Liouville Derivative Research Articles

Numerical Methods for Solving Space Fractional Partial Differential Equations Using Hadamard Finite-Part Integral Approach

We introduce a novel numerical method for solving two-sided space fractional partial differential equations in two-dimensional case. The approximation of the space fractional Riemann–Liouville derivative is based on the approximation of the Hadamard finite-part integral which has the convergence order O(h^{3- alpha }), where h is the space step size and alpha in (1, 2) is the order of Riemann–Liouville fractional derivative. Based on this scheme, we introduce a shifted finite difference method for solving space fractional partial differential equations. We obtained the error estimates with the convergence orders O(tau +h^{3-alpha }+ h^{beta }), where tau is the time step size and beta >0 is a parameter which measures the smoothness of the fractional derivatives of the solution of the equation. Unlike the numerical methods for solving space fractional partial differential equations constructed using the standard shifted Grünwald–Letnikov formula or higher order Lubich’s methods which require the solution of the equation to satisfy the homogeneous Dirichlet boundary condition to get the first-order convergence, the numerical method for solving the space fractional partial differential equation constructed using the Hadamard finite-part integral approach does not require the solution of the equation to satisfy the Dirichlet homogeneous boundary condition. Numerical results show that the experimentally determined convergence order obtained using the Hadamard finite-part integral approach for solving the space fractional partial differential equation with non-homogeneous Dirichlet boundary conditions is indeed higher than the convergence order obtained using the numerical methods constructed with the standard shifted Grünwald–Letnikov formula or Lubich’s higher order approximation schemes.

Comments on various extensions of the Riemann–Liouville fractional derivatives : About the Leibniz and chain rule properties

Starting from the Riemann–Liouville derivative, many authors have built their own notion of fractional derivative in order to avoid some classical difficulties like a non zero derivative for a constant function or a rather complicated analogue of the Leibniz relation. Discussing in full generality the existence of such operator over continuous functions, we derive some obstruction Lemma which can be used to prove the triviality of some operators as long as the linearity and the Leibniz property are preserved. As an application, we discuss some properties of the Jumarie’s fractional derivative as well as the local fractional derivative. We also discuss the chain rule property in the same perspective.

Existence of positive solutions for a semipositone fractional differential equation with perturbed term

In this work, we consider the following nonlinear fractional differential equation $$\begin{aligned} {\left\{ \begin{array}{ll} -D^{\nu } u(t)=\lambda f(t,u(t))+e(t) \ \ \ in \ \ (0,1),\\ u^{(j)}(0)=0, \ \ 0\le j \le n-2, \ \ [D^{\alpha } u(t)]_{t=1}=0,\\ \end{array}\right. } \end{aligned}$$where $$\lambda >0$$ is a parameter, $$n \ge 3$$, $$n-1< \nu < n$$, $$1 \le \alpha \le n-2$$ and $$D^{\nu }$$ stands for the standard Reimann–Liouville derivative, $$f: [0,1]\times [0,+\infty ) \longrightarrow {\mathbb {R}}$$ is sign-changing continuous function (that is, we have a so-called equation of semipositone problems). The perturbed term $$e: (0,1) \rightarrow {\mathbb {R}}$$ is measurable function and verifies some appropriate conditions. We derive some intervals of $$\lambda $$ such that the problem has positive solutions. Our study relies on Guo-Krasnoselskii fixed point theorem.

Fractional output stabilization for a class of bilinear distributed systems

In this paper, we consider the question of the state fractional spatial derivative stabilization, using Riemann–Liouville derivative of order \(\alpha \in ]0, 1[\), for a class of bilinear distributed systems. Firstly, we characterize the feedback control that ensure the strong and the weak stabilization of the fractional output. Then, we solve a minimization fractional problem. Finally, we provide an example with numerical simulations to illustrate the effectiveness of the given stabilization theorems.

Study on Fractional Differential Equations with Modified Riemann–Liouville Derivative via Kudryashov Method

Abstract In this work, the Kudryashov method is handled to find exact solutions of nonlinear fractional partial differential equations in the sense of the modified Riemann–Liouville derivative as given by Guy Jumarie. Firstly, these fractional equations can be turned into another nonlinear ordinary differential equations by fractional complex transformation. Then, the method is applied to solve the space-time fractional Symmetric Regularized Long Wave equation and the space-time fractional generalized Hirota–Satsuma coupled KdV equation. The obtained solutions include generalized hyperbolic functions solutions.

Symmetries, conservation laws and exact solutions of the time-fractional diffusivity equation via Riemann–Liouville and Caputo derivatives

ABSTRACT In this paper, we deal with the diffusivity equation, by using the Lie group analysis method. The infinitesimal generators of this equation are investigated. The concept of non-linear self-adjointness is employed to construct the conservation laws for fractional evolution equations using its Lie point symmetries. The approach is demonstrated on diffusivity equation with the Riemann–Liouville and Caputo time-fractional derivatives. It is shown that this equation is non-linearly self-adjoint and therefore desired conservation laws can be obtained using appropriate formal Lagrangian. Lie group method provides an efficient tool to solve time-fractional diffusivity equation. For this, the similarity reductions are performed with the similarity variables obtained from symmetry operators. As a result, the reduced fractional ordinary differential equations (FODEs) are deduced, and some group invariant solutions in the explicit form are obtained as well. We apply the invariant subspace method for constructing more particular solutions of the desired equation. It allows one to reduce an fractional partial differential equation to a system of non-linear FODEs.

LIE SYMMETRY ANALYSIS TO THE WEAKLY COUPLED KAUP–KUPERSHMIDT EQUATION WITH TIME FRACTIONAL ORDER

The aim of this paper is to apply the Lie group analysis method to the weakly coupled Kaup–Kupershmidt (KK) equation with time fractional order. We considered the symmetry analysis, explicit solutions to the weakly coupled time fractional KK (TF-KK) equation with Riemann–Liouville (RL) derivative. The weakly coupled TF-KK equation is reduced to a nonlinear ordinary differential equation (ODE) of fractional order. We solve the reduced fractional ODE using the sub-equation method.

Homotopy perturbation method with modified Riemann –Liouville derivative for solving fractional- time partial differential equations

Homotopy perturbation method with modified Riemann –Liouville derivative for solving fractional- time partial differential equations

Symmetry analysis of some nonlinear generalised systems of space–time fractional partial differential equations with time-dependent variable coefficients

In this paper, the Lie group analysis method is applied to carry out the Lie point symmetries of some space–time fractional systems including coupled Burgers equations, Ito’s system, coupled Korteweg–de-Vries (KdV) equations, Hirota–Satsuma coupled KdV equations and coupled nonlinear Hirota equations with time-dependent variable coefficients with the Riemann–Liouville derivative. Symmetry reductions are constructed using Lie symmetries of the systems. To the best of our knowledge, nobody has so far derived the invariants of space–time nonlinear fractional partial differential equations with time-dependent coefficients.

A Truncation Method for Solving the Time-Fractional Benjamin-Ono Equation

We deem the time-fractional Benjamin-Ono (BO) equation out of the Riemann–Liouville (RL) derivative by applying the Lie symmetry analysis (LSA). By first using prolongation theorem to investigate its similarity vectors and then using these generators to transform the time-fractional BO equation to a nonlinear ordinary differential equation (NLODE) of fractional order, we complete the solutions by utilizing the power series method (PSM).

Superconvergence points of integer and fractional derivatives of special Hermite interpolations and its applications in solving FDEs

In this paper, we study the theory of convergence and superconvergence for integer and fractional derivatives of the one-point and two-point Hermite interpolations. When considering the integer-order derivatives, exponential decay of the error is proved, and superconvergence points are located, at which the convergence rates areO(N−2) andO(N−1.5) better than the global rates for the one-point and two-point interpolations, respectively. HereNrepresents the degree of the interpolation polynomial. It is proved that theαth fractional derivative of (u−uN), withk&lt;α&lt;k+1, is bounded by its (k+1) th derivative. Furthermore, the corresponding superconvergence points are predicted for fractional derivatives, and an eigenvalue method is proposed to calculate the superconvergence points for the Riemann–Liouville derivatives. In the application of the knowledge of superconvergence points to solve FDEs, we discover that a modified collocation method makes numerical solutions much more accurate than the traditional collocation method.

Inverse Problems for a Class of Degenerate Evolution Equations with Riemann - Liouville Derivative

Abstract Unique solvability and well-posedness issues are studied for linear inverse problems with a constant unknown parameter to fractional order differential equations with Riemann – Liouvlle derivative in Banach spaces. Firstly, well-posedness criteria for the inverse problem with the Cauchy type initial conditions to the differential equation in a Banach space that solved with respect to the fractional derivative is obtained. This result is applied to search of sufficient conditions for the unique solution existence of the inverse problem for equation with linear degenerate operator at the Riemann – Liouville fractional derivative. It is shown that the presence of the matching conditions for the data of the problem excludes the possibility of the well-posedness consideration for the degenerate inverse problem with the Cauchy type condition. But for the inverse problem with the Showalter – Sidorov type conditions it is found the criteria of the well-posedness. Abstract results are used to the search of conditions of the unique solvability for an inverse problem to a class of partial differential equations of time-fractional order with polynomials of elliptic differential operators with respect to the spatial variables.

Optimal Control Problems for Certain Linear Fractional-Order Systems Given by Equations with Hilfer Derivative

Two optimal control problems are investigated for linear time-invariant systems of fractional order with lumped parameters, which dynamics is described by equations with Hilfer derivative: control problem with minimal norm and time-optimal control problem with control norm constraint. Controls are considered that are the p-integrable or essentially bounded functions. Investigation is conducted by the method of moments. Correctness and solvability conditions of the problem of moments are obtained for the problem statement considered. The optimal control problems stated are solved analytically for several particular cases and the properties of the solutions are investigated depending on fractional differentiation indices and Hilfer fractional differential operator parameters. The comparison is conducted of the results obtained with the known results for integer-order systems and fractional-order systems described by equations with Riemann–Liouville or Caputo derivative.

On Maximal Regularity for Abstract Parabolic Problems with Fractional Time Derivative

We consider initial value problems for abstract evolution equations with fractional time derivative. Concerning the Caputo derivative \(\mathbb {D}^\alpha u\), we show that certain assumptions, which are known to be sufficient to get a unique solution with a prescribed regularity, are also necessary. So we establish a maximal regularity result. We consider similar problems with the Riemann–Liouville derivative \(\partial ^\alpha u\). Here, we give a complete proof (necessity and sufficiency of the assumptions) of the corresponding maximal regularity results.

A Cauchy Type Problem for a Degenerate Equation with the Riemann–Liouville Derivative in the Sectorial Case

Under study is the unique solvability of a Cauchy type problem and a generalized Schowalter–Sidorov type problem for a class of linear inhomogeneous equations in Banach spaces with a degenerate operator at the Riemann–Liouville fractional derivative. We find an explicit form of a solution under some conditions for the pair of operators in the equation. To this end, we study a Cauchy type problem for an equation solvable with respect to the Riemann–Liouville derivative with an operator on the right-hand side which generates a resolving family of operators analytic in a sector. These abstract results are used to prove the unique solvability of an initial-boundary value problem for the Navier–Stokes system of equations of fractional order in time.

On generalized fractional integration by parts formulas and their applications to boundary value problems

Abstract In the paper, we derive a formula of integration by parts for the left-sided Hilfer derivative. Moreover, we present a formula of such a type for the right-sided Hilfer derivative. These results generalize the well-known rules of integration by parts for the Riemann–Liouville and Caputo derivatives. Finally, we use these rules to investigate some boundary value problems involving mixed fractional derivatives.

Dispersive wave solutions of the nonlinear fractional Zakhorov-Kuznetsov-Benjamin-Bona-Mahony equation and fractional symmetric regularized long wave equation

Abstract In this paper, we obtain several novelty solutions by applying the improved F-expansion method to solve the space–time fractional Zakhorov Kuznetsov Benjamin Bona Mahony (ZKBBM) equation and the space–time fractional symmetric regularized long wave (SRLW) equation. These two fractional equations have been extensively used to explain numerous forms of physical mechanism in mathematical physics and engineering. The suggested fractional complex transform and the revised Riemann–Liouville derivative is used to change the two equations to its ordinary differential equations. The exact solutions of these equations include hyperbolic and trigonometric solutions has been effectively obtained.

Lie symmetry analysis, explicit solutions and conservation laws for the time fractional Kolmogorov–Petrovskii–Piskunov equation

ABSTRACT In this paper, Lie symmetry method is applied to investigate the invariance properties of the time fractional Kolmogorov–Petrovskii–Piskunov equation with Riemann–Liouville derivative. In view of point symmetry, the vector fields for the governing equation are well constructed. And then the equation can be reduced to a fractional ODE with the help of Erdélyi–Kober operator. Moreover, a kind of explicit power series solutions of the equation is derived by virtue of the power series theory. Lastly, based on Ibragimov's method, two kinds of conservation laws for the equation are established.

Bifurcation of traveling waves and exact solutions of Kadomtsev–Petviashvili modified equal width equation with fractional temporal evolution

The article is devoted in the study of the Kadomtsev–Petviashvili modified equal width equation with fractional temporal evolution in which modified Riemann–Liouville derivative is considered. We discuss the dynamical behavior of the system using bifurcation theory in different parametric regions. We also depict the phase portraits of the traveling wave solutions and obtain explicit traveling waves including solitary wave and breaking wave solution. We apply the $$(G'/G)$$ -expansion method and the F-expansion method along with fractional complex transformation, and obtain a large variety of exact traveling wave solutions which includes solitary wave, kink-type wave, and periodic wave solutions of the equation. Finally, we demonstrate remarkable features of the traveling wave solutions via interesting figures and phase portraits.

A fast second‐order difference scheme for the space–time fractional equation

In this paper, a fast second‐order accurate difference scheme is proposed for solving the space–time fractional equation. The temporal Caputo derivative is approximated by ℱL2‐1σ formula which employs the sum‐of‐exponential approximation to the kernel function appeared in Caputo derivative. The second‐order linear spline approximation is applied to the spatial Riemann–Liouville derivative. At each time step, a fast algorithm, the preconditioned conjugate gradient normal residual method with a circulant preconditioner (PCGNR), is used to solve the resulting system that reduces the storage and computational cost significantly. The unique solvability and unconditional convergence of the difference scheme are shown by the discrete energy method. Numerical examples are given to verify numerical accuracy and efficiency of the difference schemes.