Fractional-order Problems Research Articles

Chaos control in the fractional order logistic map via impulses

In this paper the chaos control in the discrete logistic map of fractional order is obtained with an impulsive control algorithm. The underlying discrete initial value problem of fractional order is considered in terms of Caputo delta fractional difference. Every $\Delta$ steps, the state variable is instantly modified with the same impulse value, chosen from a bifurcation diagram versus impulse. It is shown that the solution of the impulsive control is bounded. The numerical results are verified via time series, histograms, and the 0-1 test. Several examples are considered

On New Solutions of Time-Fractional Wave Equations Arising in Shallow Water Wave Propagation

The primary objective of this manuscript is to obtain the approximate analytical solution of Camassa–Holm (CH), modified Camassa–Holm (mCH), and Degasperis–Procesi (DP) equations with time-fractional derivatives labeled in the Caputo sense with the help of an iterative approach called fractional reduced differential transform method (FRDTM). The main benefits of using this technique are that linearization is not required for this method and therefore it reduces complex numerical computations significantly compared to the other existing methods such as the perturbation technique, differential transform method (DTM), and Adomian decomposition method (ADM). Small size computations over other techniques are the main advantages of the proposed method. Obtained results are compared with the solutions carried out by other technique which demonstrates that the proposed method is easy to implement and takes small size computation compared to other numerical techniques while dealing with complex physical problems of fractional order arising in science and engineering.

Shifted Chebyshev schemes for solving fractional optimal control problems

Two schemes to find approximated solutions of optimal control problems of fractional order (FOCPs) are investigated. Integration and differentiation matrices were used in these schemes. These schemes used Chebyshev polynomials in the shifted case as a functional approximation. The target of the presented schemes is to convert such problems to optimization problems (OPs). Numerical examples are included, showing the strength of the schemes.

Operational-matrix-based algorithm for differential equations of fractional order with Dirichlet boundary conditions

The fractional differential equations (FDEs) are ground-breaking tools to demonstrate the complex-nature scientific systems in the form of non-linear behavior endorsed by the scientific community to develop some new and accurate mathematical methods. The main objective of this paper is the development of an extended mathematical algorithm based on the Gegenbauer wavelet method for the fractional-order problem. The Gegenbauer wavelet operational matrix with their derivative is proposed in our study. Some new operational matrices for the derivative of fractional order with Dirichlet boundary condition is purposed by introducing the piecewise function. Furthermore, a successful use to analyze the solution for the set of algebraic equations governed through the extended Gegenbauer wavelets technique is performed. Analytical solutions of the mentioned problem are effectively obtained, and a comparative study is presented. The outcomes are obtained via the modified Gegenbauer wavelet method by endorsing the accuracy and effectiveness of the mentioned technique. The convergence and error bound analysis are enclosed in our investigation. It is further verified that the algorithm is quite accurate, and an efficient mathematical tool is used to tackle the nonlinear fractional-order complex-nature problems.

An efficient analysis for N-soliton, Lump and lump–kink solutions of time-fractional (2+1)-Kadomtsev–Petviashvili equation

Solutions of the nonlinear physical problems are significant and a vital topic in real life while the soliton based algorithms are promising techniques to analyze the solutions of various nonlinear real-world problems. Herein, we are reporting some new soliton solutions including N-soliton, lump and lump–kink for (2+1)-Kadomtsev–Petviashvili of fractional-order α arising in mathematical physics. Multiple exp-function approach and bilinear form has been used to investigate the N-soliton and lump, lump–kink solutions of the discussed fractional-order problem. These methods convert the nonlinear partial differential equations into nonlinear algebraic equation having exponential functions. Furthermore, the method is oriented to the comfort of use and fitness of computer algebra systems and provides a direct and systematical solution procedure which generalizes Hirota’s perturbation scheme. Graphical representations (2D and 3D) of few specific presented multiple, lump and lump–kink solutions has been made to show the characteristics of multiple, lumps and lump–kinks as well as significant effects of α are plotted for each solution. Moreover, the amplitude of the waves is examined for various values of space variables in x and y direction as well as particular values of α as time varies. It is noticed that as α→1the solution turn into three wave and lump–kink solutions which is asserted via set of graphs. The higher values of space variable y are causing a decrease in the amplitude of the wave whereas the higher values of α are providing a higher amplitude of the wave. Hence, the attained results for nonlinear time fractional (2+1)-Kadomtsev–Petviashvili equation are endorsing the efficiency and appropriateness of multiple exp-function to analyze the N-soliton solutions. It is important to highlight that said method could be extended to diversify problem of physical nature.

Fractional power series neural network for solving delay fractional optimal control problems

ABSTRACTIn this paper, we develop a numerical method for solving the delay optimal control problems of fractional-order. The fractional derivatives are considered in the Caputo sense. The process begins with the assumption that the problem is first transformed into an equivalent problem with a fractional dynamical system without delay, using a Padé approximation. We then try to approximate the solution of the Hamiltonian conditions based on the Pontryagin minimum principle. The main feature is to implement nonlinear polynomial expansions in a neural network adaptive structure. The transfer functions of the employed neural network follow a fractional power series. The proposed technique does not use sigmoid or hyperbolic tangent nonlinear transfer functions commonly adopted in conventional neural networks at the output. Instead, linear transfer functions are employed which lead to explicit fractional power series formulae for the fractional optimal control problem. To do this, we use trial solutions for the states, Lagrange multipliers and control functions where these trial solutions are constructed by fractional power series neural network model. We then minimise the error function using an unconstrained optimisation scheme where weight parameters (or coefficients of the series) and biases associated with all neurons are unknown. Some numerical examples are given to illustrate the effectiveness of the proposed scheme.

Variational Iteration Method for Solving Boundary Value Problems of Fractional Order

This paper presents an approximate solution of ordinary boundary value problems using the variational iteration method. The fractional derivatives are described in the Caputo sense, because it allows traditional initial and boundary conditions to be included in the formulation of the problem. The solutions of proposed fractional equations are calculated in the form of convergent series with easily computable components. Also, we construct the variational iteration formula and its convergence to the exact solution for solving system of fractional order differential equations. Some examples are solved as an illustration to the method, in order to show the accuracy of the method and comparison with results the exact solution for the given test problems, which are given for comparison purpose.

Two computational approaches for solving a fractional obstacle system in Hilbert space

The primary motivation of this paper is to extend the application of the reproducing-kernel method (RKM) and the residual power series method (RPSM) to conduct a numerical investigation for a class of boundary value problems of fractional order 2α, 0<alphaleq1, concerned with obstacle, contact and unilateral problems. The RKM involves a variety of uses for emerging mathematical problems in the sciences, both for integer and non-integer (arbitrary) orders. The RPSM is combining the generalized Taylor series formula with the residual error functions. The fractional derivative is described in the Caputo sense. The representation of the analytical solution for the generalized fractional obstacle system is given by RKM with accurately computable structures in reproducing-kernel spaces. While the methodology of RPSM is based on the construction of a fractional power series expansion in rapidly convergent form and apparent sequences of solution without any restriction hypotheses. The recurrence form of the approximate function is selected by a well-posed truncated series that is proved to converge uniformly to the analytical solution. A comparative study was conducted between the obtained results by the RKM, RPSM and exact solution at different values of α. The numerical results confirm both the obtained theoretical predictions and the efficiency of the proposed methods to obtain the approximate solutions.

Existence of solutions for a higher order fractional boundary value problem posed on the half-line

The aim of this paper is to study the existence and uniqueness of solutions for a boundary value problem associated with a fractional nonlinear differential equation with higher order posed on the half-line. An appropriate continuous embedding for suitable Banach spaces are proved and the Minty–Browder theorem for monotone operators is used in the proof of existence of solutions for a boundary value problem of fractional order posed on the half-line.

Application of The Half-Sweep Egsor Iteration for Two-Point Boundary Value Problems of Fractional Order

Application of The Half-Sweep Egsor Iteration for Two-Point Boundary Value Problems of Fractional Order

On the Question of Asymptotic Integration of Singularly Perturbed Fractional-Order Problems

In this paper we consider an initial problem for systems of differential equations of fractional order with a small parameter for the derivative. Regularization problem is produced, and algorithm for normal and unique solubility of general iterative systems of differential equations with partial derivatives is given.Â

A collocation method for solving boundary value problems of fractional order

In this paper, the sinc collocation method is used to obtain the solution of the second-order fractional boundary value problems based on the conformable fractional derivative. For this purpose a theorem is proved to represent the terms having fractional derivatives in terms of sinc basis functions. To show the efficiency and accuracy of the present method, some problems are solved and the obtained solutions are compared with the approximate solutions obtained by using the other numerical methods as well as the exact solutions of the problems.

On functions without pseudo derivatives having fractional pseudo derivatives

A classical question of the fractional calculus of real-valued functions— the existence of a continuous nowhere differentiable function which has a continuous Riemann-Louville fractional derivative of any order less than one—was studied in detail by B. Ross et al. [16]). This paper deals with said question within the framework of abstract functions. Indeed, we give examples of weakly continuous vector-valued functions which fail to be pseudo differentiable but have weakly continuous fractional-pseudo derivatives of some critical order less than one. Based on these examples, it can be seen that even if an initial value problem of fractional order has a weakly continuous right-hand side, the equivalence between differential and integral form of the problem can be lost.

An efficient algorithm based on Gegenbauer wavelets for the solutions of variable-order fractional differential equations

The article is devoted to a new computational algorithm based on the Gegenbauer wavelets (GWs) to solve the linear and nonlinear variable-order fractional differential equations. The novel operational matrices for derivatives of positive integer and variable order are derived. New piecewise functions are introduced to obtain the said operational matrices. In the proposed method, the given problem via Gegenbauer wavelets is transformed to a system of algebraic equations. The obtained solutions are endorsing the accuracy and efficiency of the suggested method and are in excellent agreement with the existing literature. The convergence and error bound analysis are presented in our study to show the credibility of the computational method and support the mathematical formulation of the algorithm. The discussed problems reconfirm the appropriateness of said algorithm and perceived that the proposed algorithm is an efficient tool to tackle the nonlinear fractional order problems of complex nature.

The Multiplicity of Positive Solutions for Systems of Fractional Boundary Value Problems

This paper focuses on the multiple positive solutions for a coupled system of nonlinear boundary value problems of fractional order. Our approach is based on a fixed point theorem due to Bai and Ge. Also, an example is given to demonstrate the applicability of our main result.

Lyapunov Exponents of a Discontinuous 4D Hyperchaotic System of Integer or Fractional Order

In this paper, the dynamics of local finite-time Lyapunov exponents of a 4D hyperchaotic system of integer or fractional order with a discontinuous right-hand side and as an initial value problem, are investigated graphically. It is shown that a discontinuous system of integer or fractional order cannot be numerically integrated using methods for continuous differential equations. A possible approach for discontinuous systems is presented. To integrate the initial value problem of fractional order or integer order, the discontinuous system is continuously approximated via Filippov’s regularization and Cellina’s Theorem. The Lyapunov exponents of the approximated system of integer or fractional order are represented as a function of two variables: as a function of two parameters, or as a function of the fractional order and one parameter, respectively. The obtained three-dimensional representation leads to comprehensive conclusions regarding the nature, differences and sign of the Lyapunov exponents in both integer order and fractional order cases.

A new analysis of the Fornberg-Whitham equation pertaining to a fractional derivative with Mittag-Leffler-type kernel

The mathematical model of breaking of non-linear dispersive water waves with memory effect is very important in mathematical physics. In the present article, we examine a novel fractional extension of the non-linear Fornberg-Whitham equation occurring in wave breaking. We consider the most recent theory of differentiation involving the non-singular kernel based on the extended Mittag-Leffler-type function to modify the Fornberg-Whitham equation. We examine the existence of the solution of the non-linear Fornberg-Whitham equation of fractional order. Further, we show the uniqueness of the solution. We obtain the numerical solution of the new arbitrary order model of the non-linear Fornberg-Whitham equation with the aid of the Laplace decomposition technique. The numerical outcomes are displayed in the form of graphs and tables. The results indicate that the Laplace decomposition algorithm is a very user-friendly and reliable scheme for handling such type of non-linear problems of fractional order.

Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial

Solving differential equations of fractional (i.e., non-integer) order in an accurate, reliable and efficient way is much more difficult than in the standard integer-order case; moreover, the majority of the computational tools do not provide built-in functions for this kind of problem. In this paper, we review two of the most effective families of numerical methods for fractional-order problems, and we discuss some of the major computational issues such as the efficient treatment of the persistent memory term and the solution of the nonlinear systems involved in implicit methods. We present therefore a set of MATLAB routines specifically devised for solving three families of fractional-order problems: fractional differential equations (FDEs) (also for the non-scalar case), multi-order systems (MOSs) of FDEs and multi-term FDEs (also for the non-scalar case); some examples are provided to illustrate the use of the routines.

Numerical Solution for Initial and Boundary Value Problems of Fractional Order

Fractional calculus has been used in many fields, such as engineering, population, medicine, fluid mechanics and different fields of chemistry and physics. These fields were found to be best described using fractional differential equations (FDEs) to model their processes and equations. One of the well-known methods for solving fractional differential equations is the Shifted Legendre operational matrix (LOM) method. In this article, I proposed a numerical method based on Shifted Legendre polynomials for solving a class of fractional differential equations. A fractional order operational matrix of Legendre polynomials is also derived where the fractional derivatives are described by the Caputo derivative sense. By using the operational matrix, the initial and boundary equations are transformed into the products of several matrixes and by scattering the coefficients and the products of matrixes. I got a system of linear equations. Results obtained by using the proposed method (LOM) presented here show that the numerical method is very effective and appropriate for solving initial and boundary value problems of fractional ordinary differential equations. Moreover, some numerical examples are provided and the comparison is presented between the obtained results and those analytical results achieved that have proved the method’s validity.

Novel solution methods for initial boundary value problems of fractional order with conformable differentiation

In this work, we develop a formulation for the approximate-analytical solution of fractional partial differential equations (PDEs) by using conformable fractional derivative. Firstly, we redefine the conformable fractional Adomian decomposition method (CFADM) and conformable fractional modified homotopy perturbation method (CFMHPM). Then, we solve some initial boundary value problems (IBVP) by using the proposed methods, which can analytically solve the fractional partial differential equations (FPDE). In order to show the efficiencies of these methods, we have compared the numerical and exact solutions of the IBVP. Also, we have found out that the proposed models are very efficient and powerful techniques in finding approximate solutions for the IBVP of fractional order in the conformable sense.