Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations

We give a comparison of the efficiency of three alternative decomposition schemes for the approximate solution of multi-term fractional differential equations using the Caputo form of the fractional derivative. The schemes we compare are based on conversion of the original problem into a system of equations. We review alternative approaches and consider how the most appropriate numerical scheme may be chosen to solve a particular equation.     

On the approximate solutions for system of fractional integro-differential equations using Chebyshev pseudo-spectral method

In this paper, we implement Chebyshev pseudo-spectral method for solving numerically system of linear and non-linear fractional integro-differential equations of Volterra type. The proposed technique is based on the new derived formula of the Caputo fractional derivative. The suggested method reduces this type of systems to the solution of system of linear or non-linear algebraic equations. We give the convergence analysis and derive an upper bound of the error for the derived formula. To demonstrate the validity and applicability of the suggested method, some test examples are given. Also, we present a comparison with the previous work using the homotopy perturbation method.     

Operational solution of fractional differential equations

The main goal of this paper is to solve fractional differential equations by means of an operational calculus. Our calculus is based on a modified shift operator which acts on an abstract space of formal Laurent series. We adopt Weyl’s definition of derivatives of fractional order.     

A rational approximate solution for generalized pantograph‐delay differential equations

In this paper, a new rational approximation based on a rational interpolation and collocation method is proposed for the solutions of generalized pantograph equations. A comprehensive error analysis is provided. The first part of the error analysis gives an upper bound for the absolute error. The second part is based on residual error procedure that estimates the absolute error. Some numerical examples are given to illustrate the method. The theoretical results support the numerical results. Copyright © 2015 John Wiley & Sons, Ltd.     

Fundamental solution of a system of fractional partial differential equations

We consider a system of Riemann-Liouville fractional partial differential equations with constant coefficients and obtain a general representation of solutions in a rectangular domain. The asymptotic behavior and other properties of the fundamental solution are studied.     

On the concept of solution for fractional differential equations with uncertainty

We consider a differential equation of fractional order with uncertainty and present the concept of solution. It extends, for example, the cases of first order ordinary differential equations and of differential equations with uncertainty. Some examples are presented.     

Conditions for the existence of a solution of a system of differential equations close to an approximate solution

We consider nonautonomous systems of differential equations and state conditions for the existence of an exact solution in a neighborhood of an approximate one by analyzing the linear system of the first approximation in a neighborhood of the constructed approximate solution. We present conditions for the existence of a bounded solution of a linear inhomogeneous system of differential equations.     

The approximate solution of integro‐ differential equations

We consider a class of partial integro‐differential equations with appropriate conditions and its corresponding equation in the field of Mikusiński operators. As is usual in numerical analysis, we construct the corresponding difference equation, determine its solution, analyze its character and treat it as the approximate solution of the considered problem. We also estimate the error of approximation.     

Computational scheme for solving nonlinear fractional stochastic differential equations with delay

Abstract

This paper studies the numerical solution of fractional stochastic delay differential equations driven by Brownian motion. The proposed algorithm is based on linear B-spline interpolation. The convergence and the numerical performance of the method are analyzed. The technique is adopted for determining the statistical indicators of stochastic responses of fractional Langevin and Mackey-Glass models with stochastic excitations.     

On the concept and existence of solution for impulsive fractional differential equations

This paper is motivated from some recent papers treating the problem of the existence of a solution for impulsive differential equations with fractional derivative. We firstly show that the formula of solutions in cited papers are incorrect. Secondly, we reconsider a class of impulsive fractional differential equations and introduce a correct formula of solutions for a impulsive Cauchy problem with Caputo fractional derivative. Further, some sufficient conditions for existence of the solutions are established by applying fixed point methods. Some examples are given to illustrate the results.     

Numerical solution of fractional order differential equations by extrapolation

We present an extrapolation type algorithm for the numerical solution of fractional order differential equations. It is based on the new result that the sequence of approximate solutions of these equations, computed by means of a recently published algorithm by Diethelm [6], possesses an asymptotic expansion with respect to the stepsize. From this we conclude that the application of extrapolation is justified, and we obtain a very efficient differential equation solver with practically no additional numerical costs. This is also illustrated by a number of numerical examples. This revised version was published online in August 2006 with corrections to the Cover Date.     

Fast solvers for finite difference scheme of two-dimensional time-space fractional differential equations

Numerical Algorithms - Generally, solving linear systems from finite difference alternating direction implicit scheme of two-dimensional time-space fractional differential equations with Gaussian...     

A fractional characteristic method for solving fractional partial differential equations

The method of characteristics has played a very important role in mathematical physics. Previously, it has been employed to solve the initial value problem for partial differential equations of first order. In this work, we propose a new fractional characteristic method and use it to solve some fractional partial differential equations.     

On the approximate solution of the singular problem of Cauchy for ordinary differential equations

In the present paper an approximate solution of the singular problem of Cauchy for the ordinary differential equation of mth order is constructed and, by the method of finite differences, sufficient conditions are found for the convergence to the exact solution when the mesh width tends to zero.     

New existence results of positive solution for a class of nonlinear fractional differential equations

In this article, the existence and uniqueness of positive solution for a class of nonlinear fractional differential equations is proved by constructing the upper and lower control functions of the nonlinear term without any monotone requirement. Our main method to the problem is the method of upper and lower solutions and Schauder fixed point theorem. Finally, we give an example to illuminate our results.     

Comments on the concept of existence of solution for impulsive fractional differential equations

In some recent works dealing with the existence of solutions for impulsive fractional differential equations, it is pointed out that the concept of solutions for such equations in some preceding papers is incorrect. In support of this claim, the authors of these papers begin with a counterexample. The objective of this note to indicate the mistake in these counterexamples and show the plausibility of the previous results.     

Existence of the mild solution for some fractional differential equations with nonlocal conditions

We are concerned in this paper with the existence of mild solutions to the Cauchy Problem for the fractional differential equation with nonlocal conditions: D ^q x(t)=Ax(t)+t ⁿ f(t,x(t),Bx(t)), t∈[0,T], n∈ℤ⁺, x(0)+g(x)=x ₀, where 0<q<1, A is the infinitesimal generator of a C ₀-semigroup of bounded linear operators on a Banach space X.     

Numerical solution of differential algebraic equations using a multiquadric approximation scheme

The objective of this paper is to solve differential algebraic equations using a multiquadric approximation scheme. Therefore, we present the notation and basic definitions of the Hessenberg forms of the differential algebraic equations. In addition, we present the properties of the proposed multiquadric approximation scheme and its advantages, which include using data points in arbitrary locations with arbitrary ordering. Moreover, error estimation and the run time of the method are also considered. Finally some experiments were performed to illustrate the high accuracy and efficiency of the proposed method, even when the data points are scattered and have a closed metric.