Liouville Fractional Derivative Research Articles

Numeric Fem’s Solution for Space-Time Diffusion Partial Differential Equations with Caputo–Fabrizion and Riemann–Liouville Fractional Order’s Derivatives

Abstract In this paper, we use the finite element method to solve the fractional space-time diffusion equation over finite fields. This equation is obtained from the standard diffusion equation by replacing the first temporal derivative with the new fractional derivative recently introduced by Caputo and Fabrizion and the second spatial derivative with the Riemann–Liouville fractional derivative. The existence and uniqueness of the numerical solution and the result of error estimation are given. Numerical examples are used to support the theoretical results.

Existence and uniqueness of blow-up solution to a fully fractional thermostat model

In this article, we deal with a fully fractional thermostat model in the settings of the Riemann–Liouville fractional derivatives. The equivalences between the fractional differential equations and the corresponding Volterra–Fredholm integral equations are rigorously derived. By choosing an appropriate weighted Banach space of continuous functions, we employ two standard fixed-point theorems, Leray–Schauder alternative and Banach contraction principle, to establish the existence of blow-up solutions — the unbounded mathematical solutions in the operational interval. Furthermore, an implicit numerical scheme based on the right product rectangle rule is presented, which provides the numerical approximation of the obtained solution. Some examples are provided to validate our theoretical findings, along with numerical simulations of the solutions.

Further physical study about solution structures for the fractional equal width equation (EW) in shallow water hydrodynamics

In this article, with the help of Matlab R2021a software, the exact solutions of the EW are successfully examined by Sub-Equation method and the unified F-expansion method. The main novelty of this paper lies in the six aspects: (1) Singular periodic wave U-shaped solition is discovered by the sub equation method which did not happen in previous papers [35–37]. (2) More and more intensive multi-periodic M-shape wave solutions are also discovered by the unified F-expansion method which did not happen in previous papers [32,34–37]. (3) The dynamic wave behaviours about the got results and the effects of fractionality α are illustrated by Figs. 1–7. The fractional parameter α masters the appearance of the solitons and makes the involved alphabetic shaped solitions due to the increase in the fractionality number α in Figs. 1–7. (4) The effect about free parameters c,k can be seen from Figs. 8–11 in this paper. If only the increase of wave velocity parameter c or wave number parameter k, the energy trend of the wave is always positive and intensive. This also never happens in previous literatures [34–37]. (5) The Jumarie’s modified Riemann–Liouville and beta fractional and conformable derivatives are tested for accuracy in Figs. 12 and 13. We seem firstly compare the relations and distinctions among different fractional order derivatives in EW model. (6) In our best cognitive, the planar dynamical system about EW model whose phase portraits are also first studied. Furthermore, phase portraits for the space–time fractional equal width equation are plotted, and the corresponding sensitivity and chaotic behaviors are analyzed. The geometrical representations of the EW provide the corporal information to explain the physically phenomenons.

The Hölder Regularity for Abstract Fractional Differential Equation with Applications to Rayleigh–Stokes Problems

In this paper, we studied the Hölder regularities of solutions to an abstract fractional differential equation, which is regarded as an abstract version of fractional Rayleigh–Stokes problems, rising up to describing a non-Newtonian fluid with a Riemann–Liouville fractional derivative. The purpose of this article was to establish the Hölder regularities of mild solutions, classical solutions, and strict solutions. We introduced an interpolation space in terms of an analytic resolvent to lower the spatial regularity of initial value data. By virtue of the properties of analytic resolvent and the interpolation space, the Hölder regularities were obtained. As applications, the main conclusions were applied to the regularities of fractional Rayleigh–Stokes problems.

Approximate symmetry of time-fractional partial differential equations with a small parameter

In the sense of Riemann–Liouville fractional derivative, an approximate symmetry method of (1+1)-dimensional time-fractional partial differential equations (PDEs) with a small parameter is proposed. By first inserting the power series expansion of dependent variable in the small parameter into the fractional PDEs and then separating the equation with respect to the small parameter to get a coupled system of fractional PDEs, we define the pth-order approximate symmetries by the Lie symmetries of the first p+1 coupled fractional PDEs. After performing symmetry reductions for the coupled system of fractional PDEs, we find that zero-order reduced equation coincides with the fractional ordinary differential equation corresponding to the fractional PDE without containing the small parameter while the solutions of higher-order reduced equation can be obtained by solving linear variable coefficient fractional ordinary differential equations. We first exemplify the whole procedure of computing approximate symmetry by studying a time-fractional KdV equation with a small parameter. Specifically, we obtain the higher-order approximate symmetries and perform approximate symmetry reductions by means of an optimal system of one-dimensional approximate Lie subalgebras. Moreover, we construct an explicit power series solution and show the dynamic behaviors of the truncated power series solutions by the evolutional figures. Then in order to make the proposed approximate symmetry method be further convincing, we study an anomalous diffusion equation with a small parameter and construct an analytical solution in terms of the Mittag-Leffler function.

Existence, uniqueness and Ulam stability results for a mixed-type fractional differential equations with p-Laplacian operator

In this paper, we study a nonlinear fractional p-Laplacian boundary value problem containing both left Riemann–Liouville and right Caputo fractional derivatives with initial and integral conditions. Some new results on the existence and uniqueness of a solution for the model are obtained as well as the Ulam stability of the solutions. Two examples are provided to show the applicability of our results.

Mittag-Leffler-Type Stability of BAM Neural Networks Modeled by the Generalized Proportional Riemann–Liouville Fractional Derivative

The main goal of the paper is to use a generalized proportional Riemann–Liouville fractional derivative (GPRLFD) to model BAM neural networks and to study some stability properties of the equilibrium. Initially, several properties of the GPRLFD are proved, such as the fractional derivative of a squared function. Additionally, some comparison results for GPRLFD are provided. Two types of equilibrium of the BAM model with GPRLFD are defined. In connection with the applied fractional derivative and its singularity at the initial time, the Mittag-Leffler exponential stability in time of the equilibrium is introduced and studied. An example is given, illustrating the meaning of the equilibrium as well as its stability properties.

An alternating direction implicit compact finite difference scheme for the multi-term time-fractional mixed diffusion and diffusion–wave equation

An alternating direction implicit(ADI) compact finite difference scheme for the two-dimensional multi-term time-fractional mixed diffusion and diffusion–wave equation is given in this paper. By using the Riemann–Liouville fractional integral operator on both sides of the original equation, we obtain a time-fractional integro–differential equation with the order of the highest derivative being one. Using the weighted and shifted Grünwald formulas of Riemann–Liouville fractional derivative and fractional integral, and the Crank–Nicolson approximation, a temporal second-order approximation is obtained for the equivalent integrodifferential system. In the spatial direction, a fourth-order compact approximation is employed to give a fully discrete scheme. Using the splitting techniques for higher dimensional problems, we derive the fully discrete ADI Crank–Nicolson scheme. With the positive definiteness of the related time discrete coefficients and spatial operators, the stability and convergence (second-order in time and fourth-order in space) in the discrete L2-norm are proved by using energy method. Two numerical examples are given to demonstrate the efficiency and accuracy of the proposed method.

Similarity Reductions, Power Series Solutions, and Conservation Laws of the Time-Fractional Mikhailov–Novikov–Wang System

The current study presents a comprehensive Lie symmetry analysis for the time-fractional Mikhailov–Novikov–Wang (MNW) system with the Riemann–Liouville fractional derivative. The corresponding simplified equations with the Erdélyi–Kober fractional derivative are constructed by group invariant solutions. Furthermore, we obtain explicit solutions with the help of the power series method and show the dynamical behavior via evolutional figures. Finally, by means of Ibragimov’s new conservation theorem, the conservation laws are derived for the system.

Weighted fractional stochastic integro-differential equation with infinite delay

In this paper, we consider a weighted fractional stochastic integro-differential equation with infinite delay and nonzero initial values involving a Riemann–Liouville fractional derivative of order 1/2<alpha <1. The existence of a mild solution is investigated using fractional calculus, stochastic analysis, and the fixed point theorem. An example is also provided to illustrate the obtained result.

On the study of the positive solutions of a BVP under $$\psi $$-Riemann–Liouville fractional derivative via upper and lower solution method

On the study of the positive solutions of a BVP under $$\psi $$-Riemann–Liouville fractional derivative via upper and lower solution method

A vacuum solution of modified Einstein equations based on fractional calculus

In this work, we construct a modified version of the Einstein field equations for a vacuum and spherically symmetric spacetime in terms of the Riemann–Liouville fractional derivative. The main difference between our approach and other works is that we ensure that both the classical differential equations and the classical solutions are exactly recovered in the limit when the fractional parameter is turned off. We assume that the fractional equations are valid inside and near the horizon radius and match the classical solution at the horizon. Our approach resembles the Herrera–Witten strategy (Adv High Energy Phys 2018:3839103, 2018, https://doi.org/10.1155/2018/3839103, arXiv:1806.07143 [gr-qc]), where the authors constructed an alternative black hole solution by assuming that inside the horizon the spacetime is hyperbolically symmetric and matches the classical spherically symmetric exterior solution at one point at the horizon. We obtain that, depending on the value of the fractional parameter, the solutions can be interpreted as a regular black hole or a gravastar. As a final step, we compute the fractional curvature scalars and show that the solution is regular everywhere inside the horizon.

Real order total variation with applications to the loss functions in learning schemes

Loss functions are an essential part in modern data-driven approaches, such as bi-level training scheme and machine learnings. In this paper, we propose a loss function consisting of a [Formula: see text]-order (an)-isotropic total variation semi-norms [Formula: see text], [Formula: see text], defined via the Riemann–Liouville (RL) fractional derivative. We focus on studying key theoretical properties, such as the lower semi-continuity and compactness with respect to both the function and the order of derivative [Formula: see text], of such loss functions.

Sharp error estimates for spatial-temporal finite difference approximations to fractional sub-diffusion equation without regularity assumption on the exact solution

Finite difference method as a popular numerical method has been widely used to solve fractional diffusion equations. In the general spatial error analyses, an assumption $$u\in C^{4}({\bar{\varOmega }})$$ is needed to preserve $${\mathcal {O}}(h^{2})$$ convergence when using central finite difference scheme to solve fractional sub-diffusion equation with Laplace operator, but this assumption is somewhat strong, where u is the exact solution and h is the mesh size. In this paper, a novel analysis technique is proposed to show that the spatial convergence rate can reach $${\mathcal {O}}(h^{\min (\sigma +\frac{1}{2}-\epsilon ,2)})$$ in both $$l^{2}$$ -norm and $$l^{\infty }$$ -norm in one-dimensional domain when the initial value and source term are both in $${\dot{H}}^{\sigma }(\varOmega )$$ but without any regularity assumption on the exact solution, where $$\sigma \ge 0$$ and $$\epsilon >0$$ being arbitrarily small. After making slight modifications on the scheme, acting on the initial value and source term, the spatial convergence rate can be improved to $${\mathcal {O}}(h^{2})$$ in $$l^{2}$$ -norm and $${\mathcal {O}}(h^{\min (\sigma +\frac{3}{2}-\epsilon ,2)})$$ in $$l^{\infty }$$ -norm. It is worth mentioning that our spatial error analysis is applicable to high dimensional cube domain by using the properties of tensor product. Moreover, two kinds of averaged schemes are provided to approximate the Riemann–Liouville fractional derivative, and $${\mathcal {O}}(\tau ^{2})$$ convergence is obtained for all $$\alpha \in (0,1)$$ . Finally, some numerical experiments verify the effectiveness of the built theory.

Homotopy Analysis Method using Jumarie’s Approach for Multi-Dimensional Nonlinear Schrödinger Equations

In this paper, we suggest a fractional functional for the Homotopy Analysis Method (HAM) to solve the nonlinear fractional order partial differential equations with fractional order initial conditions by using the modified Riemann–Liouville fractional derivative proposed by G. Jumarie. Graphs have been plotted for different values of α , which clearly reflect the reliability of proposed scheme.

Numerical approach for the fractional order cable model with theoretical analyses

This study, considers the fractional order cable model (FCM) in the sense of Riemann–Liouville fractional derivatives (R-LFD). We use a modified implicit finite difference approximation to solve the FCM numerically. The Fourier series approach is used to examine the proposed scheme’s theoretical analysis, including stability and convergence. The scheme is shown to be unconditionally stable, and the approximate solution converges to the exact solution. To demonstrate the application and feasibility of the proposed approach, a numerical example is provided.

Symmetrized fractional total variation for signal and image analysis

We introduce and study a variational model for signal and image analysis based on Riemann–Liouville fractional derivatives. Both the one-dimensional and two-dimensional cases are studied. The model exploits a quadratic fitting data term together with both right and left Riemann–Liouville fractional derivatives as regularizing terms, with the aim of achieving an orientation-independent analysis.

Breakage-Dependent Fractional Plasticity Model for Sands

Particle breakage could significantly influence the shear strength, volumetric deformation, stress–dilatancy relationship, and plastic-flow rule of crushable sands. A breakage-dependent fractional-plasticity (BDFP) model was proposed in this paper to capture the evolution of the stress–strain relationship for crushable sands by using a general plastic-flow rule, which is obtained from the Riemann–Liouville fractional derivative on the modified Cam-Clay yield surface and the incorporation of the breakage-dependent void-ratio–pressure state parameter in the framework of a kinetic breakage-dependent critical state line. The BDFP model can well predict the state- and breakage-dependent strength and deformation of crushable sands. Specifically under higher confining pressures, the prediction for the evolution of the volumetric strain of crushable sands by the BDFP model is much better than that by the model not considering particle breakage.

The General Fractional Integrals and Derivatives on a Finite Interval

The general fractional integrals and derivatives considered so far in the Fractional Calculus literature have been defined for the functions on the real positive semi-axis. The main contribution of this paper is in introducing the general fractional integrals and derivatives of the functions on a finite interval. As in the case of the Riemann–Liouville fractional integrals and derivatives on a finite interval, we define both the left- and the right-sided operators and investigate their interconnections. The main results presented in the paper are the 1st and the 2nd fundamental theorems of Fractional Calculus formulated for the general fractional integrals and derivatives of the functions on a finite interval as well as the formulas for integration by parts that involve the general fractional integrals and derivatives.

A fast Euler–Maruyama method for Riemann–Liouville stochastic fractional nonlinear differential equations

In this paper, based on the sum-of-exponentials approximation, a fast Euler–Maruyama (EM) method is constructed to solve a kind of multi-term Riemann–Liouville stochastic fractional differential equations. Then the strong convergence order min{1−αm,0.5} of the proposed EM method is proved with Riemann–Liouville fractional derivatives’ orders satisfying 0<α1<α2<⋯<αm<1. Finally, two numerical examples are given to support the theoretical results and show the powerful computational performance of the fast EM method.