Riemann Liouville Time-fractional Derivative Research Articles

Lie symmetry analysis of (2+1)-dimensional time fractional Kadomtsev-Petviashvili equation

In this paper, Lie symmetry analysis method is applied to the (2+1)-dimensional time fractional Kadomtsev-Petviashvili (KP) equation with the mixed derivative of Riemann-Liouville time-fractional derivative and integer-order $x$-derivative. We obtained all the Lie symmetries admitted by the KP equation and used them to reduce the (2+1)-dimensional fractional partial differential equation with Riemann-Liouville fractional derivative to some (1+1)-dimensional fractional partial differential equations with Erd\'{e}lyi-Kober fractional derivative or Riemann-Liouville fractional derivative, thereby getting some exact solutions of the reduced equations. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equation studied.

Invariant analysis, invariant subspace method and conservation laws of the (2+1)-dimensional mixed fractional Broer–Kaup–Kupershmidt system

In this paper, we investigate the (2+1)-dimensional mixed fractional Broer–Kaup–Kupershmidt system (MFBKKS) with Riemann–Liouville time fractional derivative and integer order y-derivative. This system models dispersive and nonlinear long gravity waves propagating in shallow water in two horizontal directions with time memories. Lie symmetry analysis and invariant subspace method are distinctly employed to construct the exact solutions to the MFBKKS. Initially, the Lie algebras admitted by MFBKKS are obtained with the help of Lie symmetry analysis. Then, we establish the commutative table, adjoint relations and adjoint transformation matrix. Specifically, the one-dimensional optimal system is established correspondingly, and symmetry reduction is performed. In particular, (2+1)-dimensional MFBKKS is reduced to (1+1)-dimensional mixed fractional system using Erdélyi–Kober fractional differential operator. Further, the power series solution of MFBKKS is constructed via power series method. By applying invariant subspace method, we obtain more exact solutions of MFBKKS. In addition, the conservation laws are derived by new Noether theorem. Finally, the three-dimensional diagrams of some obtained solutions are demonstrated utilizing Matlab for visualization, and some of the calculations were verified using computational packages Maple for symbolic computation.

Analysis and numerical approximation of the fractional-order two-dimensional diffusion-wave equation

Non-local fractional derivatives are generally more effective in mimicking real-world phenomena and offer more precise representations of physical entities, such as the oscillation of earthquakes and the behavior of polymers. This study aims to solve the 2D fractional-order diffusion-wave equation using the Riemann–Liouville time-fractional derivative. The fractional-order diffusion-wave equation is solved using the modified implicit approach based on the Riemann–Liouville integral sense. The theoretical analysis is investigated for the suggested scheme, such as stability, consistency, and convergence, by using Fourier series analysis. The scheme is shown to be unconditionally stable, and the approximate solution is consistent and convergent to the exact result. A numerical example is provided to demonstrate that the technique is more workable and feasible.

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

The Rayleigh–Stokes Problem for a Heated Generalized Second-Grade Fluid with Fractional Derivative: An Implicit Scheme via Riemann–Liouville Integral

The goal of this study is to use the fast algorithm to solve the Rayleigh–Stokes problem for heated generalized second-grade fluid (RSP-HGSGF) with Riemann–Liouville time fractional derivative using the fast algorithm. The modified implicit scheme, which is formulated by the Riemann–Liouville integral formula and applied to the fractional RSP-HGSGF, is proposed. Numerical experiments will be carried out to demonstrate that the scheme is simple to implement, and the results will reveal the best way to implement the suggested technique. The proposed scheme’s stability and convergence will be examined using the Fourier series. The method is stable, and the approximation solution approaches the exact solution. A numerical demonstration will be provided to demonstrate the applicability and viability of the suggested strategy.

Time-Space Fractional Diffusion Problems: Existence, Decay Estimates and Blow-Up of Solutions

The aim of this paper is to study the following time-space fractional diffusion problem ∂tβu+(-Δ)αu+(-Δ)α∂tβu=λf(x,u)+g(x,t)inΩ×R+,u(x,t)=0in(RN\\Ω)×R+,u(x,0)=u0(x)inΩ,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} \\displaystyle \\partial _t^\\beta u+(-\\Delta )^\\alpha u+(-\\Delta )^\\alpha \\partial _t^\\beta u=\\lambda f(x,u) +g(x,t) &{}\ ext{ in } \\Omega \ imes {\\mathbb {R}}^{+},\\\\ u(x,t)=0\\ \\ &{}\ ext{ in } ({\\mathbb {R}}^N{\\setminus }\\Omega )\ imes {\\mathbb {R}}^+,\\\\ u(x,0)=u_0(x)\\ &{}\ ext{ in } \\Omega ,\\\\ \\end{array}\\right. } \\end{aligned}$$\\end{document} where Omega subset {mathbb {R}}^N is a bounded domain with Lipschitz boundary, (-Delta )^{alpha } is the fractional Laplace operator with 0<alpha <1, partial _t^{beta } is the Riemann-Liouville time fractional derivative with 0<beta <1, lambda is a positive parameter, f:Omega times {mathbb {R}}rightarrow {mathbb {R}} is a continuous function, and gin L^2(0,infty ;L^2(Omega )). Under natural assumptions, the global and local existence of solutions are obtained by applying the Galerkin method. Then, by virtue of a differential inequality technique, we give a decay estimate of solutions. Moreover, the blow-up property of solutions is also investigated.

Numerical analysis of variable-order fractional KdV-Burgers-Kuramoto equation

&lt;abstract&gt;&lt;p&gt;In this paper, a fully discrete local discontinuous Galerkin finite element method is proposed to solve the KdV-Burgers-Kuramoto equation with variable-order Riemann-Liouville time fractional derivative. The method proposed in this paper is based on the finite difference method in time and local discontinuous Galerkin method in space. For all $ \epsilon(t)\in (0, 1) $ with variable order, we prove the scheme is unconditional stable and convergent. Finally, numerical examples are provided to verify the theoretical analysis and the order of convergence for the proposed method.&lt;/p&gt;&lt;/abstract&gt;

An Implicit Numerical Approach for 2D Rayleigh Stokes Problem for a Heated Generalized Second Grade Fluid with Fractional Derivative

In this research work, our aim is to use the fast algorithm to solve the Rayleigh–Stokes problem for heated generalized second-grade fluid (RSP-HGSGF) involving Riemann–Liouville time fractional derivative. We suggest the modified implicit scheme formulated in the Riemann–Liouville integral sense and the scheme can be applied to the fractional RSP-HGSGF. Numerical experiments will be conducted, to show that the scheme is stress-free to implement, and the outcomes reveal the ideal execution of the suggested technique. The Fourier series will be used to examine the proposed scheme stability and convergence. The technique is stable, and the approximation solution converges to the exact result. To demonstrate the applicability and viability of the suggested strategy, a numerical demonstration will be provided.

Inverse problems of determining an order of fractional Riemann-Liouville time-fractional derivative for the subdiffusion equation in RN

An initial-boundary value problem for the subdiffusion equation with an elliptic operator A(D) in RN is studied in the article. Existence and uniqueness theorems for the problem under study are proved by the Fourier method. Considering the order of the Riemann-Liouville time-fractional derivative as an unknown parameter, an inverse problem of determining this parameter is investigated. Likewise, the initial-boundary value problem was considered in the case of replacing the operator A(D) with its power Aσ.Then, existence and uniqueness theorems were proved for the solution of the inverse problem of determining the order of the fractional derivative and the power σ.

Invariant analysis and conservation laws of the time-fractional [formula omitted]-family peakon equations

In this paper, we further extend the theories of Lie symmetry group and conservation law to study the time-fractional b-family peakon equations. The main distinction of the equations with the usual time-fractional partial differential equations is the mixed derivative of Riemann-Liouville time-fractional derivative and integer-order x-derivative. Thus we first give a prolongation formula of the infinitesimal generator for the case of mixed derivative, then after finding the Lie symmetries, we use them to transform the equations into fractional and integer-order ordinary differential equations respectively. Some exact solutions and power series solutions are constructed. Finally, a general conservation law formula is given based on the idea of nonlinear self-adjointness and some nontrivial conservation laws of the equations are presented.

Symmetry structure of multi-dimensional time-fractional partial differential equations

In this paper, we concentrate on the Lie symmetry structure of a system of multi-dimensional time-fractional partial differential equations (PDEs). Specifically, we first give an explicit prolongation formula involving Riemann–Liouville time-fractional derivative for the Lie infinitesimal generator in multi-dimensional case, and then show that the infinitesimal generator has an elegant structure. Furthermore, we present two simple conditions to determine the infinitesimal generators where one is a system of linear time-fractional PDEs, the other is a system of integer-order PDEs and plays the dominant role in finding the infinitesimal generators. We study three time-fractional PDEs to illustrate the efficiencies of the results.

Numerical-analytical method for solving boundary value problem for the generalized moisture transport equation

The paper studies qualitatively new equations of moisture transfer, which generalize the Aller and Aller-Lykov equations. The generalization contributes to revealing in the original equations the specific features of the studied massifs, their structure, physical properties, processes occurring in them through the introduction of the notion of the rates of change of the fractal dimension. We have obtained solutions to the constant coefficient difference equations as a system arising when using the method of lines for the equations with a Riemann-Liouville time fractional derivative with boundary conditions of the first kind. A priori estimates are obtained that imply convergence of the obtained solutions to systems of ordinary differential equations with variable fractional coefficients. Numerical tests have been carried out to confirm theoretical results of the study.

Higher-Order Symmetries of a Time-Fractional Anomalous Diffusion Equation

Higher-order symmetries are constructed for a linear anomalous diffusion equation with the Riemann–Liouville time-fractional derivative of order α∈(0,1)∪(1,2). It is proved that the equation in question has infinite sequences of nontrivial higher-order symmetries that are generated by two local recursion operators. It is also shown that some of the obtained higher-order symmetries can be rewritten as fractional-order symmetries, and corresponding fractional-order recursion operators are presented. The proposed approach for finding higher-order symmetries is applicable for a wide class of linear fractional differential equations.

Periodic property of the time-fractional Kundu–Mukherjee–Naskar equation

The Kundu–Mukherjee–Naskar equation (KMN) can model the data transmission in birefringent fibers through molecules or pulses in a few Femto-seconds. A fractional modification with Riemann-Liouville time-fractional derivative is suggested to take into account the discontinuous period, and the wave-train description is used to reduce the equation to a fractional nonlinear Schrödinger equation, which is further converted to a damping Duffing equation. Effective modification of the homotopy perturbation method is suggested to deal with the damping term.

Групповая классификация, инвариантные решения и законы сохранения нелинейного двумерного ортотропного уравнения фильтрации с дробной производной Римана-Лиувилля по времени

Рассматривается нелинейное двумерное ортотропное уравнение фильтрации с дробной производной Римана-Лиувилля по времени. Доказывается, что такое уравнение может допускать группы точечных преобразований только линейно-автономного типа. Решается задача групповой классификации рассматриваемого уравнения по его точечным симметриям относительно коэффициентов пьезопроводности, являющихся функциями квадрата модуля градиента давления. Доказывается, что если порядок дробного дифференцирования меньше единицы, основная допускаемая уравнением группа точечных преобразований является четырехпараметрической и расширяется до пятипараметрической в изотропном случае. Для степенных зависимостей коэффициентов пьезопроводности допускаемая группа становится пятипараметрической в ортотропном случае и шестипараметрической в изотропном случае. Также выделяется специальный вид степенной зависимости коэффициентов, не имеющий аналога в случае классического уравнения фильтрации, при котором происходит дополнительное расширение группы оператором проективного преобразования. Для уравнения с порядком дробного дифференцирования $\alpha \in (1,2)$ размерности всех допускаемых групп оказываются на единицу больше за счет допускаемого оператора, соответствующего сдвигу решения на дополнительное частное решение этого уравнения. На основе проведенной групповой классификации для соответствующих алгебр Ли инфинитезимальных операторов групп точечных преобразований, допускаемых различными видами рассматриваемого нелинейного ортотропного дробно-дифференциального уравнения, выписываются представления инвариантно-групповых решений, соответствующие оптимальным системам двумерных подалгебр. Приводятся примеры уравнений, получающихся в результате симметрийной редукции исходного дробно-дифференциального уравнения, а также некоторые их решения. Доказывается, что рассматриваемое дробно-дифференциальное уравнение фильтрации является нелинейно самосопряженным, что дает возможность строить его законы сохранения. Соответствующие компоненты всех найденных сохраняющихся векторов приводятся в явном виде.

Time-fractional dependence of the shear force in some beam type problems with negative Young modulus

Under the Euler–Lagrange equation, a time-fractional dependence for the shear force in a beam in the absence of a transverse load is analysed. The main assumption is that the beam is composed by a material with a negative Young modulus. The existence of such materials is documented in several recent studies. A strategy adopting analytical calculations involving Laplace transforms concludes that, up to a known constant, the shear force can be computed as the Riemann–Liouville time-fractional derivative of order 32 of the deflection of the beam. This identity presents an interesting connection with the Bagley–Torvik equation.

Approximate symmetry group classification for a nonlinear fractional filtration equation of diffusion-wave type

A one-dimensional nonlinear fractional filtration equation with the Riemann–Liouville time-fractional derivative is proposed for modeling fluid flow through a porous medium. This equation is derived under an assumption that the fluid has a fractional equation of state in which the fluid density depends on the time-fractional derivative of pressure. The obtained equation belongs to the diffusion-wave type of equations. A case when the order of fractional differentiation is close to an integer number is considered, and a small parameter is introduced into the fractional filtration equation under consideration. An expansion of the Riemann–Liouville time-fractional derivative into the series with respect to this small parameter is obtained. Using this expansion, a first-order approximation of the derived fractional filtration equation is performed. Next, the problem of approximate Lie point symmetry group classification for this approximate nonlinear filtration equation with a small parameter is studied. It is shown that approximate symmetry groups admitted by different realizations of the approximate filtration equation have much more dimensions than the corresponding exact Lie point symmetry groups admitted by unperturbed fractional diffusion-wave equations. Obtained classification results permit to construct approximate invariant solutions for the considered nonlinear time-fractional filtration equations.

A compact ADI Crank–Nicolson difference scheme for the two-dimensional time fractional subdiffusion equation

ABSTRACTIn this paper, a compact alternating direction implicit (ADI) Crank–Nicolson difference scheme is proposed and analysed for the solution of two-dimensional time fractional subdiffusion equation. The Riemann–Liouville time fractional derivative is approximated by the weighted and shifted Grünwald difference operator and the spatial derivative is discretized by a fourth-order compact finite difference method. The stability and convergence of the difference scheme are discussed and theoretically proven by using the energy method. Finally, numerical experiments are carried out to show that the numerical results are in good agreement with the theoretical analysis.

Parameters identification for the unsteady helical flows of a generalized Oldroyd-B fluid model based on its numerical solution

In this paper, we mainly consider the problem of parameter identification for the unsteady helical flows of a generalized Oldroyd-B fluid between two infinitely long coaxial circular cylinders. The implicit finite difference method is employed to obtain the numerical solution for the direct problem. By means of the Levenberg–Marquardt method, the numerical inversion for identifying the three parameters of the model is implemented simultaneously, that is, the Riemann–Liouville time-fractional derivative α, relaxation time λ, and retardation time λr. To test the validity of the proposed numerical method, varied initial guesses and the observed data, whether or not they involve random error, have been used to verify the reliability of the estimation. This paper provides an efficient method to obtain the estimated value of the unknown parameters for the generalized non-Newtonian fluids model.

Regional gradient controllability of sub-diffusion processes

By sub-diffusion systems we mean to replace the first order time derivative of normal diffusion system with a Riemann–Liouville time fractional derivative of order α∈(0,1). In this paper, we first introduce a new definition of regional gradient controllability for the sub-diffusion system with the control inputs appearing in the differential equations as distributed inputs, which recovers the usual definition of regional gradient controllability as α→1. The sufficient and necessary conditions on actuators to achieve the gradient controllability in a given subregion of the whole domain are presented. An approach to guarantee the regional gradient controllability of the problems studied within the considered subregion using minimum energy control effort is presented. Several examples are presented in the end to illustrate the effectiveness of our results, where zone actuators, pointwise actuators or filament actuators are respectively considered.