Numerical approaches to fractional calculus and fractional ordinary differential equation

Nowadays, fractional calculus are used to model various different phenomena in nature, but due to the non-local property of the fractional derivative, it still remains a lot of improvements in the present numerical approaches. In this paper, some new numerical approaches based on piecewise interpolation for fractional calculus, and some new improved approaches based on the Simpson method for the fractional differential equations are proposed. We use higher order piecewise interpolation polynomial to approximate the fractional integral and fractional derivatives, and use the Simpson method to design a higher order algorithm for the fractional differential equations. Error analyses and stability analyses are also given, and the numerical results show that these constructed numerical approaches are efficient.     

A fractional calculus approach to nonlocal elasticity

If the attenuation function of strain is expressed as a power law, the formalism of fractional calculus may be used to handle Eringen nonlocal elastic model. Aim of the present paper is to provide a mechanical interpretation to this nonlocal fractional elastic model by showing that it is equivalent to a discrete, point-spring model. A one-dimensional geometry is considered; the static, kinematic and constitutive equations are presented and the governing fractional differential equation highlighted. Two numerical procedures to solve the fractional equation are finally implemented and applied to study the strain field in a finite bar under given edge displacements.     

Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach

This article presents a new approach to dealing with time dependent quantities such as autocorrelation function of harmonic and anharmonic systems using coherent states and partial differential equations. The approach that is normally used to evaluate dynamical quantities involves formidable operator algebra. That operator algebra becomes insurmountable when employing Morse oscillator coherent states. This problem becomes even more complicated in case of Morse oscillator as it tends to exhibit divergent dynamics. This approach employs linear partial differential equations, some of which may be solved exactly and analytically, thereby avoiding the cumbersome noncommutative algebra required to manipulate coherent states of Morse oscillator. Additionally, the arising integrals while using the herein presented method feature stability and high numerical efficiency. The correctness, applicability, and utility of the above approach are tested by reproducing the partition and optical autocorrelation function of the harmonic oscillator. A closed-form expression for the equilibrium canonical partition function of the Morse oscillator is derived using its coherent states and partial differential equations. Also, a nonequilibrium autocorrelation function expression for weak electron–phonon coupling in condensed systems is derived for displaced Morse oscillator in electronic state. Finally, the utility of the method is demonstrated through further simplifying the Morse oscillator partition function or autocorrelation function expressions reported by other researchers in unevaluated form of second-order derivative exponential. Comparison with exact dynamics shows identical results.     

Explicit solutions of a certain class of differential equations by means of fractional calculus

Judging by the remarkably large number of recent publications on Fractional Calculus and Its Applications in several widely diverse areas of mathematical, physical, and engineering sciences, the current popularity and importance of the subject of fractional calculus cannot be overemphasized. Motivated by some of these interesting developments, many authors have recently demonstrated the usefulness of fractional calculus in the derivation of explicit particular solutions of a number of linear ordinary and partial differential equations of the second and higher orders. The main object of the present paper is to show how several recent contributions on this subject, involving a certain class of ordinary differential equations, can be obtained (in a unified manner) by suitably applying some general theorems on explicit particular solutions of a family of linear ordinary fractional differintegral equations.     

Chaos in discrete fractional difference equations

Recently, the discrete fractional calculus (DFC) is receiving attention due to its potential applications in the mathematical modelling of real-world phenomena with memory effects. In the present paper, the chaotic behaviour of fractional difference equations for the tent map, Gauss map and 2x(mod 1) map are studied numerically. We analyse the chaotic behaviour of these fractional difference equations and compare them with their integer counterparts. It is observed that fractional difference equations for the Gauss and tent maps are more stable compared to their integer-order version.     

Singular fractional evolution differential equations

We give an existence-uniqueness result for linear and nonlinear time fractional evolution equations with singularities in corresponding norm in extended Colombeau algebra of generalized functions using fractional analog for Duhamel principle. Paper deals with some nonlinear models with singularities appearing in viscoelasticity and in anomalous processes which have met great interest among researchers who consider them as a challenge in recent years.     

Numerical solution of fractional differential equations via a Volterra integral equation approach

The main focus of this paper is to present a numerical method for the solution of fractional differential equations. In this method, the properties of the Caputo derivative are used to reduce the given fractional differential equation into a Volterra integral equation. The entire domain is divided into several small domains, and by collocating the integral equation at two adjacent points a system of two algebraic equations in two unknowns is obtained. The method is applied to solve linear and nonlinear fractional differential equations. Also the error analysis is presented. Some examples are given and the numerical simulations are also provided to illustrate the effectiveness of the new method.     

A matrix approach for partial differential equations with Riesz space fractional derivatives

Fractional partial differential equations are emerging in many scientific fields and their numerical solution is becoming a fundamental topic. In this paper we consider the Riesz fractional derivative operator and its discretization by fractional centered differences. The resulting matrix is studied, with an interesting result on a connection between the decay behavior of its entries and the short memory principle from fractional calculus. The Shift-and-Invert method is then applied to approximate the solution of the partial differential equation as the action of the matrix exponential on a suitable vector which mimics the given initial conditions. The numerical results confirm the good approximation quality and encourage the use of the proposed approach.     

Fractional vector calculus and fractional Maxwell’s equations

The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. We solve some problems of consistent formulations of FVC by using a fractional generalization of the Fundamental Theorem of Calculus. We define the differential and integral vector operations. The fractional Green’s, Stokes’ and Gauss’s theorems are formulated. The proofs of these theorems are realized for simplest regions. A fractional generalization of exterior differential calculus of differential forms is discussed. Fractional nonlocal Maxwell’s equations and the corresponding fractional wave equations are considered.     

Analytical approach for the fractional differential equations by using the extended tanh method

In this study, we consider analytical solutions of space–time fractional derivative foam drainage equation, the nonlinear Korteweg–de Vries equation with time and space-fractional derivatives and time-fractional reaction–diffusion equation by using the extended tanh method. The fractional derivatives are defined in the modified Riemann–Liouville context. As a result, various exact analytical solutions consisting of trigonometric function solutions, kink-shaped soliton solutions and new exact solitary wave solutions are obtained.     

Numerical solution of fractional differential equations by using fractional B-spline

In this paper, we present fractional B-spline collocation method for the numerical solution of fractional differential equations. We consider this method for solving linear fractional differential equations which involve Caputo-type fractional derivatives. The numerical results demonstrate that the method is efficient and quite accurate and it requires relatively less computational work. For this reason one can conclude that this method has advantage on other methods and hence demonstrates the importance of this work.     

Random solutions to a system of fractional differential equations via the Hadamard fractional derivative

In this work, we use a new random fixed point theorem in vector metric spaces due to Sinacer et al. [M.L. Sinacer et al., Random Oper. Stoch. Equ. 24, 93 (2016)] to prove the existence of solutions and the compactness of solution sets of a random system of fractional differential equations via the Hadamard-type derivative. The existence, modification and stochastically continuity of an M²-solution are also proved.     

Exact solutions to the time-fractional differential equations via local fractional derivatives

This article utilizes the local fractional derivative and the exp-function method to construct the exact solutions of nonlinear time-fractional differential equations (FDEs). For illustrating the validity of the method, it is applied to the time-fractional Camassa–Holm equation and the time-fractional-generalized fifth-order KdV equation. Moreover, the exact solutions are obtained for the equations which are formed by different parameter values related to the time-fractional-generalized fifth-order KdV equation. This method is an reliable and efficient mathematical tool for solving FDEs and it can be applied to other non-linear FDEs.     

Exact solutions for nonlinear partial fractional differential equations

In this article,we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations.We use the improved(G’/G)-expansion function method to calculate the exact solutions to the time-and space-fractional derivative foam drainage equation and the time-and space-fractional derivative nonlinear KdV equation.This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations.     

Initialized fractional differential equations with Riemann-Liouville fractional-order derivative

The initial value problem of fractional differential equations and its solving method are studied in this paper. Firstly, for easy understanding, a different version of the initialized operator theory is presented for Riemann-Liouville’s fractional-order derivative, addressing the initial history in a straightforward form. Then, the initial value problem of a single-term fractional differential equation is converted to an equivalent integral equation, a form that is easy for both theoretical and numerical analysis, and two illustrative examples are given for checking the correctness of the integral equation. Finally, the counter-example proposed in a recent paper, which claims that the initialized operator theory results in wrong solution of a fractional differential equation, is checked again carefully. It is found that solving the equivalent integral equation gives the exact solution, and the reason behind the result of the counter-example is that the calculation therein is based on the conventional Laplace transform for fractional-order derivative, not on the initialized operator theory. The counter-example can be served as a physical model of creep phenomena for some viscoelastic materials, and it is found that it fits experimental curves well.     

Partial differential equations possessing Frobenius integrable decompositions

Frobenius integrable decompositions are introduced for partial differential equations. A procedure is provided for determining a class of partial differential equations of polynomial type, which possess specified Frobenius integrable decompositions. Two concrete examples with logarithmic derivative Bäcklund transformations are given, and the presented partial differential equations are transformed into Frobenius integrable ordinary differential equations with cubic nonlinearity. The resulting solutions are illustrated to describe the solution phenomena shared with the KdV and potential KdV equations.     

A quantum algebra approach to discrete equations on uniform lattices

A quantum algebra method for deducing the symmetries of discrete equations on uniform lattices is proposed. In principle, such a procedure can be applied to discretizations in a single coordinate (space or time) and the symmetries obtained in this way are indeed differential-difference operators. Firstly, the method is illustrated on two known examples that have been also analysed from the usual Lie symmetry approach: a uniform space lattice discretization of the (1+1) free heat-Schrödinger equation associated to a quantum Schrödinger algebra, and a discrete space (1+1) wave equation provided by a quantumso(2, 2) algebra. Furthermore, we construct a discrete space (2+1) wave equation from a new quantumso(3, 2) algebra, to show that this method is useful in higher dimensions. Time discretizations are also commented.     

Paradox of enrichment: A fractional differential approach with memory

The paradox of enrichment (PoE) proposed by Rosenzweig [M. Rosenzweig, The paradox of enrichment, Science 171 (1971) 385–387] is still a fundamental problem in ecology. Most of the solutions have been proposed at an individual species level of organization and solutions at community level are lacking. Knowledge of how learning and memory modify behavioral responses to species is a key factor in making a crucial link between species and community levels. PoE resolution via these two organizational levels can be interpreted as a microscopic- and macroscopic-level solution. Fractional derivatives provide an excellent tool for describing this memory and the hereditary properties of various materials and processes. The derivatives can be physically interpreted via two time scales that are considered simultaneously: the ideal, equably flowing homogeneous local time, and the cosmic (inhomogeneous) non-local time. Several mechanisms and theories have been proposed to resolve the PoE problem, but a universally accepted theory is still lacking because most studies have focused on local effects and ignored non-local effects, which capture memory. Here we formulate the fractional counterpart of the Rosenzweig model and analyze the stability behavior of a system. We conclude that there is a threshold for the memory effect parameter beyond which the Rosenzweig model is stable and may be used as a potential agent to resolve PoE from a new perspective via fractional differential equations.     

Exponential integrators for time–fractional partial differential equations

Time–fractional partial differential equations can be numerically solved by first discretizing with respect to the spatial derivatives and hence applying a time–step integrator. An exponential integrator for fractional differential equations is proposed to overcome the stability issues due to the stiffness in the resulting semi–discrete system. Convergence properties and the main implementation issues are studied. The advantages of the proposed method are illustrated by means of some test problems.     

Solving systems of fractional differential equations by homotopy-perturbation method

In this Letter, approximate analytical solutions of systems of Fractional Differential Equations (FDEs) are derived by the Homotopy-Perturbation Method (HPM). The fractional derivatives are described in the Caputo sense. The solutions are obtained in the form of rapidly convergent infinite series with easily computable terms. Numerical results reveal that HPM is very effective and simple for obtaining approximate solutions of nonlinear systems of FDEs.