Riemann-Liouville Derivative Research Articles

The investigation of travelling wave solutions to some coupled space-time fractional differential equations in fluids

Solutions of nonlinear coupled fractional differential equations (FDEs), namely, the space-time fractional coupled Burgers’ equations, coupled MKdV equations, and coupled KdV equations are obtained by the improved fractional sub-equation method in the sense of Jumarie’s modified Riemann-Liouville derivative. The form of these solutions suggests ansatz suitable for finding solutions of much more general FDEs of coupled type.

On the time-fractional coupled Burger equation: Lie symmetry reductions, approximate solutions and conservation laws

In this paper, the time-fractional coupled Burger equation under Riemann-Liouville derivative is systematically analyzed. Firstly, the Lie point symmetry is obtained by applying the Lie symmetry analysis; then by utilizing the above acquired Lie point symmetry, the similarity reductions are obtained. Through similarity reductions, the coupled time-fractional Boussinesq-Burgers equation is reduced to nonlinear fractional ordinary differential equations (FODEs), with Erdélyi-Kober fractional differential operator. Then the power series method and q-homotopy analysis method are used to obtain the approximate solutions of coupled time-fractional Boussinesq-Burgers equation in the sense of the Caputo fractional derivative. Finally, the conservation laws are derived by using Noether theorem.

Cauchy type problems for fractional differential equations

While it is known that one can consider the Cauchy problem for evolution equations with Caputo derivatives, the situation for the initial value problems for the Riemann–Liouville derivatives is less understood. In this paper, we propose new type initial, inner, and inner-boundary value problems for fractional differential equations with the Riemann–Liouville derivatives. The results on the existence and uniqueness are proved, and conditions on the solvability are found. The well-posedness of the new type of initial, inner, and inner-boundary conditions is also discussed. Moreover, we give explicit formulas for the solutions. As an application fractional partial differential equations for general positive operators are studied.

Invariance Analysis, Exact Solution and Conservation Laws of (2 + 1) Dim Fractional Kadomtsev-Petviashvili (KP) System

In this work, a Lie group reduction for a (2 + 1) dimensional fractional Kadomtsev-Petviashvili (KP) system is determined by using the Lie symmetry method with Riemann Liouville derivative. After reducing the system into a two-dimensional nonlinear fractional partial differential system (NLFPDEs), the power series (PS) method is applied to obtain the exact solution. Further the obtained power series solution is analyzed for convergence. Then, using the new conservation theorem with a generalized Noether’s operator, the conservation laws of the KP system are obtained.

On the soliton solutions for an intrinsic fractional discrete nonlinear electrical transmission line

This paper focuses on finding soliton solutions for an intrinsic fractional discrete nonlinear electrical transmission lattice. Our investigation is based on the fact that for a realistic system, the electrical characteristics of a capacitor (and an inductor via skin effect) should include a fractional-order time derivative. In this respect for the model under consideration, we derive a fractional nonlinear partial differential equation for the voltage dynamics by applying the Kirchhoff’s laws. It is realized that the behavior of new soliton solutions obtained is influenced by the fractional-order time derivative as well as the coupling values. The fractional order also modifies the propagation velocity of the voltage wave notwithstanding their structure and tends to set up localized structure for low coupling parameter values. However, for a high value of the coupling parameter, the fractional order is less seen on the shapes of the new solitary solutions that are analytically derived. Several methods such as the Kudryashov method, the $$(G'/G)$$ -expansion method, the Jacobi elliptical functions method and the Weierstrass elliptic function expansion method led us to derive these solitary solutions while using the modified Riemann–Liouville derivatives in addition to the fractional complex transform. An insight into the overall dynamics of our network is provided through the analysis of the phase portraits.

Mittag–Leffler stability and finite-time control for a fractional-order hydraulic turbine governing system with mechanical time delay: An linear matrix inequalitie approach

This article mainly studies the Mittag–Leffler stability and finite-time control of a time-delay fractional-order hydraulic turbine governing system. First, properties of the Riemann–Liouville derivative and some important lemmas are introduced. Second, considering the mechanical time delay of the main servomotor, the mathematical model of a fractional-order hydraulic turbine governing system with mechanical time delay is presented. Then, based on Mittag–Leffler stability theorem, a suitable sliding surface and finite-time controller are designed for the hydraulic turbine governing system. The system stability is confirmed, and the stability condition is given in the form of linear matrix inequalities. Finally, the traditional proportional–integral–derivative control method and an existing sliding mode control method are selected to verify the effectiveness and robustness of the proposed method. This study also provides a new approach for the stability analysis of the time-delay fractional-order hydraulic turbine governing system.

Existence and Stability for a Nonlinear Coupledp-Laplacian System of Fractional Differential Equations

In this paper, we study the nonlinear coupled system of equations with fractional integral boundary conditions involving the Caputo fractional derivative of ordersθ1 and θ2and Riemann–Liouville derivative of ordersϱ1 and ϱ2with thep-Laplacian operator, wheren−1<θ1,θ2,ϱ1,ϱ2≤n, andn≥3. With the help of two Green’s functionsGϱ1w,ℑ,Gϱ2w,ℑ, the considered coupled system is changed to an integral system. Since topological degree theory is more applicable in nonlinear dynamical problems, the existence and uniqueness of the suggested coupled system are treated using this technique, and we find appropriate conditions for positive solutions to the proposed problem. Moreover, necessary conditions are highlighted for the Hyer–Ulam stability of the solution for the specified fractional differential problems. To confirm the theoretical analysis, we provide an example at the end.

Invariant solutions of fractional-order spatio-temporal partial differential equations

Abstract This paper considers two categories of fractional-order population growth models, where a time component is defined by Riemann–Liouville derivatives. These models are studied under the Lie symmetry approach, and we reduce the fractional partial differential equations to nonlinear ordinary differential equations. Subsequently, solutions of the latter are determined numerically or with the aid of Laplace transforms. Graphical representations for integral and trigonometric solutions are presented. A key feature of these models is the connection between spatial patterning of organisms versus competitive coexistence.

Extremal solutions of nonlinear functional discontinuous fractional equations

This paper is devoted to prove the existence of extremal solutions of fractional equation with Riemann–Liouville derivative. The existence follows from the method of lower and upper solutions. Some jumps in the derivative of these functions are allowed. It is important to point out that a discontinuous and functional dependence on the nonlinear part of the equation with respect to the solution is allowed. The construction of the Green’s function related to the linear part of the equation coupled to spectral theory is fundamental to deduce the results.

Analysis of nonlinear fractional diffusion equations with a Riemann-liouville derivative

<p style='text-indent:20px;'>In this paper, we consider a nonlinear fractional diffusion equations with a Riemann-Liouville derivative. First, we establish the global existence and uniqueness of mild solutions under some assumptions on the input data. Some regularity results for the mild solution and its derivatives of fractional orders are also derived. Our key idea is to combine the theories of Mittag-Leffler functions, Banach fixed point theorem and some Sobolev embeddings.</p>

Optical singular and dark solitons to the (2 + 1)-dimensional time–space fractional nonlinear Schrödinger equation

In this study, the (2+1)-dimensional nonlinear Schrödinger equation with fractional-order derivative in both time and space is considered within the modified Riemann–Liouville derivatives. This fractional-order equation is newly derived by using the fractional variational principle and the semi-inverse method. By introducing the fractional complex transform, the (G′∕G)-expansion method, the tanh–coth method, and the modified simple equation method are used to obtain the explicit analytical solutions of this equation. As a result, optical singular and dark soliton solutions are obtained. The behaviours of the soliton solutions are expressed by 2D and 3D graphs.

Some results for initial value problem of nonlinear fractional equation in Sobolev space

This paper focuses on the existence and uniqueness of solutions for certain types of fractional differential equations involving Riemann–Liouville derivative. The main results are obtained by some fixed point theorems in Sobolev space.

Дифференциальные игры дробного порядка с распределенными параметрами

Изучена задача преследования в дифференциальных играх дробного порядка с распределенными параметрами. Частные дробные производные по времени и пространственные переменные рассматриваются в смысле Римана-Лиувилля, при аппроксимации применяется формула Грюн-вальда-Летникова. Рассматривается задача попадания в некоторой положительной окраине терминального множества. Для решения применяется способ конечных различий. Аппроксимируются дробные производные Римана-Лиувилля по пространственным переменным на отрезке с помощью формулы Грюнвальда-Летникова. По достаточному признаку существования дробной производной получена разностная аппроксимация производной дробного порядка по времени. Аппроксимируя дифференциальную игру на явную разностную, получаем дискретную игру. Сформулирована соответствующая задача преследования для дискретной игры, полученная с помощью аппроксимации непрерывной игры. Определено понятие возможности завершения преследования, дискретной игры в смысле точного поимки. Получены достаточные условия для завершения преследования. При этом показано, что порядок аппроксимации по времени равен единице, а по пространственным переменным — двум. Доказано, если в дискретной игре по заданному начальному положению возможно завершение преследования в смысле точного поимки, то в непрерывной игре из соответствующего начального положения возможно завершение преследования. Предложена структура построения управлений преследования, которая обеспечит завершение игры за конечное время. Методы, применяемые для этой задачи, могут быть использованы для изучения дифференциальных игр, описываемых более общими уравнениями дробного порядка.

An Energy Stable and Maximum Bound Preserving Scheme with Variable Time Steps for Time Fractional Allen--Cahn Equation

In this work, we propose a Crank-Nicolson-type scheme with variable steps for the time fractional Allen-Cahn equation. The proposed scheme is shown to be unconditionally stable (in a variational energy sense), and is maximum bound preserving. Interestingly, the discrete energy stability result obtained in this paper can recover the classical energy dissipation law when the fractional order $\alpha \rightarrow 1.$ That is, our scheme can asymptotically preserve the energy dissipation law in the $\alpha \rightarrow 1$ limit. This seems to be the first work on variable time-stepping scheme that can preserve both the energy stability and the maximum bound principle. Our Crank-Nicolson scheme is build upon a reformulated problem associated with the Riemann-Liouville derivative. As a by product, we build up a reversible transformation between the L1-type formula of the Riemann-Liouville derivative and a new L1-type formula of the Caputo derivative, with the help of a class of discrete orthogonal convolution kernels. This is the first time such a \textit{discrete} transformation is established between two discrete fractional derivatives. We finally present several numerical examples with an adaptive time-stepping strategy to show the effectiveness of the proposed scheme.

Image Denoising Based on Fractional Gradient Vector Flow and Overlapping Group Sparsity as Priors.

In this paper, a new regularization term in the form of L1-norm based fractional gradient vector flow (LF-GGVF) is presented for the task of image denoising. A fractional order variational method is formulated, which is then utilized for estimating the proposed LF-GGVF. Overlapping group sparsity along with LF-GGVF is used as priors in image denoising optimization framework. The Riemann-Liouville derivative is used for approximating the fractional order derivatives present in the optimization framework. Its role in the framework helps in boosting the denoising performance. The numerical optimization is performed in an alternating manner using the well-known alternating direction method of multipliers (ADMM) and split Bregman techniques. The resulting system of linear equations is then solved using an efficient numerical scheme. A variety of simulated data that includes test images contaminated by additive white Gaussian noise are used for experimental validation. The results of numerical solutions obtained from experimental work demonstrate that the performance of the proposed approach in terms of noise suppression and edge preservation is better when compared with that of several other methods.

Numerical Schemes for Time-Space Fractional Vibration Equations

In this paper, we present a numerical scheme and an alternating direction implicit (ADI) scheme for the one-dimensional and two-dimensional time-space fractional vibration equations (FVEs), respectively. Firstly, the considered time-space FVEs are equivalently transformed into their partial integro-differential forms with the classical first order integrals and the Riemann-Liouville derivative. This transformation can weaken the smoothness requirement in time when discretizing the partial integro-differential problems. Secondly, we use the Crank-Nicolson technique combined with the midpoint formula, the weighted and shifted Grunwald difference formula and the second order convolution quadrature formula to deal with the temporal discretizations. Meanwhile, the classical central difference formula and fractional central difference formula are applied to approximate the second order derivative and the Riesz derivative in spatial direction, respectively. Further, an ADI scheme is constructed for the two-dimensional case. Then, the convergence and unconditional stability of the proposed schemes are proved rigorously. Both of the schemes are convergent with the second order accuracy in time and space. Finally, two numerical examples are given to support the theoretical results.

Stability Analysis of Two Kinds of Fractional-Order Neural Networks Based on Lyapunov Method

At present, the theory and application of fractional-order neural networks remain in the exploratory stage. We study the asymptotic stability of fractional-order neural networks with Riemann-Liouville (R-L) derivatives. For non-delayed and delayed systems, we propose an asymptotic stability criterion based on the combination of the Lyapunov method and linear matrix inequality (LMI) method. The highlights include the following: (1) for fractional-order neural networks with time delay, the existence and uniqueness of solutions are proven by using matrix analysis theory and contraction mapping theorem, and (2) based on the unique solution, a suitable Lyapunov functional is constructed. Based on the inequality theorem and LMI method, two sets of asymptotic stability criteria for fractional-order neural networks are proven, which avoids the difficulty of solving the fractional derivative by the Leibniz law. Finally, the results are verified using numerical simulations.

Exact solutions and numerical simulations of time-fractional Fokker-Plank equation for special stochastic process

‎In this paper, a type of time-fractional Fokker-Planck equation (FPE) of the OrnsteinUhlenbeck process is solved via Riemann-Liouville and Caputo derivatives. An analytical method based on symmetry operators is used for finding reduced form and exact solutions of the equation. A numerical simulation based on the M¨untz-Legendre polynomials is applied in order to find some approximated solutions of the equation.

A posteriori error estimates of spectral method for the fractional optimal control problems with non-homogeneous initial conditions

<abstract><p>In this paper we consider an optimal control problem governed by a space-time fractional diffusion equation with non-homogeneous initial conditions. A spectral method is proposed to discretize the problem in both time and space directions. The contribution of the paper is threefold: (1) A discussion and better understanding of the initial conditions for fractional differential equations with Riemann-Liouville and Caputo derivatives are presented. (2) A posteriori error estimates are obtained for both the state and the control approximations. (3) Numerical experiments are performed to verify that the obtained a posteriori error estimates are reliable.</p></abstract>

Stability and existence the solution for a coupled system of hybrid fractional differential equation with uniqueness

In this literature, we are investigating the existence and uniqueness of solutions for a coupled system of hybrid differential equations with p-Laplacian operator involving the fractional derivatives Caputo and Riemann–Liouville derivatives of various orders. For this purpose, the proposed problem will be converted into integral equations by using two Green functions for ], where . The existence of a solution is proved by using topological degree and Leray–Schauder fixed point theorems. Uniqueness of solution is obtained with the help of Banach principle. Some adequate conditions for Hyers–Ulam- stability of the solution also are investigated. To verify the results. An expressive example will be presented to clarify the results.