Riemann-Liouville Derivative Research Articles

On inverse problems for uncoupled space-time fractional operators involving time-dependent coefficients

We study the uncoupled space-time fractional operators involving time-dependent coefficients and formulate the corresponding inverse problems. Our goal is to determine the variable coefficients from the exterior partial measurements of the Dirichlet-to-Neumann map. We exploit the integration by parts formula for Riemann-Liouville and Caputo derivatives to derive the Runge approximation property for our space-time fractional operator based on the unique continuation property of the fractional Laplacian. This enables us to extend early unique determination results for space-fractional but time-local operators to the space-time fractional case.

Direct and inverse problems for a parabolic-hyperbolic equation involving Riemann–Liouville derivatives

Direct and inverse problems for a parabolic-hyperbolic equation involving Riemann–Liouville derivatives

CONSTRUCTION OF CLOSED FORM SOLITON SOLUTIONS TO THE SPACE-TIME FRACTIONAL SYMMETRIC REGULARIZED LONG WAVE EQUATION USING TWO RELIABLE METHODS

The employment of nonlinear fractional partial differential equations (NLFPDEs) is not limited to branches of mathematics entirely but also applicable in other science fields such as biology, physics and engineering. This paper derives some solitary wave solutions for the space-time fractional symmetric regularized long wave (SRLW) equation by means of the improved [Formula: see text]-expansion approach and the [Formula: see text]-expansion method. We use the definition of the Jumarie’s modified Riemann–Liouville derivative to handle the fractional derivatives appearing in this equation. Diverse types of soliton solutions are successfully expressed on the form of rational, hyperbolic, trigonometric, and complex functions. We extract kink wave, internal solitary wave, and solitary wave solutions. The performances of the proposed methods are compared with each other. Moreover, we compare the constructed results with some published solutions. The long behaviors of the obtained solutions are plotted in 2D and 3D figures. The resulting outcomes point out that the used techniques promise to empower us to deal with more NLFPDEs arising in mathematical physics.

On two backward problems with Dzherbashian-Nersesian operator

<abstract><p>We investigate the initial-boundary value problems for a fourth-order differential equation within the powerful fractional Dzherbashian-Nersesian operator (FDNO). Boundary conditions considered in this manuscript are of the Samarskii-Ionkin type. The solutions obtained here are based on a series expansion using Riesz basis in a space corresponding to a non-self-adjoint spectral problem. Conditional to some regularity, consistency, alongside orthogonality dependence, the existence and uniqueness of the obtained solutions are exhibited by using Fourier method. Acquired results here are more general than those obtained by making use of conventional fractional operators such as fractional Riemann-Liouville derivative (FRLD), fractional Caputo derivative (FCD) and fractional Hilfer derivative (FHD).</p></abstract>

On two-order fractional boundary value problem with generalized Riemann-Liouville derivative

On two-order fractional boundary value problem with generalized Riemann-Liouville derivative

Total approximate controllability of non-instantaneous impulsive fractional differential systems involving Riemann-Liouville derivatives of order <inline-formula><tex-math id="M1">$ \beta \in (1,2) $</tex-math></inline-formula>

This artifact introduces the concept of total approximate controllability in order to seek approximate controllability of the impulsive systems at break points in addition to the final point. The chosen system of inspection is a nonlinear fractional differential system affected by non-instantaneous impulses. The system is governed by Riemann-Liouville derivatives of higher-order with fixed lower limits. The appropriate integral-type initial conditions are determined differently depending on the impulsive functions. Firstly, mild solution of the concerned system is constructed, followed by a sufficient set of assumptions required to manifest the existence, uniqueness, and controllability results. Next, the definition of total approximate controllability is interposed. Further, the total approximate controllability of the concerned fractional system is established using Riemann-Liouville fractional resolvent, appropriately defined interval-wise Nemytskii operators, and an iterative technique. Lastly, an illustration is put forward to validate the proposed methodology.

Homomorphic filtering for the image enhancement based on fractional-order derivative and genetic algorithm

The main aim of image enhancement is to improve the visual quality or appearance of an image. This article presents an image enhancement method based on Grunwald-Letnikov, Riemann-Liouville fractional-order derivatives and genetic algorithm to boost the homomorphic filtering performance. Homomorphic filtering is used to attenuate the contribution made by the illumination and amplify the reflectance components of an image. This work uses a fractional-order derivative to enhance the mid- and high-frequencies and preserve the low-frequencies. The enhancement of the image depends on the parameters required for the homomorphic filter function and fractional-order value, which are not the same for all types of images. Hence, the genetic algorithm is applied, which automatically determines these parameters by optimizing the fitness function. The capability of the proposed approach is evaluated using performance metrics such as information entropy, average gradient, and contrast improvement index on different sizes of images. An average improvement in information entropy of 6.5%, average gradient of 52%, and contrast improvement index of 75%, respectively, are achieved for standard, medical images and images with low contrast and non-uniform illumination conditions. Also, the proposed method outperforms the existing methods by producing a better visual appearance of the image.

Describing Water Wave Propagation Using the G′G2–Expansion Method

In the present study, our focus is to obtain the different analytical solutions to the space–time fractional Bogoyavlenskii equation in the sense of the Jumaries-modified Riemann–Liouville derivative and to the conformable time–fractional-modified nonlinear Schrödinger equation that describes the fluctuation of sea waves and the propagation of water waves in ocean engineering, respectively. The G′G2–expansion method is applied to investigate the dynamics of solitons in relation to governing models. Moreover, the restriction conditions for the existence of solutions are reported. In addition, we note that the accomplished solutions are useful to the description of wave fluctuation and the wave propagation survey and are also significant for experimental and numerical verification in ocean engineering.

Approximation Solution for Fuzzy Fractional-Order Partial Differential Equations

In this article, the authors study the comparison of the generalization differential transform method (DTM) and fuzzy variational iteration method (VIM) applied to determining the approximate analytic solutions of fuzzy fractional KdV, K(2,2) and mKdV equations. Furthermore, we establish the approximation solution two-and three-dimensional fuzzy time-fractional telegraphic equations via the fuzzy reduced differential transform method (RDTM). Finding an exact or closed-approximation solution to a differential equation is possible via fuzzy RDTM. Finally, we present the fuzzy fractional variational homotopy perturbation iteration method (VHPIM) with a modified Riemann-Liouville derivative to solve the fuzzy fractional diffusion equation (FDE). Using this approach achieves a rapidly convergent sequence that approaches the exact solution of the equation. The proposed methods are investigated based on fuzzy fractional derivatives with some illustrative examples. The results reveal that the schemes are highly effective for obtaining the solutions to fuzzy fractional partial differential equations.

On continuity properties of the improved conformable fractional derivatives

The conformable fractional derivative has been introduced to extend the familiar limit definition of the classical derivative. Despite having many advantages compared to other fractional derivatives such as satisfying nice properties as classical derivative and easy to solve numerically, it also has disadvantages as it gives large error compared to Riemann-Liouville and Caputo fractional derivatives. Modified types of conformable derivatives have been proposed to overcome the shortcoming. The improved conformal fractional derivatives are declared to be better approximations of Riemann-Liouville and Caputo derivatives in terms of physical behavior. In this paper, properties concerning continuity of the improved conformable fractional derivative are investigated. We prove the relation between -differentiable and continuity of a function and corresponding interior extremum theorem. We also prove the properties close to Rolle’s Theorem and Mean Value Theorem for the improved conformable fractional derivatives.

Notes on Convergence Results for Parabolic Equations with Riemann–Liouville Derivatives

Fractional diffusion equations have applications in various fields and in this paper we consider a fractional diffusion equation with a Riemann–Liouville derivative. The main objective is to investigate the convergence of solutions of the problem when the fractional order tends to 1−. Under some suitable conditions on the Cauchy data, we prove the convergence results in a reasonable sense.

Symmetry Analysis, Invariant Solutions, and Conservation Laws of Fractional KdV-Like Equation

In this paper, Lie symmetries of time-fractional KdV-Like equation with Riemann-Liouville derivative are performed. With the aid of infinitesimal symmetries, the vector fields and symmetry reductions of the equation are constructed, respectively; as a result, the invariant solutions are acquired in one case; we show that the KdV-like equation can be reduced to a fractional ordinary differential equation (FODE) which is connected with the Erdélyi-Kober functional derivative; for this kind of reduced form, we use the power series method for extracting the explicit solutions in the form of power series solution. Finally, Ibragimov’s theorem has been employed to construct the conservation laws.

Existence and partial approximate controllability of nonlinear Riemann–Liouville fractional systems of higher order

This article studies the existence and partial approximate controllability of higher order nonlocal semilinear fractional differential equations with Riemann–Liouville derivatives avoiding Lipschitz assumptions of nonlinear operator and nonlocal functions. To derive the existence result, we make approximate systems corresponding to the original system. For this, we construct the mild solutions in terms of fractional resolvent. Then, we prove the partial approximate controllability of the nonlinear system by using the obtained existence result. Finally, we give an example to illustrate the established theory.

Systems of Riemann–Liouville Fractional Differential Equations with ρ-Laplacian Operators and Nonlocal Coupled Boundary Conditions

In this paper, we study the existence of positive solutions for a system of fractional differential equations with ρ-Laplacian operators, Riemann–Liouville derivatives of diverse orders and general nonlinearities which depend on several fractional integrals of differing orders, supplemented with nonlocal coupled boundary conditions containing Riemann–Stieltjes integrals and varied fractional derivatives. The nonlinearities from the system are continuous nonnegative functions and they can be singular in the time variable. We write equivalently this problem as a system of integral equations, and then we associate an operator for which we are looking for its fixed points. The main results are based on the Guo–Krasnosel’skii fixed point theorem of cone expansion and compression of norm type.

Inverse Problem of Determining an Order of the Riemann-Liouville Time-Fractional Derivative

Inverse Problem of Determining an Order of the Riemann-Liouville Time-Fractional Derivative

Fractional derivatives as weighted average of historical values: an application to COVID-19 in Brazil

The memory effect is an interesting tool that can be seen in fractional differential equations. To show this clearly, in this paper we prove that the Caputo derivative of a function 𝑓, as well as the Riemann-Liouville integral and derivative, are proportional to a weighted average of the historical values of 𝑓 or 𝑓 ′. For this, we use the statistical expectation of functions, whose random variable follows a beta distribution. Moreover, through the respective probability density functions, for each operator we specify the weight of the historical values of the function to determine its current value, according to the values of the fractional order of the derivative. Furthermore, to prove the effectiveness of the memory effect to describe real phenomena, we compared a classic model with its fractional version to model COVID-19 in Brazil.

A unified way to solve IVPs and IBVPs for the time-fractional diffusion-wave equation

The time-fractional diffusion-wave equation is revisited, where the time derivative is of order 2 nu and 0 < nu le 1. The behaviour of the equation is ‘diffusion-like’ (respectively, ‘wave-like’) when 0 < nu le frac{1}{2} (respectively, frac{1}{2} < nu le 1). Two types of time-fractional derivatives are considered, namely the Caputo and Riemann-Liouville derivatives. Initial value problems and initial-boundary value problems are studied and handled in a unified way using an embedding method. A two-parameter auxiliary function is introduced and its properties are investigated. The time-fractional diffusion equation is used to generate a new family of probability distributions, and that includes the normal distribution as a particular case.

A simulation expressivity of the quenching phenomenon in a two-sided space-fractional diffusion equation

The aims of this paper are to investigate and propose a numerical approximation for a quenching type diffusion problem associated with a two-sided Riemann-Liouville space-fractional derivative. The approach adopts weighted Grünwald formulas for suitable spatial discretization. An implicit Crank-Nicolson scheme combined with adaptive technology is then implemented for a temporal integration. Monotonicity, positivity preservation and linearized stability are proved under suitable constraints on spatial and temporal discretization parameters. Two specially designed simulation experiments are presented for illustrating and outreaching properties of the numerical method constructed. Connections between the two-sided fractional differential operator and critical values including quenching time, critical length and quenching location are investigated.

Recovering a Space-Dependent Source Term in the Fractional Diffusion Equation with the Riemann–Liouville Derivative

This research determines an unknown source term in the fractional diffusion equation with the Riemann–Liouville derivative. This problem is ill-posed. Conditional stability for the inverse source problem can be given. Further, a fractional Tikhonov regularization method was applied to regularize the inverse source problem. In the theoretical results, we propose a priori and a posteriori regularization parameter choice rules and obtain the convergence estimates.

A Review on Integral Transforms of Fractional Integral and Derivative

In this paper, we present a review of the integral transformations of fractional calculus, namely the Riemann-Liouville integral, the Riemann-Liouville derivative, and the Caputo derivative.