Riemann-Liouville Fractional Derivative Research Articles

Lie Symmetry Analysis, Power Series Solutions and Conservation Laws of (2+1)-Dimensional Time Fractional Modified Bogoyavlenskii–Schiff Equations

In this paper, Lie symmetry analysis method is applied to the (2+1)-dimensional time fractional modified Bogoyavlenskii–Schiff equations, which is an important model in physics. The one-dimensional optimal system composed by the obtained Lie symmetries is utilized to reduce the system of (2+1)-dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to the system of (1+1)-dimensional fractional partial differential equations with Erdélyi–Kober fractional derivative. Then the power series method is applied to derive explicit power series solutions for the reduced system. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equations studied.

Lie symmetry scheme to the generalized Korteweg–de Vries equation with Riemann–Liouville fractional derivative

The Korteweg–de Vries (KdV) equation is an essential model to characterize shallow water waves in fluid mechanics. Here, we investigated the generalized time and time-space fractional KdV equation with fractional derivative of Riemann–Liouville. At the beginning of, we applied the fractional Lie symmetry scheme to derive their symmetry, respectively. We found that the vector fields of these considered equations decrease as the independent variables fractionalize. Subsequently, the one-parameter Lie transformation groups of these concerned models were yielded. At the same time, they can be reduced into fractional order ordinary differential equations with the Erdélyi–Kober fractional operators. Finally, by obtaining the nonlinear self-adjointness, conservation laws of the generalized time-space fractional KdV equation were also found. These good results provide a basis for us to further understand the phenomenon of shallow water waves.

A note on Kirchhoff type boundary value problem involving Riemann-Liouville fractional derivative

In this paper, we study some nonlinear Kirchhoff boundary value problems of fractional differential equations involving Riemann Liouville operator. Under appropriate assumptions on the functions in the given problem, we establish the existence of solutions using variational methods combined with the mountain pass theorem. Moreover, an illustrative example is presented to prove the validity of the main result.

A New Nonlinear Integral Inequality with a Tempered Ψ–Hilfer Fractional Integral and Its Application to a Class of Tempered Ψ–Caputo Fractional Differential Equations

In this paper, the tempered Ψ–Riemann–Liouville fractional derivative and the tempered Ψ–Caputo fractional derivative of order n−1<α<n∈N are introduced for Cn−1–functions. A nonlinear version of the second Henry–Gronwall inequality for integral inequalities with the tempered Ψ–Hilfer fractional integral is derived. By using this inequality, an existence and uniqueness result and a sufficient condition for the non-existence of blow-up solutions of nonlinear tempered Ψ–Caputo fractional differential equations are proved. Illustrative examples are given.

Comparative study of integer-order and fractional-order artificial neural networks: Application for mathematical function generation

This paper investigates the impact of fractional derivatives on the activation functions of an artificial neural network (ANN). Based on the results and analysis, a three-layer backpropagation neural network model with fractional and integer derivatives employed in the activation function and fractional gradient descent method backpropagation learning algorithm has been proposed. Specifically, three perceptrons have been proposed based on fractional and integer derivatives applied to activation functions and learning algorithms. They are fractional derivative activation function (FDAF) perceptron, integer derivative activation function (IDAF)-fractional derivative learning algorithm (FDLA) perceptron, and fractional derivative learning algorithm (FDLA) perceptron. The Riemann-Liouville (RL) fractional derivative, Grunwald-Letnikov (GL) derivative, Caputo-Fabrizio (CF) fractional derivative, Caputo (C) fractional derivative, and Atangana-Baleanu (AB) fractional derivative have been employed. The impact of these derivatives’ fractional-order (FO) is investigated in a wide range from 0.1−0.9 on testing mean square error (MSE) and time required to train the perceptron. FO-based and IO-based perceptrons are compared with the help of performance metrics such as testing MSE and the time required to train the models. The training and testing simulation results illustrate that Caputo-Fabrizio derivative-based perceptrons outperform other fractional derivatives. Also, the performance of the FO-based perceptron is better in terms of the least MSE.

Analytic Solutions for Hilfer Type Fractional Langevin Equations with Variable Coefficients in a Weighted Space

In this work, analytic solutions of initial value problems for fractional Langevin equations involving Hilfer fractional derivatives and variable coefficients are studied. Firstly, the equivalence of an initial value problem to an integral equation is proved. Secondly, the existence and uniqueness of solutions for the above initial value problem are obtained without a contractive assumption. Finally, a formula of explicit solutions for the proposed problem is derived. By using similar arguments, corresponding conclusions for the case involving Riemann–Liouville fractional derivatives and variable coefficients are obtained. Moreover, the nonlinear case is discussed. Two examples are provided to illustrate theoretical results.

Lie Symmetry Analysis and Conservation Laws of Fractional Benjamin–Ono Equation

In this paper, the fractional Benjamin–Ono differential equation with a Riemann–Liouville fractional derivative is considered using the Lie symmetry analysis method. Two symmetries admitted by the equation are obtained. Then, the equation is reduced to a fractional ordinary differential equation with an Erdélyi–Kober fractional derivative by one of the symmetries. Finally, conservation laws for the equations are constructed using the new conservation theorem.

On the generalized time fractional reaction–diffusion equation: Lie symmetries, exact solutions and conservation laws

In this paper, Lie symmetry analysis method is applied to the generalized time fractional reaction–diffusion equation. We obtain a conditional symmetric group and some conservation laws of the governing equation. The obtained Lie symmetries are used to reduce the studied fractional partial differential equation to some fractional ordinary differential equations with Riemann–Liouville fractional derivative or Erdélyi-Kober fractional derivative. Furthermore, we obtained asymptotic stable solutions and convergent power series solutions for the reduced equations. The dynamic behavior of these exact solutions is presented graphically.

A Solution of Fractal Dynamic Pharmacokinetics Problem

There are numerous uses for fractional differential equations in engineering, physics, and technology. A path for solving the fractal differential equations was examined, and its homogeneous form of was introduced. The notions of Riemann-Liouville fractional derivatives served as the foundation for the journey.. The solutions of the linear non-homogeneous fractal differential equations are given in detail.

An isochronous stress ratio logarithmic strain curve based clay creep model considering the effects of hardening and damage

Creep is one of the typical mechanical properties of clay, and studying the creep mechanical properties of clay is of great significance to construction projects in clay sites. This study conducted creep tests on Chengdu clay and found that the soil mass underwent elastic deformation, decay creep deformation, steady-state creep deformation, and accelerated creep deformation. The isochronous stress ratio-logarithmic strain curves and their mathematical models were proposed to thoroughly analyze clay creep mechanical properties. Creep automatic feature points, such as linear elastic extreme point, initial yield point, long-term strength point, and plastic point, were identified on the curve. Considering the hardening and damage effects during creep loading, linear elastic and viscoelastic elements considering the time-dependent damage, a viscoplastic element considering the load hardening effect, and viscoplastic and plastic elements considering the load damage effect were established based on the element model and the Riemann–Liouville fractional derivative. Based on the mechanical properties of the whole clay creep process, the creep mechanical feature points, and the established element model, a clay creep model was proposed considering the hardening and damage effects. The rationality and regularity of the creep model were verified using the creep test data. This research accurately revealed the creep mechanical properties of clay and facilitated soil deformation prediction, thus providing technical guidance and references for construction projects in clay sites.

Extended Conformable K-Hypergeometric Function and Its Application

The extended conformable k-hypergeometric function finds various applications in physics due to its ability to describe complex mathematical relationships arising in different physical scenarios. Here are a few instances of its uses in physics, including nuclear physics, fluid dynamics, quantum mechanics, and astronomy. The main objectives of this paper are to introduce the extended conformable k-hypergeometric and confluent hypergeometric functions by utilizing the new definition of the α,k-beta function and studying its important properties, like integral representation, summation formula, derivative formula, transform formula, and generating function. Also, introduce the extension of the Riemann–Liouville fractional derivative and establish some results related to the newly defined fractional operator, such as the Mellin transform and relations to extended α,k-hypergeometric functions.

On the solution of the Cauchy problem for a higher-order equation with the fractional Riemann-Liouville derivative

In this article, a representation of the solution of the Cauchy problem for an equation of high even order with a fractional derivative in the Riemann-Liouville sense of order greater than one is obtained. The uniqueness of the solution is proved.

A fractional stochastic SPEIQR epidemic model in switching network for COVID-19

The rapid spread of COVID-19 globally is the cause of panic and billions of dollars in losses and some public policies have been implemented to contain the spread of the virus. It is highly significant to establish mathematical models to describe the spread of the pandemic. In this paper, a fractional stochastic susceptible–protected–exposed–infected–quarantined–recovered (SPEIQR) epidemic model in switching network is proposed to investigate COVID-19 and other diseases in the future, in which the switching connections in the propagation network of viruses based on a Bernoulli random variable are given to represent the policy change during COVID-19. The effects of environmental fluctuations are also considered by introducing the multiplicative noise terms into the disease transmission coefficient to obtain a more realistic mathematical model. Furthermore, the Riemann–Liouville fractional derivative is used to describe the dynamics of disease transmission process which is dependent on the history information of the stochastic system. Some qualitative properties of the model are derived and some sufficient conditions are obtained to ensure the local extinction with probability one of the disease. Finally, we use the epidemic data of three countries (Italy, Germany and Switzerland) and three cities in China (Hubei, Hunan and Henan) from January 22 to July 20 to demonstrate the validity of the derived results and the availability of the proposed models. This work provides some theoretical results to deeply understand the dynamics of COVID-19 and regulate the disease dynamics.

Investigation of the Time Fractional Higher-Dimensional Nonlinear Modified Equation of Wave Propagation

In this article, we analyzed the time fractional higher-dimensional nonlinear modified model of wave propagation, namely the (3 + 1)-dimensional Benjamin–Bona–Mahony-type equation. The fractional sense was defined by the classical Riemann–Liouville fractional derivative. We derived firstly the existence of symmetry of the time fractional higher-dimensional equation. Next, we constructed the one-dimensional optimal system to the time fractional higher-dimensional nonlinear modified model of wave propagation. Subsequently, it was reduced into the lower-dimensional fractional differential equation. Meanwhile, on the basis of the reduced equation, we obtained its similarity solution. Through a series of analyses of the time fractional high-dimensional model and the results of the above obtained, we can gain a further understanding of its essence.

Reachability of time-varying fractional dynamical systems with Riemann-Liouville fractional derivative

Reachability of time-varying fractional dynamical systems with Riemann-Liouville fractional derivative

Khalouta transform and applications to Caputo-fractional differential equations

The paper aims to utilize an integral transform, specifically the Khalouta transform, an abstraction of various integral transforms, to address fractional differential equations using both Riemann-Liouville and Caputo fractional derivative. We discuss some results and the existence of this integral transform. In addition, we also discuss the duality between Shehu transform and Khalouta transform. The numerical examples are provided to confirm the applicability and correctness of the proposed method for solving fractional differential equations.2010 Mathematics ClassificationPrimary 92B05, 92C60; Secondary 26A33.

Invariant analysis, exact solutions, and conservation laws of time fractional thin liquid film equations

This present paper investigates Lie symmetry analysis, one-dimensional optimal system, exact solutions and conservation laws of the (2 + 1)-dimensional time fractional thin liquid film equations (TFTLFE) with Riemann–Liouville fractional derivative. Explicitly, we obtain six vector fields and the one-dimensional optimal system admitted by TFTLFE. Then, we perform the symmetry reductions with the help of Erdélyi–Kober fractional differential operator and (2 + 1)-dimensional TFTLFE is reduced into (1 + 1)-dimensional fractional partial differential equations (FPDEs). Additionally, by means of compound variable transformation and the power series expansion method, the solution of reduced FPDEs is obtained and its convergence is verified. Moreover, we derive other solutions for the reduced equations taking advantage of the invariant subspace method. Furthermore, the conservation laws are also established utilizing generalized Noether's theorem. Finally, we construct the exact solution using the method of conservation laws.

Analysis of the convective heat transfer through straight fin by using the Riemann-Liouville type fractional derivative: Probed by machine learning

This work aims to analyze the transfer of heat through new fractional-order convective straight fins by using the Riemann-Liouville type fractional derivatives. The convection through the fins is considered in such a way that the thermal conductivity depends on the temperature. The transformed fractional-order problems are constituted through an optimization problem in such a way that the L2 norm remains minimal. The objective functions are further analyzed with the hybrid Cuckoo search (HCS) algorithm that use the artificial neural network (ANN) mechanism. The impacts of the fractional parameter β, the thermo-geometric parameter of fin ψ, and dimensionless thermal conductivity α are explained through figures and tables. The fin efficiency during the whole process is explained with larger values of ψ. It is found that the larger values of ψ decline the fin efficacy. The fractional parameter declines the thermal profile as we approach the integer order. The convergence of HCS algorithm is performed in each case study. The residual error touches E−14 for the integer order of α. The present results are validated through Table 6 by comparing with HPM, VIM and LHPM, while the error for HCS-ANN touches E−13. This proves that the proposed HCS is efficient.

Stability on a boundary problem with RL-Fractional derivative in the sense of Atangana-Baleanu of variable-order

<abstract><p>In this paper, we study the existence and stability of solutions in connection to a non-local multiterm boundary value problem (BVP) with differential equations equipped with the Riemann-Liouville (RL) fractional derivative in the sense of Atangana-Baleanu of variable-order. The results about the existence property are investigated and proved via Krasnoselskii's fixed point theorem. Note that all theorems in the present research are studied based on piece-wise constant functions defined on generalized intervals. We shall convert our main BVP with the RL-fractional derivative of the Atangana-Baleanu type of variable-order to an equivalent BVP of constant order of the RL-Atangana-Baleanu derivative. In the next step, we examine the Ulam-Hyers stability for the supposed variable-order RL-Atangana-Baleanu BVP. Finally, we provide some examples to validate that our results are applicable.</p></abstract>

The Rishi Transform method for solving multi-high order fractional differential equations with constant coefficients

<abstract> <p>In this paper, we suggest the Rishi transform, which may be used to find the analytic (exact) solution to multi-high-order linear fractional differential equations, where the Riemann-Liouville and Caputo fractional derivatives are used. We first developed the Rishi transform of foundational mathematical functions for this purpose and then described the important characteristics of the Rishi transform, which may be applied to solve ordinary differential equations and fractional differential equations. Following that, we found an exact solution to a particular example of fractional differential equations. We looked at four numerical problems and solved them all step by step to demonstrate the value of the Rishi transform. The results show that the suggested novel transform, "Rishi Transform, " yields exact solutions to multi-higher-order fractional differential equations without doing complicated calculation work.</p> </abstract>