the Grünwald–Letnikov definition are introduced for the time derivatives. In addition, Chapman–Enskog analysis is applied to recover the FDAEs. To test the validity of the model, three numerical examples are carried out. In addition, a comparative study of the proposed model and the classical implicit finite difference scheme is also conducted. The numerical results show that the model is suitable for simulating FDAEs.

Optimization for Software Implementation of Fractional Calculus Numerical Methods in an Embedded System.

In this article, some practical software optimization methods for implementations of fractional order backward difference, sum, and differintegral operator based on Grünwald–Letnikov definition are presented. These numerical algorithms are of great interest in the context of the evaluation of fractional-order differential equations in embedded systems, due to their more convenient form compared to Caputo and Riemann–Liouville definitions or Laplace transforms, based on the discrete convolution operation. A well-known difficulty relates to the non-locality of the operator, implying continually increasing numbers of processed samples, which may reach the limits of available memory or lead to exceeding the desired computation time. In the study presented here, several promising software optimization techniques were analyzed and tested in the evaluation of the variable fractional-order backward difference and derivative on two different Arm® Cortex®-M architectures. Reductions in computation times of up to 75% and 87% were achieved compared to the initial implementation, depending on the type of Arm® core.

An algorithm for fractional Schrödinger equation in case of Morse potential

Based on methods of numerical integration and Riemann–Liouville definition of the fractional derivatives, we find a numerical algorithm to find solutions of the time independent fractional Schrödinger equation for Morse potential or the quantum oscillator potential in one dimension, and the iteration formula is applied for multiple values of the fractional parameter of the space dependent fractional Schrödinger equation and multiple values of energy. We define and use the dimensionless form of the space dependent fractional Schrödinger equation of Morse potential. We employ the iteration formula of the time independent fractional Schrödinger equation of Morse potential to find the wave functions in the case of hydrogen chloride and hydrogen fluoride molecules for a certain value of the fractional parameter of the space dependent fractional Schrödinger equation and for many values of the dimensionless energy of each molecule.

New Analytical Solutions of Conformable Time Fractional Bad and Good Modified Boussinesq Equations

Abstract The main purpose of this article is to obtain the new solutions of fractional bad and good modified Boussinesq equations with the aid of auxiliary equation method, which can be considered as a model of shallow water waves. By using the conformable wave transform and chain rule, nonlinear fractional partial differential equations are converted into nonlinear ordinary differential equations. This is an important impact because both Caputo definition and Riemann–Liouville definition do not satisfy the chain rule. By using conformable fractional derivatives, reliable solutions can be achieved for conformable fractional partial differential equations.

Generation and solutions to the time-space fractional coupled Navier-Stokes equations

In this paper, a Lagrangian of the coupled Navier-Stokes equations is proposed based on the semi-inverse method. The fractional derivatives in the sense of Riemann-Liouville definition are used to replace the classical derivatives in the Lagrangian. Then the fractional Euler-Lagrange equation can be derived with the help of the fractional variational principles. The Agrawal?s method is devot?ed to lead to the time-space fractional coupled Navier-Stokes equations from the above Euler-Lagrange equation. The solution of the time-space fractional coupled Navier-Stokes equations is obtained by means of RPS algorithm. The numerical results are presented by using exact solutions.

Fractional-Order System Identification and Equalization in Massive MIMO Systems

In this paper, a new methodology for the identification and equalization of massive Multiple-Input Multiple-Output (MIMO) channels in fifth generation (5G) wireless communications is proposed. Channel equalization is performed in state-space, having estimated the channel using the fractional-order Multivariable Output Error State Space (MOESP) system identification algorithm. The proposed fractional-order algorithm is an improvement over its integer-order counterpart that is currently used for state-space channel identification/estimation and state-space channel equalization. When dealing with fractional-order calculus our work adopts the Riemann-Liouville definition. Our numerical results of channel identification having used a chirp signal to excite our massive MIMO system show a lower mean square error (MSE) in channel estimation using the proposed fractional-order MOESP identification algorithm when compared to integer-order MOESP identification algorithm of increased order. Furthermore, following equalization, transmission using Binary phase shift keying (BPSK), Quadrature phase shift keying (QPSK) and 256-Quadrature amplitude modulation (256-QAM) signals show that the symbol error rate (SER) as a function of signal to noise ratio (SNR) performance of the fractional-order equalization algorithm compares to that of the integer-order equalization algorithm. The proposed channel identification algorithm also provides a more parsimonious solution for modelling multipath fading, thus enabling the design of fractional-order equalizers. In addition, they may be used in other applications where high order filtering and more complex control algorithms which are difficult to tune would be needed. Finally, the work has other technological applications where there is a requirement for modelling and control of propagation processes in dispersive media.

Description of selected pneumatic elements and systems using fractional calculus

The paper presents the application of fractional calculus to describe the dynamics of selected pneumatic elements and systems. In the construction of mathematical models of the analysed dynamic systems, the Riemann-Liouville definition of differintegral of non- integer order was used. For the analysed model, transfer function of integer and non-integer order was determined. Functions describing characteristics in frequency domains were determined, whereas the characteristics of the elements and systems were obtained by means of computer simulation. MATLAB programme were used for the simulation research.

English

We introduce fractional-order Bessel functions (FBFs) to obtain an approximate solution for various kinds of differential equations. Our main aim is to consider the new functions based on Bessel polynomials to the fractional calculus. To calculate derivatives and integrals, we use Caputo fractional derivatives and Riemann-Liouville fractional integral definitions. Then, operational matrices of fractional-order derivatives and integration for FBFs are derived. Also, we discuss an error estimate between the computed approximations and the exact solution and apply it in some examples. Applications are given to three model problems to demonstrate the effectiveness of the proposed method.

New Wave Solutions of Time-Fractional Coupled Boussinesq–Whitham–Broer–Kaup Equation as A Model of Water Waves

The main purpose of this paper is to obtain the wave solutions of conformable time fractional Boussinesq–Whitham–Broer–Kaup equation arising as a model of shallow water waves. For this aim, the authors employed auxiliary equation method which is based on a nonlinear ordinary differential equation. By using conformable wave transform and chain rule, a nonlinear fractional partial differential equation is converted to a nonlinear ordinary differential equation. This is a significant impact because neither Caputo definition nor Riemann–Liouville definition satisfies the chain rule. While the exact solutions of the fractional partial derivatives cannot be obtained due to the existing drawbacks of Caputo or Riemann–Liouville definitions, the reliable solutions can be achieved for the equations defined by conformable fractional derivatives.

The General Kelvin Model and Poynting Model Based on the General Fractional Calculus

In this paper, the Riemann-Liouville fractional-order derivative definition without nonsingular power-law kernel is used as mathematical tool to describe the rheology models. Two fractional order coupling models which are the general Kelvin-Voigt and Poynting-Thomson are gradually discussed through Laplace transform method and Mittag-Leffler function. Meanwhile, the creep compliances and relaxation modulus via the Riemann-Liouville general fractional order derivative are also given. The models via the classical calculus could be regarded as a special situation compared with the two improved models proposed in this paper.

Analysis and Modeling of Fractional-Order Buck Converter Based on Riemann-Liouville Derivative

In previous studies, researchers used the fractional definition of Caputo to study fractional-order power converter. However, it is found that the model based on Caputo fractional definition is inconsistent with the actual situation. The fractional definition is crucial for the application of fractional-order theory into electric engineering, which directly influences the accuracy of model and operational characteristics of power converter. This paper analyzes fractional-order Buck converter according to the fractional definition of Riemann-Liouville. Firstly, the converter model operating in continuous conduction mode (CCM) is established under Riemann-Liouville fractional definition, and it shows that voltage gain and inductor current are relative with the order, but they are independent of order in the model of Caputo fractional definition. Secondly, the different steady-state characteristics of CCM Buck converter are researched under the models of Caputo and Riemann-Liouville derivative. Finally, the feasible simulations and experiments verify that the model derived by Riemann-Liouville fractional definition is more accurate and suitable for analyzing Buck converter.

New idea of Atangana and Baleanu fractional derivatives to human blood flow in nanofluids.

Applications of fractional derivatives are rare for blood flow problems, more exactly in nanofluids. The old definitions published in the literature for fractional derivatives, such as Riemann-Liouville definition, are rarely used by the researchers now; instead, they like to use the new definition introduced by Atangana and Baleanu quite recently. Therefore, in this article, a new idea of Atangana and Baleanu for fractional derivatives possessing a non-local and non-singular kernel has been applied to blood of nanofluids. Blood is considered as a base fluid, and single-wall carbon nanotubes are suspended in blood as nanoparticles in order to make a nanofluid. The magnetic effect with Lorentz force is also taken. The modelled problem is first written in the dimensionless form and later on solved by using an integral transform of Laplace. The effects of embedded parameters are shown in various plots on blood flow and temperature. The heart transfer rate is computed numerically in a tabular form. The results showed that Atangana and Baleanu fractional parameter slow down the blood motion, whereas increasing nanoparticles' volume fraction causes a significant increase in the heat transfer rate.

A study on discrete and discrete fractional pharmacokinetics-pharmacodynamics models for tumor growth and anti-cancer effects

Abstract We study the discrete and discrete fractional representation of a pharmacokinetics - pharmacodynamics (PK-PD) model describing tumor growth and anti-cancer effects in continuous time considering a time scale h ℕ 0 h $h\mathbb{N}_0^h$ , where h > 0. Since the measurements of the tumor volume in mice were taken daily, we consider h = 1 and obtain the model in discrete time (i.e. daily). We then continue with fractionalizing the discrete nabla operator to obtain the model as a system of nabla fractional difference equations. The nabla fractional difference operator is considered in the sense of Riemann-Liouville definition of the fractional derivative. In order to solve the fractional discrete system analytically we state and prove some theorems in the theory of discrete fractional calculus. For the data fitting purpose, we use a new developed method which is known as an improved version of the partial sum method to estimate the parameters for discrete and discrete fractional models. Sensitivity analysis is conducted to incorporate uncertainty/noise into the model. We employ both frequentist approach and Bayesian method to construct 90 percent confidence intervals for the parameters. Lastly, for the purpose of practicality, we test the discrete models for their efficiency and illustrate their current limitations for application.

Fitting of experimental data using a fractional Kalman-like observer

In this paper, a fractional order Kalman filter (FOKF) is presented, this is based on a system expressed by fractional differential equations according to the Riemann–Liouville definition. In order to get the best fitting of the FOKF, the cuckoo search optimization algorithm (CS) was used. The purpose of using the CS algorithm is to optimize the order of the observer, the fractional Riccati equation and the FOKF tuning parameters. The Grünwald–Letnikov approximation was used to compute the numerical solution of the FOKF. To show the effectiveness of the proposed FOKF, four examples are presented, the brain activity, the cutaneous potential recordings of a pregnant woman, the earthquake acceleration, and the Chua’s circuit response.

Research on Time-Space Fractional Model for Gravity Waves in Baroclinic Atmosphere

The research of gravity solitary waves movement is of great significance to the study of ocean and atmosphere. Baroclinic atmosphere is a complex atmosphere, and it is closer to the real atmosphere. Thus, the study of gravity waves in complex atmosphere motion is becoming increasingly essential. Deriving fractional partial differential equation models to describe various waves in the atmosphere and ocean can open up a new window for us to understand the fluid movement more deeply. Generally, the time fractional equations are obtained to reflect the nonlinear waves and few space-time fractional equations are involved. In this paper, using multiscale analysis and perturbation method, from the basic dynamic multivariable equations under the baroclinic atmosphere, the integer order mKdV equation is derived to describe the gravity solitary waves which occur in the baroclinic atmosphere. Next, employing the semi-inverse and variational method, we get a new model under the Riemann-Liouville derivative definition, i.e., space-time fractional mKdV (STFmKdV) equation. Furthermore, the symmetry analysis and the nonlinear self-adjointness of STFmKdV equation are carried out and the conservation laws are analyzed. Finally, adopting the exp(-Φ(ξ)) method, we obtain five different solutions of STFmKdV equation by considering the different cases of the parameters (η,σ). Particularly, we study the formation and evolution of gravity solitary waves by considering the fractional derivatives of nonlinear terms.

A new definition of fractional derivative

In this paper, a new fractional derivative of the Caputo type is proposed and some basic properties are studied. The form of the definition shows that the new derivative is the natural extension of the Caputo one, and that it yields the Caputo derivative with designated memory length. By adaptively changing the memory length, the new definition is capable of capturing local memory effect in a distinct way, which is critical in modelling complex systems where the short memory properties has to be considered. Another attractive property of the new derivative is that it is naturally associated with the Riemann–Liouville definition and as a result, the well established Grünwald–Letnikov approach for numerically solving the fractional differential equation can be readily embedded to approximate the solution of differential equation that involves the new derivatives. Numerical simulations demonstrate the changeable memory effect of the new definition.

Single image fog removal algorithm in spatial domain using fractional order anisotropic diffusion

This paper presents a novel image defogging algorithm using fractional-order anisotropic diffusion equation. The proposed algorithm uses the airlight map extracted from the foggy model as the initial image in the anisotropic diffusion process. The iterative diffusion process improves this airlight map. The anisotropic diffusion process is generalized to the order of any real number between [1, 2) using the Riemann-Liouville definition of the fractional order derivatives. The formulation of the iterative process is carried out in the spatial domain to have a simple and computationally efficient implementation. Simulation results validate that the proposed algorithm is outperforming over few of the existing algorithms. The comparison study is carried out using different metrics like contrast gain, colorfulness index, contrast-to-noise ratio and visible edges ratio.

A novel approach of fractional-order time delay system modeling based on Haar wavelet

In this paper, fractional-order time delay system modeling is presented using Haar wavelet operational matrix of integration. Therefore, it does not require any prior knowledge of transfer function structure or partial information about fractional differentiation order. It allows the estimation of the implicit time delay parameter together with other model parameters by utilizing new delay operational matrix of Haar wavelet based on Riemann-Liouville definition. The proposed technique reduces the complexity of identification by converting the complex fractional calculus equations into simple algebra. The efficacy of the approach is verified on various integer and non-integer (fractional) order systems in simulation. For realistic condition, proposed method is verified in the presence of noise in simulation and also demonstrated on the real-time process control temperature system.