A new operational matrix based on Bernoulli wavelets for solving fractional delay differential equations

In this research, a Bernoulli wavelet operational matrix of fractional integration is presented. Bernoulli wavelets and their properties are employed for deriving a general procedure for forming this matrix. The application of the proposed operational matrix for solving the fractional delay differential equations is explained. Also, upper bound for the error of operational matrix of the fractional integration is given. This operational matrix is utilized to transform the problem to a set of algebraic equations with unknown Bernoulli wavelet coefficients. Several numerical examples are solved to demonstrate the validity and applicability of the presented technique.     

Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations

Haar wavelet operational matrix has been widely applied in system analysis, system identification, optimal control and numerical solution of integral and differential equations. In the present paper we derive the Haar wavelet operational matrix of the fractional order integration, and use it to solve the fractional order differential equations including the Bagley-Torvik, Ricatti and composite fractional oscillation equations. The results obtained are in good agreement with the existing ones in open literatures and it is shown that the technique introduced here is robust and easy to apply.     

分数阶积分微分方程的近似解

本文给出了分数阶积分微分方程的一种新的解法.利用未知函数的泰功多项式展开将分数阶积分微分方程近拟转化为一个涉及未知函数及其n阶导数的线性方程组.数值例子表明该方法的有效性.     

A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order

In this paper we present a computational method for solving a class of nonlinear Fredholm integro-differential equations of fractional order which is based on CAS (Cosine And Sine) wavelets. The CAS wavelet operational matrix of fractional integration is derived and used to transform the equation to a system of algebraic equations. Some examples are included to demonstrate the validity and applicability of the technique.     

Numerical solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach

In this paper, a numerical method is presented to obtain and analyze the behavior of numerical solutions of distributed order fractional differential equations of the general form in the time domain with the Caputo fractional derivative. The suggested method is based on the Müntz–Legendre wavelet approximation. We derive a new operational vector for the Riemann–Liouville fractional integral of the Müntz–Legendre wavelets by using the Laplace transform method. Applying this operational vector and collocation method in our approach, the problem can be reduced to a system of linear and nonlinear algebraic equations. The arising system can be solved by the Newton method. Discussion on the error bound and convergence analysis for the proposed method is presented. Finally, seven test problems are considered to compare our results with other well‐known methods used for solving these problems. The results in the tabulated tables highlighted that the proposed method is an efficient mathematical tool for analyzing distributed order fractional differential equations of the general form.     

Numerical solution of the fractional Bagley‐Torvik equation by using hybrid functions approximation

In this paper, a new numerical method for solving the fractional Bagley‐Torvik equation is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block‐pulse functions and Bernoulli polynomials are presented. The Riemann‐Liouville fractional integral operator for hybrid functions is introduced. This operator is then utilized to reduce the solution of the initial and boundary value problems for the fractional Bagley‐Torvik differential equation to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2015 John Wiley & Sons, Ltd.     

A numerical scheme based on Bernoulli wavelets and collocation method for solving fractional partial differential equations with Dirichlet boundary conditions

In this paper, an efficient and accurate numerical method is presented for solving two types of fractional partial differential equations. The fractional derivative is described in the Caputo sense. Our approach is based on Bernoulli wavelets collocation techniques together with the fractional integral operator, described in the Riemann‐Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations, which greatly simplifies the problem. By using Newton's iterative method, this system is solved and the solution of fractional partial differential equations is achieved. Some results concerning the error analysis are obtained. The validity and applicability of the method are demonstrated by solving four numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions much easier.     

A numerical method for solving boundary value problems for fractional differential equations

A numerical scheme, based on the Haar wavelet operational matrices of integration for solving linear two-point and multi-point boundary value problems for fractional differential equations is presented. The operational matrices are utilized to reduce the fractional differential equation to system of algebraic equations. Numerical examples are provided to demonstrate the accuracy and efficiency and simplicity of the method.     

Generalized wavelet quasilinearization method for solving population growth model of fractional order

The primary aim of this study is to introduce and develop a generalized wavelet method together with the quasilinearization technique to solve the Volterra's population growth model of fractional order. Unlike the existing operational matrix methods based on orthogonal functions, we formulate the wavelet operational matrices of general order integration without using the block pulse functions. Consequently, the governing problem is transformed into an equivalent system of algebraic equations, which can be tackled with any classical method. The applicability of the proposed method is demonstrated via an illustrative comparison of the numerical outcomes with those found by other known methods. The experimental outcomes demonstrate that the proposed method is fast, accurate, simple, and computationally reliable.     

Numerical solution of fractional differential equations using cubic B-spline wavelet collocation method

Physical processes with memory and hereditary properties can be best described by fractional differential equations due to the memory effect of fractional derivatives. For that reason reliable and efficient techniques for the solution of fractional differential equations are needed. Our aim is to generalize the wavelet collocation method to fractional differential equations using cubic B-spline wavelet. Analytical expressions of fractional derivatives in Caputo sense for cubic B-spline functions are presented. The main characteristic of the approach is that it converts such problems into a system of algebraic equations which is suitable for computer programming. It not only simplifies the problem but also speeds up the computation. Numerical results demonstrate the validity and applicability of the method to solve fractional differential equation.     

Analysis of differential equations of fractional order

This paper provides a robust convergence checking method for nonlinear differential equations of fractional order with consideration of homotopy perturbation technique. The differential operators are taken in the Caputo sense. Some theorems to prove the existence and uniqueness of the series solutions are presented. Results show that the proposed theoretical analysis is accurate.     

Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet

In this paper, we first construct the second kind Chebyshev wavelet. Then we present a computational method based on the second kind Chebyshev wavelet for solving a class of nonlinear Fredholm integro-differential equations of fractional order. The second kind Chebyshev wavelet operational matrix of fractional integration is derived and used to transform the equation to a system of algebraic equations. The method is illustrated by applications and the results obtained are compared with the existing ones in open literature. Moreover, comparing the methodology with the known technique shows that the present approach is more efficient and more accurate.     

Numerical approach to differential equations of fractional order

In this paper, the variational iteration method and the Adomian decomposition method are implemented to give approximate solutions for linear and nonlinear systems of differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper presents a numerical comparison between the two methods for solving systems of fractional differential equations. Numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order.     

Analytical approximations for a population growth model with fractional order

In this paper, we apply the homotopy analysis method (HAM) to solve the fractional Volterra’s model for population growth of a species in a closed system. This technique is extended to give solutions for nonlinear fractional integro–differential equations. The whole HAM solution procedure for nonlinear fractional differential equations is established. Further, the accurate analytical approximations are obtained for the first time, which are valid and convergent for all time t. This indicates the validity and great potential of the homotopy analysis method for solving nonlinear fractional integro–differential equations.     

Discretization algorithm for fractional order integral by Haar wavelet approximation

A discretization algorithm is proposed by Haar wavelet approximation theory for the fractional order integral. In this paper, the integration time is divided into two parts, one presents the effect of the past sampled data, calculated by the iterative method, and the other presents the effect of the recent sampled data at a fixed time interval, calculated by the Haar wavelet. This method can reduce the amount of the stored data effectively and be applied to the design of discrete-time fractional order PID controllers. Finally, several numerical examples and simulation results are given to illustrate the validity of this discretization algorithm.     

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

In this paper, shifted Legendre polynomials will be used for constructing the numerical solution for a class of multiterm variable‐order fractional differential equations. In the proposed method, the shifted Legendre operational matrix of the fractional variable‐order derivatives will be investigated. The fundamental problem is reduced to an algebraic system of equations using the constructed matrix and the collocation technique, which can be solved numerically. The error estimate of the proposed method is investigated. Some numerical examples are presented to prove the applicability, generality, and accuracy of the suggested method.     

Numerical methods for nonlinear partial differential equations of fractional order

In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. Numerical results show that the two approaches are easy to implement and accurate when applied to partial differential equations of fractional order.     

A new operational matrix of fractional order integration for the Chebyshev wavelets and its application for nonlinear fractional Van der Pol oscillator equation

In this paper, an efficient and accurate computational method based on the Chebyshev wavelets (CWs) together with spectral Galerkin method is proposed for solving a class of nonlinear multi-order fractional differential equations (NMFDEs). To do this, a new operational matrix of fractional order integration in the Riemann–Liouville sense for the CWs is derived. Hat functions (HFs) and the collocation method are employed to derive a general procedure for forming this matrix. By using the CWs and their operational matrix of fractional order integration and Galerkin method, the problems under consideration are transformed into corresponding nonlinear systems of algebraic equations, which can be simply solved. Moreover, a new technique for computing nonlinear terms in such problems is presented. Convergence of the CWs expansion in one dimension is investigated. Furthermore, the efficiency and accuracy of the proposed method are shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient. As a useful application, the proposed method is applied to obtain an approximate solution for the fractional order Van der Pol oscillator (VPO) equation.     

Lie symmetry analysis to Fisher''s equation with time fractional order

The aim of this letter is to apply the Lie group analysis method to the Fisher''s equation with time fractional order. We considered the symmetry analysis, explicit solutions to the time fractional Fisher''s(TFF) equations with Riemann-Liouville (R-L) derivative. The time fractional Fisher''s is reduced to respective nonlinear ordinary differential equation(ODE) of fractional order. We solve the reduced fractional ODE using an explicit power series method.     

Numerical comparison of methods for solving linear differential equations of fractional order

In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving linear differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper will present a numerical comparison between the two methods and a conventional method such as the fractional difference method for solving linear differential equations of fractional order. The numerical results demonstrates that the new methods are quite accurate and readily implemented.