Linear Multi-term Fractional Differential Equations Research Articles

Necessary and sufficient condition for the existence of solutions to two-point boundary value problem of fuzzy linear multi-term fractional differential equations

Necessary and sufficient condition for the existence of solutions to two-point boundary value problem of fuzzy linear multi-term fractional differential equations

Analysis of fractional differential equations with fractional derivative of generalized Mittag-Leffler kernel

In this paper, we study classes of linear and nonlinear multi-term fractional differential equations involving a fractional derivative with generalized Mittag-Leffler kernel. Estimates of fractional derivatives at extreme points are first obtained and then implemented to derive new comparison principles for related linear equations. These comparison principles are used to analyze the solutions of the linear multi-term equations, where norm estimates of solutions, uniqueness and several comparison results are established. For the nonlinear problem, we apply the Banach fixed point theorem to establish the existence of a unique solution.

A fractional spectral collocation method for general Caputo two-point boundary value problems

In this paper we consider a fractional spectral collocation method for solving linear multi-term fractional differential equations involving Caputo-type fractional derivative. By using integral equation method, the boundary value problem of the fractional differential equation is reformulated as an equivalent Volterra-Fredholm integral equation with weakly singular kernels. Then we apply the fractional spectral collocation method to solve the resulting equation. We ensure that the approximate equation has a unique solution and the approximate solution arrives at the optimal convergence order in the infinite norm. At last, numerical examples are given to confirm the theoretical results.

Novel numerical solution to the fractional neutron point kinetic equation in nuclear reactor dynamics

In this work, a novel numerical solution to modified Fractional Neutron Point Kinetic (FNPK) equations is presented. The method is based on a numerical solution to linear multi-term fractional differential equations taking from scientific literature. Differential-integral operators of fractional order are numerically solved with the novel method. The impact of the order of the operators has been assessed during the process of order reduction of the fractional differential-integral equation. The numerical solution is applied to case with sinusoidal reactivity, and different values of the anomalous diffusion order are used to study the effect on the neutron density. The results of the neutron density behavior obtained with this proposed numerical novel solution were compared against the classical neutron point kinetics equations and with other results from scientific literature. The comparison showed a clear improvement of the numerical results when using a fractional differential-integral operator instead of an only fractional differential operator.

New fractional Lanczos vector polynomials and their application to system of Abel–Volterra integral equations and fractional differential equations

In this paper, the recursive approach of the τ-method is developed to construct new fractional order canonical polynomials for solving systems of Abel–Volterra integral equations. Due to the singular behavior of solutions of these equations, the existing spectral approaches suffer from low accuracy. To overcome this drawback we use Müntz polynomials as basis functions which give remarkable approximation to functions with singular behavior at origin and state Tau approximation to the exact solution based on a new sequence of basis vector canonical polynomials that is generated by a simple recursive formula in terms of fractional order Müntz polynomials. The efficiency and simplicity of the proposed method are illustrated by some examples. Convergence analysis of the method is also discussed. The paper is closed by providing application of this method to a linear multi-term fractional differential equations.

Constructive Existence of (1,1)-Solutions to Two-Point Value Problems for Fuzzy Linear Multiterm Fractional Differential Equations

In this paper, we consider the following two-point boundary value problems of fuzzy linear fractional differential equations: (Dc1,1αy)(t)⊕b(t)⊗(Dc1,1βy)(t)⊕c(t)⊗y(t)=f(t), t∈(0,1), y(0)=y0 and y(1)=y1, where b,c∈C(I), b(t),c(t)≥0, y,f∈C(I,RF), I=[0,1], y0,y1∈RF and 1<β<α≤2. Our existence result is based on Banach fixed point theorem and the approximate solution of our problem is obtained by applying the Haar wavelet operational matrix.

Numerical solution of the conformable differential equations via shifted Legendre polynomials

ABSTRACTIn this paper, we are concerned with the linear and nonlinear multi-term fractional differential equations. Firstly, a new approximate formula of the conformable fractional derivative is derived. The proposed formula is based on the shifted Legendre polynomials. The spectral Legendre collocation method is presented for solving multi-term fractional differential equations. Here, the fractional derivatives are described in the conformable sense. Convergence analysis and estimate an error upper bound of the obtained approximate formula are given. The validity and the applicability of the proposed technique are shown by numerical examples.

Representation of solution of initial value problem for fuzzy linear multi-term fractional differential equation with continuous variable coefficient

We consider the representation of solutions of the initial value problems of fuzzy linear multi-term in-homogeneous fractional differential equations with continuous variable coefficients.

Numerical treatment of non-integer order partial differential equations by omitting discretization of data

In this manuscript, we derived the shifted Jacobi operational matrices of fractional derivatives and integration which are applied for numerical solution of general linear multi-term fractional partial differential equations (FPDEs). A new approach implementing shifted Jacobi operational matrix without using the shifted Jacobi collocation technique is introduced for the numerical solution of multi-term FPDEs. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. Further the proposed method needs no discretization of data. The proposed method is applied for solving linear multi-term FPDEs subject to initial conditions and the approximate solutions are obtained for some tested problems. Special attention is given to the comparison of the numerical results obtained by the considered method with those found by other known methods like homotopy analysis (HAM) method. For computation purposes, we use Matlab 2016.

Smoothing transformation and spline collocation for linear fractional boundary value problems

We construct and justify a high order method for the numerical solution of multi-point boundary value problems for linear multi-term fractional differential equations involving Caputo-type fractional derivatives. Using an integral equation reformulation of the boundary value problem we first regularize the solution by a suitable smoothing transformation. After that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid. Optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established. To illustrate the reliability of the proposed method some numerical results are given.

Modified spline collocation for linear fractional differential equations

We propose and analyze a class of high order methods for the numerical solution of initial value problems for linear multi-term fractional differential equations involving Caputo-type fractional derivatives. Using an integral equation reformulation of the initial value problem we first regularize the solution by a suitable smoothing transformation. After that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid. Optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established. Theoretical results are verified by some numerical examples.

Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems

We present a direct solution technique for approximating linear multiterm fractional differential equations (FDEs) on semi-infinite interval, using generalized Laguerre polynomials. We derive the operational matrix of Caputo fractional derivative of the generalized Laguerre polynomials which is applied together with generalized Laguerre tau approximation for implementing a spectral solution of linear multiterm FDEs on semi-infinite interval subject to initial conditions. The generalized Laguerre pseudo-spectral approximation based on the generalized Laguerre operational matrix is investigated to reduce the nonlinear multiterm FDEs and its initial conditions to nonlinear algebraic system, thus greatly simplifying the problem. Through several numerical examples, we confirm the accuracy and performance of the proposed spectral algorithms. Indeed, the methods yield accurate results, and the exact solutions are achieved for some tested problems.

On a Generalized Laguerre Operational Matrix of Fractional Integration

A new operational matrix of fractional integration of arbitrary order for generalized Laguerre polynomials is derived. The fractional integration is described in the Riemann-Liouville sense. This operational matrix is applied together with generalized Laguerre tau method for solving general linear multiterm fractional differential equations (FDEs). The method has the advantage of obtaining the solution in terms of the generalized Laguerre parameter. In addition, only a small dimension of generalized Laguerre operational matrix is needed to obtain a satisfactory result. Illustrative examples reveal that the proposed method is very effective and convenient for linear multiterm FDEs on a semi-infinite interval.

A Modified Generalized Laguerre Spectral Method for Fractional Differential Equations on the Half Line

This paper deals with modified generalized Laguerre spectral tau and collocation methods for solving linear and nonlinear multiterm fractional differential equations (FDEs) on the half line. A new formula expressing the Caputo fractional derivatives of modified generalized Laguerre polynomials of any degree and for any fractional order in terms of the modified generalized Laguerre polynomials themselves is derived. An efficient direct solver technique is proposed for solving the linear multiterm FDEs with constant coefficients on the half line using a modified generalized Laguerre tau method. The spatial approximation with its Caputo fractional derivatives is based on modified generalized Laguerre polynomialsLi(α,β)(x)withx∈Λ=(0,∞),α>−1, andβ>0, andiis the polynomial degree. We implement and develop the modified generalized Laguerre collocation method based on the modified generalized Laguerre-Gauss points which is used as collocation nodes for solving nonlinear multiterm FDEs on the half line.

A new modified generalized Laguerre operational matrix of fractional integration for solving fractional differential equations on the half line

In this paper, we derived a new operational matrix of fractional integration of arbitrary order for modified generalized Laguerre polynomials. The fractional integration is described in the Riemann-Liouville sense. This operational matrix is applied together with the modified generalized Laguerre tau method for solving general linear multi-term fractional differential equations (FDEs). Only small dimension of a modified generalized Laguerre operational matrix is needed to obtain a satisfactory result. Illustrative examples reveal that the present method is very effective and convenient for linear multi-term FDEs on a semi-infinite interval.

STABILITY-PRESERVING HIGH-ORDER METHODS FOR MULTITERM FRACTIONAL DIFFERENTIAL EQUATIONS

The numerical approximation of linear multiterm fractional differential equations is investigated. Convolution quadratures based on Runge–Kutta methods together with formulas for the efficient inversion of the Laplace transform are considered to provide highly accurate numerical solutions. Implementation issues are discussed and good stability properties are shown. The effectiveness of the algorithm is analyzed by means of some numerical experiments.

Piecewise polynomial collocation for linear boundary value problems of fractional differential equations

We consider a class of boundary value problems for linear multi-term fractional differential equations which involve Caputo-type fractional derivatives. Using an integral equation reformulation of the boundary value problem, some regularity properties of the exact solution are derived. Based on these properties, the numerical solution of boundary value problems by piecewise polynomial collocation methods is discussed. In particular, we study the attainable order of convergence of proposed algorithms and show how the convergence rate depends on the choice of the grid and collocation points. Theoretical results are verified by two numerical examples.

Finite Element Method for Linear Multiterm Fractional Differential Equations

We consider the linear multiterm fractional differential equation (fDE). Existence and uniqueness of the solution of such equation are discussed. We apply the finite element method (FEM) to obtain the numerical solution of this equation using Galerkin approach. A comparison, through examples, between our techniques and other previous numerical methods is established.

A new Jacobi operational matrix: An application for solving fractional differential equations

In this paper, we derived the shifted Jacobi operational matrix (JOM) of fractional derivatives which is applied together with spectral tau method for numerical solution of general linear multi-term fractional differential equations (FDEs). A new approach implementing shifted Jacobi operational matrix in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of nonlinear multi-term FDEs. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. The proposed methods are applied for solving linear and nonlinear multi-term FDEs subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.

Spline collocation methods for linear multi-term fractional differential equations

Some regularity properties of the solution of linear multi-term fractional differential equations are derived. Based on these properties, the numerical solution of such equations by piecewise polynomial collocation methods is discussed. The results obtained in this paper extend the results of Pedas and Tamme (2011) [15] where we have assumed that in the fractional differential equation the order of the highest derivative of the unknown function is an integer. In the present paper, we study the attainable order of convergence of spline collocation methods for solving general linear fractional differential equations using Caputo form of the fractional derivatives and show how the convergence rate depends on the choice of the grid and collocation points. Theoretical results are verified by some numerical examples.