Linear Fractional Differential Equations Research Articles

Mittag–Leffler stability of numerical solutions to time fractional ODEs

The asymptotic stable region and long-time decay rate of solutions to linear homogeneous Caputo time fractional ordinary differential equations (F-ODEs) are known to be completely determined by the eigenvalues of the coefficient matrix. Very different from the exponential decay of solutions to classical ODEs, solutions of F-ODEs decay only polynomially, leading to the so-called Mittag-Leffler stability, which was already extended to semi-linear F-ODEs with small perturbations. This work is mainly devoted to the qualitative analysis of the long-time behavior of numerical solutions. By applying the singularity analysis of generating functions developed by Flajolet and Odlyzko (SIAM J. Disc. Math. 3(2), 216–240, 1990), we are able to prove that both $$\mathcal {L}$$ 1 scheme and strong A-stable fractional linear multistep methods (F-LMMs) can preserve the numerical Mittag-Leffler stability for linear homogeneous F-ODEs exactly as in the continuous case. Through an improved estimate of the discrete fractional resolvent operator, we show that strong A-stable F-LMMs are also Mittag-Leffler stable for semi-linear F-ODEs under small perturbations. For the numerical schemes based on $$\alpha$$ -difference approximation to Caputo derivative, we establish the Mittag-Leffler stability for semi-linear problems by making use of properties of the Poisson transformation. The new results and analyses provide not only the rigorous justifications and explanations of the Mittag-Leffler stability of numerical solutions with exact decay rate, but also establish some close connection between the continuous and discrete F-ODEs. Numerical experiments are presented for several typical time fractional evolutional equations, including time fractional sub-diffusion equations and semi-linear F-ODEs. All the numerical results exhibit the typical long-time polynomial decay rate, which is fully consistent with our theoretical predictions.

HADAMARD FRACTIONAL CALCULUS ON TIME SCALES

This study defines a Hadamard fractional sum by use of the time-scale theory. Then a [Formula: see text]-fractional difference is given and fundamental theorems are proved. Initial value problems of fractional difference equations are presented and their equivalent fractional sum equations are provided. The discrete Mittag-Leffler function solutions of linear fractional difference equations are obtained. It can be concluded that the new discrete fractional calculus of Hadamard type is well defined.

On linear fractional differential equations with variable coefficients

We study and solve linear ordinary differential equations, with fractional order derivatives of either Riemann–Liouville or Caputo types, and with variable coefficients which are either integrable or continuous functions. In each case, the solution is given explicitly by a convergent infinite series involving compositions of fractional integrals, and its uniqueness is proved in suitable function spaces using the Banach fixed point theorem. As a special case, we consider the case of constant coefficients, whose solutions can be expressed by using the multivariate Mittag–Leffler function. Some illustrative examples with potential applications are provided.

Local Fractional Aboodh Transform and its Applications to Solve Linear Local Fractional Differential Equations

In this work we focus on presenting a method for solving local fractional differential equations. This method based on the combination of the Aboodh transform with the local fractional derivative (we can call it local fractional Aboodh transform), where we have provided some important results and properties. We concluded this work by providing illustrative examples, through which we focused on solving some linear local fractional differential equations in order to obtain nondifferential analytical solutions.

Steklov problem for a linear ordinary fractional delay differential equation with the Riemann-Liouville derivative

This paper studies a nonlocal boundary value problem with Steklov’s conditions of the first type for a linear ordinary delay differential equation of a fractional order with constant coefficients. The Green’s function of the problem with its properties is found. The solution to the problem is obtained explicitly in terms of the Green’s function. A condition for the unique solvability of the problem is found, as well as the conditions under which the solvability condition is satisfied. The existence and uniqueness theorem is proved using the representation of the Green’s function and its properties, as well as the representation of the fundamental solution to the equation and its properties. The question of eigenvalues is investigated. The theorem on the finiteness of the number of eigenvalues is proved using the notation of the solution in terms of the generalized Wright function, as well as the asymptotic properties of the generalized Wright function as λ → ∞ and λ → −∞.

Numerical study of multi-order fractional differential equations with constant and variable coefficients

In this manuscript, a numerical method based on the conjunction of Paraskevopoulos's algorithm and operational matrices is developed to solve numerically the multi-order linear and nonlinear fractional differential equations. By means of this conjunction, the multi-order problem is decomposed into a system of differential equations of fractional order which are then solved by employing operational matrices approach. The accuracy and efficiency of the method is examined by taking some examples. In addition, the numerical results presented in Pak et al. [Analytical solutions of linear inhomogeneous fractional differential equation with continuous variable coefficients. Adv Differ Equ. 2019;2019(256):1–22] are improved in our work.

Investigation to analytic solutions of modified conformable time–space fractional mixed partial differential equations

In this paper, the invariant subspace method (ISM) is developed to obtain the exact solution of linear and nonlinear time and space fractional mixed partial differential equations involving modified conformable fractional derivative (MCFD). Moreover, the method is successfully extended to the coupled system of modified conformable fractional differential equations. Variety of time-space fractional mixed PDEs are considered and solved to illustrate the established result of ISM. Finally, a comparison between exact solution of fractional PDE in the sense of MCFD and Caputo fractional derivative (Ca-FD) is presented through some examples of time–space fractional mixed partial differential equations.

A Numerical Method for Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

Nowadays, fractional derivative is used to model various problems in science and engineering. In this paper, a new numerical method to approximate the generalized Hattaf fractional derivative involving a nonsingular kernel is proposed. This derivative included several forms existing in the literature such as the Caputo–Fabrizio fractional derivative, the Atangana–Baleanu fractional derivative, and the weighted Atangana–Baleanu fractional derivative. The new proposed method is based on Lagrange polynomial interpolation, and it is applied to solve linear and nonlinear fractional differential equations (FDEs). In addition, the error made during the approximation of the FDEs using our proposed method is analyzed. By comparing the approximate and exact solutions, it is noticed that the new numerical method is very efficient and converges very quickly to the exact solution. Furthermore, our proposed numerical method is also applied to nonlinear systems of FDEs in virology.

Two-point nonlocal nonlinear fractional boundary value problem with Caputo derivative: Analysis and numerical solution

Abstract This work presents the existential and unique results for the solution to a kind of high-order fractional nonlinear differential equations involving Caputo fractional derivative. The boundary condition is of the integral type, which entangles both starting and ending points of the domain. First, the unique exact solution is extracted in terms of Green’s function for the linear fractional differential equation, and then Banach contraction mapping theorem is applied to prove the main result in the case of the general nonlinear source term. Then, our main result is demonstrated by an illustrative example, which shows its legitimacy and applicability. Furthermore, numerical-based semi-analytical technique has been presented to approximate the unique solution to the desired order of precision.

Exact Solvability Conditions for the Non-Local Initial Value Problem for Systems of Linear Fractional Functional Differential Equations

The exact conditions sufficient for the unique solvability of the initial value problem for a system of linear fractional functional differential equations determined by isotone operators are established. In a sense, the conditions obtained are optimal. The method of the test elements intended for the estimation of the spectral radius of a linear operator is used. The unique solution is presented by the Neumann’s series. All theoretical investigations are shown in the examples. A pantograph-type model from electrodynamics is studied.

Finite-time H∞ control of linear singular fractional differential equations with time-varying delay

Abstract In this paper, we propose an efficient analytical approach based on fractional calculus and singularity value theory to designing the finite-time $H_\infty $ controller for linear singular fractional differential equations with time-varying delay. By introducing new fractional-order $H_\infty $ norm, the state feedback controller is designed to guarantee that the closed-loop system is singular, impulse-free and finite-time stable with prescribed $H_\infty $ performance. New sufficient conditions for designing the $H_\infty $ finite-time controller are presented. The results of this paper improve the corresponding ones of integer-order singular systems with time-varying delay. Finally, a numerical example demonstrates the validity and effectiveness of the proposed theoretical results.

A new Chelyshkov matrix method to solve linear and nonlinear fractional delay differential equations with error analysis

A new Chelyshkov matrix method to solve linear and nonlinear fractional delay differential equations with error analysis

Mikusiński's operational calculus for Prabhakar fractional calculus

ABSTRACT The operational calculus method of Mikusiński is a powerful yet underappreciated theory, enabling differential equations to be solved using abstract algebra. We interpret the fractional integrals and derivatives of Prabhakar type within Mikusiński's theory, by defining them on appropriate function spaces and using judiciously their fundamental properties such as semigroup and series formulae. This theoretical framework enables explicit solutions to be found, for the first time, for a general class of linear Prabhakar fractional differential equations, with the solutions expressed as double or multiple infinite series.

Numerical Solution of Linear Fractional Differential Equation with Delay Through Finite Difference Method

This article addresses a new numerical method to find a numerical solution of the linear delay differential equation of fractional order , the fractional derivatives described in the Caputo sense. The new approach is to approximating second and third derivatives. A backward finite difference method is used. Besides, the composite Trapezoidal rule is used in the Caputo definition to match the integral term. The accuracy and convergence of the prescribed technique are explained. The results are shown through numerical examples.

Application of modified extended tanh method in solving fractional order coupled wave equations

In this paper, our purpose is to analyse the dynamical behaviour of the solitary waves of coupled wave equations, namely fractional order modified Volterra’s equations. Here, the local fractional derivative viz. beta derivative has been used and the solutions are analysed for different values of the parameter β. The fractional order modified Volterra’s equations consist of two non linear fractional partial differential equations. Fractional complex transformation has been applied to convert fractional order non linear partial differential equations into integer order non linear ordinary differential equations. The modified extended tanh method is used to find ample solitary wave solutions of fractional order modified Volterra’s equations. The solutions of modified Volterra’s equations have been given in the sense of Riccati equation. Proposed method has been utilized to pluck out periodic singular soliton and kink shape solution, which are illustrated by 2D and 3D plots.

Numerical solution of fractional differential equations with Caputo derivative by using numerical fractional predict\u2013correct technique

Fractional differential equations have recently demonstrated their importance in a variety of fields, including medicine, applied sciences, and engineering. The main objective of this study is to propose an Adams-type multistep method for solving differential equations of fractional order. The method is developed by implementing the Lagrange interpolation and taking into account the idea of the Adams–Moulton method for fractional case. The fractional derivative applied in this study is in the Caputo derivative operator. The analysis of the proposed method is presented in terms of order of the method, order of accuracy, and convergence analysis, with the proposed method being proved to converge. The stability of the method is also examined, where the stability regions appear to be symmetric to the real axis for various values of α. In order to validate the competency of the proposed method, several numerical examples for solving linear and nonlinear fractional differential equations are included. The method will be presented in the numerical predict–correct technique for the condition where alpha in (0,1), in which α represents the order of fractional derivatives of D^{alpha }y(t).

The Mixed Boundary Value Problems and Chebyshev Collocation Method for Caputo-Type Fractional Ordinary Differential Equations

The boundary value problem (BVP) for the varying coefficient linear Caputo-type fractional differential equation subject to the mixed boundary conditions on the interval 0≤x≤1 was considered. First, the BVP was converted into an equivalent differential–integral equation merging the boundary conditions. Then, the shifted Chebyshev polynomials and the collocation method were used to solve the differential–integral equation. Varying coefficients were also decomposed into the truncated shifted Chebyshev series such that calculations of integrals were only for polynomials and can be carried out exactly. Finally, numerical examples were examined and effectiveness of the proposed method was verified.

Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann–Liouville Sense

In this paper, we first consider the general fractional derivatives of arbitrary order defined in the Riemann–Liouville sense. In particular, we deduce an explicit form of their null space and prove the second fundamental theorem of fractional calculus that leads to a closed form formula for their projector operator. These results allow us to formulate the natural initial conditions for the fractional differential equations with the general fractional derivatives of arbitrary order in the Riemann–Liouville sense. In the second part of the paper, we develop an operational calculus of the Mikusiński type for the general fractional derivatives of arbitrary order in the Riemann–Liouville sense and apply it for derivation of an explicit form of solutions to the Cauchy problems for the single- and multi-term linear fractional differential equations with these derivatives. The solutions are provided in form of the convolution series generated by the kernels of the corresponding general fractional integrals.

USER-ORIENTED INTELLIGENCE MINING UNDER THE EXISTENCE OF SOLUTIONS TO INTEGRAL BOUNDARY VALUE PROBLEMS FOR FUZZY PARTIAL FRACTIONAL DIFFERENTIAL EQUATIONS

In order to accurately perceive user intention and improve the reliability of user intention information mining, the existence of solution of integral boundary value problem of fuzzy partial fractional differential equation is used to mine the information of user intention. Firstly, the periodic boundary value problem of fractional order fuzzy linear differential equation, periodic boundary value problem of fractional order fuzzy nonlinear differential equation and periodic boundary value problem of fractional order fuzzy coupled differential system are described in this research. The existence of the solution of the integral boundary value problem of the fuzzy partial fractional order differential equation is proved. Then a user-oriented information mining framework and an evaluation model based on the existence of solutions for the integral boundary value problem of fuzzy partial fractional differential equations are constructed. Finally, this research makes a case study and a comprehensive evaluation of user-oriented information mining based on the existence of solutions of integral boundary value problems for fuzzy partial fractional differential equations. The results show that the method based on the existence of the solution of the integral boundary value problem of fuzzy partial fractional order differential equation is feasible and scientific for the information mining of user intention. It is also concluded that this method is suitable for the modelling and solving of intelligence mining with complicated and unclear user intention. The research provides a good guidance for information mining oriented to user intention.

A new Bell function approach to solve linear fractional differential equations

In this study, a new collocation approach based on the Bell polynomials is presented to solve linear fractional differential equations. The aim of this study is to find the approximate solutions of linear fractional differential equations in the truncated Bell series. Firstly, fractional Bell functions are defined by generalizing the Bell polynomials with fractional powers. The generalized Bell functions are expressed in matrix form. Secondly, by using the Caputo derivative, the derivatives of the assumed solution form are transformed to the matrix forms. The matrix forms of the desired solution and its derivatives are substituted in the equation. By using equally spaced collocation points, the linear fractional equation is reduced to a system of linear algebraic equations. The obtained system is written in the compact form. The matrix forms of the conditions in the problem are formed by using the matrix form of the desired solution. Hence, a new linear algebraic system is obtained by means of the previous system and the system of the conditions. The obtained last system is solved and thus its solutions give the coefficients of the desired solution. The error and convergence analysis of the method is studied. Lastly, to show the applicability and efficiency of the technique, the method is applied to numerical examples.