Linear Fractional Differential Equations Research Articles

On artificial neural networks approach with new cost functions

In this manuscript, the artificial neural networks approach involving generalized sigmoid function as a cost function, and three-layered feed-forward architecture is considered as an iterative scheme for solving linear fractional order ordinary differential equations. The supervised back-propagation type learning algorithm based on the gradient descent method, is able to approximate this a problem on a given arbitrary interval to any desired degree of accuracy. To be more precise, some test problems are also given with the comparison to the simulation and numerical results given by another usual method.

A unified spectral method for FPDEs with two-sided derivatives; Part II: Stability, and error analysis

We present the stability and error analysis of the unified Petrov–Galerkin spectral method, developed in [1], for linear fractional partial differential equations with two-sided derivatives and constant coefficients in any (1+d)-dimensional space-time hypercube, d=1,2,3,⋯, subject to homogeneous Dirichlet initial/boundary conditions. Specifically, we prove the existence and uniqueness of the weak form and perform the corresponding stability and error analysis of the proposed method. Finally, we perform several numerical simulations to compare the theoretical and computational rates of convergence.

Conformable Laplace Transform of Fractional Differential Equations

In this paper, we use the conformable fractional derivative to discuss some fractional linear differential equations with constant coefficients. By applying some similar arguments to the theory of ordinary differential equations, we establish a sufficient condition to guarantee the reliability of solving constant coefficient fractional differential equations by the conformable Laplace transform method. Finally, the analytical solution for a class of fractional models associated with the logistic model, the von Foerster model and the Bertalanffy model is presented graphically for various fractional orders. The solution of the corresponding classical model is recovered as a particular case.

Determination of Order in Linear Fractional Differential Equations

Abstract The order of fractional differential equations (FDEs) has been proved to be of great importance in an accurate simulation of the system under study. In this paper, the orders of some classes of linear FDEs are determined by using the asymptotic behaviour of their solutions. Specifically, it is demonstrated that the decay rate of the solutions is influenced by the order of fractional derivatives. Numerical investigations are conducted into the proven formulas.

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.

Some notes on properties of the matrix Mittag-Leffler function

We have come across with publications where some properties of the matrix exponential were incorrectly extended to the matrix Mittag-Leffler function, and then used as a key tool to solve certain linear matrix fractional differential equations. The main purpose of these notes is to give some clarifications on properties of the matrix Mittag-Leffler function, by explaining in detail why some identities do not hold and by providing a list of (valid) properties of this function. A sufficient condition to identify very special cases of pairs of matrices satisfying the semigroup property is given as well as examples illustrating the results.

Hybrid Method for Solution of Fractional Order Linear Differential Equation with Variable Coefficients

Abstract In this paper, we proposed a new analytical hybrid methods for the solution of conformable fractional differential equations (CFDE), which are based on the recently proposed conformable fractional derivative (CFD) in R. Khalil, M. Al Horani, A. Yusuf and M. Sababhed, A New definition of fractional derivative, J. Comput. Appl. 264 (2014). Moreover, we use the method of variation of parameters and reduction of order based on CFD, for the CFDE. Furthermore, to show the efficiency of the proposed analytical hybrid method, some examples are also presented.

A generalization of the Mittag\u2013Leffler function and solution of system of fractional differential equations

The solutions of system of linear fractional differential equations of incommensurate orders are considered and analytic expressions for the solutions are given by using the Laplace transform and multi-variable Mittag–Leffler functions of matrix arguments. We verify the result with numeric solutions of an example. The results show that the Mittag–Leffler functions are important tools for analysis of a fractional system. The analytic solutions obtained are easy to program and are approximated by symbolic computation software such as MATHEMATICA.

System of linear fractional differential equations and the Mittag-Leffler functions with matrix variable

The solution for system of linear fractional differential equations is derived in terms of the Mittag-Leffler functions with matrix variable. Three different methods for calculating the Mittag-Leffler functions with matrix variable are obtained with the help of inverse Laplace transform, Jordan canonical matrix and minimal polynomial, respectively. The solution for system of linear first-order differential equations is obtained as a special case. The results show that the Mittag-Leffler functions with matrix variable are powerful tools for solving system of linear fractional differential equations.

Variational iteration method combined with new transform to solve fractional partial differential equations

The aim of this paper is to combined the variational iteration method with Aboodh transform method to solve linear and nonlinear fractional partial differential equations. Some illustrative examples are given as the linear and nonlinear fractional Klein-Gordon equations and the time fractional diffusion equation. The results reveal that this method is very effective, simple and can be applied to other physical differential equations with fractional order. The fractional derivative is taken in the Caputo sense.

Theories and applications of the inverse fractional natural transform method

In this paper, we propose a new method called the inverse fractional natural transform method (IFNTM). We present theoretical results and apply them to obtain approximate solutions of linear fractional ordinary differential equations (LFODEs) and partial differential equations (LFPDEs). The fractional derivatives are described in the Caputo sense. The algorithm described in this study is expected to be further employed to solve similar linear problems in fractional calculus.

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English

Fractional collocation methods for multi-term linear and nonlinear fractional differential equations with variable coefficients on the half line

ABSTRACTIn this paper, two new and efficient fractional spectral collocation schemes are proposed to solve the multi-term linear and nonlinear fractional differential equations (FDEs) with variable coefficients on the half line, both for the Riemann–Liouville and Caputo fractional operators. Our main task is to find corresponding fractional pseudospectral differentiation matrices (F-PSDMs) which are computed by two different methods. The first one is based on the three-term recurrence relations and the derivative recurrence relations of the generalized Laguerre polynomials. The second one is based on the direct fractional integral and derivative relations of the generalized Laguerre polynomials. Based on the F-PSDMs, the corresponding spectral collocation schemes are developed. By taking suitable parameter involved in the methods, we can significantly enhance the performance of these schemes. Moreover, both schemes are easy to extend to solve the variable-order FDEs. Numerical examples are presented, which show the efficiency and spectral accuracy of our methods.

Boundary value problem for linear and nonlinear fractional differential equations

In this paper, we are interested in the boundary conditions u(0)=au(1),u′(0)=bu′(1) for linear fractional differential equation 0cDtαu(t)+λu(t)=0. Via Laplace transform and inverse Laplace transform, we obtain eigenvalues and eigenfunctions. Furthermore, we study the same boundary conditions for nonlinear fractional differential equation 0cDtαu(t)+f(t,u(t))=0. Combining the obtained eigenvalues and eigenfunctions and the improved Leray–Schauder degree, we prove that there exists at least one nontrivial solution for nonlinear boundary value problem.

Representation of solution of a Riemann–Liouville fractional differential equation with pure delay

This paper derives a representation of a solution to the initial value problem for a linear fractional delay differential equation with Riemann–Liouville derivative. We apply the method of variation of constants to obtain the representation of a solution via a delayed Mittag-Leffler type matrix function.

Transforming linear FDEs with rational orders into ODEs by modified differential operator multiplication method

This paper presents a modified differential operator multiplication (DOM) method for solving a certain class of fractional differential equations (FDEs), with emphasis on linear oscillators subjected to periodic excitation. The main idea of DOM is to transform the considered FDEs with rational order r/m into rth-order ordinary differential equations (ODEs), herein r and m are positive integers. The transformation is realized by differentiating the FDEs stepwise with fractional-order r/m, until the rth-order derivative is reached due to the accumulative property for FDs. In the modified method, differently, first-order ODEs are deduced by transforming the FDEs with order r/m into 1/m. In addition, we introduce auxiliary state space variables to represent the FDs of inhomogeneous terms such as triangle functions. Exact and explicit first-order ODEs are deduced and solved by the Runge–Kutta algorithm. To validate the modified DOM, we solve the fractional Kelvin-Voigt equation and harmonically excited oscillators with visco-elastic dampings. The presented method can provide highly accurate solution, at the expense of linearly increasing computational resources as the solution domain expands. Such high computation accuracy and efficiency would potentially enable the presented method to become a benchmark comparison for solving certain FDEs, especially harmonically excited fractional oscillators.

ELZAKI PROJECTED DIFFERENTIAL TRANSFORM METHOD FOR FRACTIONAL ORDER SYSTEM OF LINEAR AND NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATION

The pivotal aim of this paper is to propose an efficient computational technique, namely, Elzaki fractional projected differential transform method (EFPDTM) to solve the system of linear and nonlinear fractional differential equations. In the EFPDTM process, we investigate the behavior of independent variables for convergent series solution in admissible range. The EFPDTM manipulates and controls the series solution, which rapidly converges to the exact solution in a large admissible domain in a very efficient way. The solution procedure and explanation show the flexible efficiency of the EFPDTM, compared to the other existing classical techniques for solving the system of linear and nonlinear fractional differential equations.

Solutions of Some Linear Fractional Partial Differential Equations in Mathematical Physics

In this article, we use double Laplace transform method to find solution of general linear fractional partial differential equation in terms of Mittag-Leffler function subject to the initial and boundary conditions. The efficiency of the method is illustrated by considering fractional wave and diffusion equations, Klein-Gordon equation, Burger’s equation, Fokker-Planck equation, KdV equation, and KdV-Burger’s equation of mathematical physics.

Efficient Mittag-Leffler Collocation Method for Solving Linear and Nonlinear Fractional Differential Equations

In this paper, a new approximation method for fractional differential equations based on Mittag-Leffler function is developed. Finite Mittag-Leffler function and its fractional-order derivatives are investigated. An efficient technique for solving linear and nonlinear fractional order differential equations is developed. The proposed method combines Mittag-Leffler collocation method and optimization technique. Error estimation of the approximation is stated and proved. We present numerical results and comparisons of previous treatments to demonstrate the efficiency and applicability of the proposed method. Making use of small number of unknowns, the resulting solution converges to the exact one in the linear case and it has a very small error in the nonlinear case.

Fractional-order Legendre–Laguerre functions and their applications in fractional partial differential equations

In this paper, we consider a new fractional function based on Legendre and Laguerre polynomials for solving a class of linear and nonlinear time-space fractional partial differential equations with variable coefficients. The concept of the fractional derivative is utilized in the Caputo sense. The idea of solving these problems is based on operational and pseudo-operational matrices of integer and fractional order integration with collocation method. We convert the problem to a system of algebraic equations by applying the operational matrices, pseudo-operational matrices and collocation method. Also, we calculate the upper bound for the error of integral operational matrix of the fractional order. We illustrated the efficiency and the applicability of the approach by considering several numerical examples in the format of table and graph. We also describe the physical application of some examples.