Matrix Of Fractional Order Integration Research Articles

Operational Matrix of Fractional Order Integration and Its Application to Solve Fractional Differential Equations (FDEs) Using Haar Wavelet Collocation Method (HWCM)

Wavelets play an essential part in numerical analysis. In this study, a novel numerical technique to solve fractional differential equations (FDEs) corresponding to initial conditions is presented using Haar wavelet approximations. Haar wavelet is first presented with an operational matrix of fractional order integration. Then, illustrative examples are presented to signify the validity and applicability of the proposed method.

A New Method to Solve Dual Systems of Fractional Integro-Differential Equations by Legendre Wavelets

The method that will be presented, is numerical solution based on the Legendre wavelets for solving dual systems of fractional integro-differential equations (FIDEs). First of all we make the operational matrix of fractional order integration. The application of this matrix is transforming FIDEs to a system of algebric equations. By this changing, we are able to solve it by a simple solution. In this way, the Legendre wavelets and their operator matrix are the most important keys of our solution. After explaining the method we test on some illustrative examples which numerical solutions of these examples demonstrate the validity and applicability of suggested method.

Numerical Solution of Nonlinear Fractional Bratu Equation with Hybrid Method

This work, we studied the Bratu differential equation of the fractional-order, numerically via introducing a hybrid method. This method is a combination of the Chebyshev polynomials and the block-pulse wavelets matrix of fractional order integration concerning the Caputo sense. Our approach transforms the nonlinear differential equation into a nonlinear algebraic system. We analyzed various forms of the fractional Bratu equation with different parameters of the equation and its fractional derivative orders. Besides, we show that the introduced method is convergent. We present the obtained results in tables and graphs. Based on these results, we can see the accuracy and convergence of the solutions. Since the Bratu equation is nonlinear so, based on the results obtained, we can say that the used method in approximating the solutions has acceptable accuracy and performance.

Numerical solution of strongly nonlinear full fractional duffing equation

In this work, based on the operational matrix of fractional order integration, we present an approach for the numerical solution of strongly nonlinear full fractional Duffing equation. For this goal, with respect to the Caputo derivative, we use the block-pulse wavelets matrix of fractional order integration. To obtain this purpose, the errors are analyzed. The method has been considered by some numerical examples and variations in coefficients as well as in the derivative of the equation too. We examine the exact solutions for the equation under study in three different categories including, polynomials, exponential and sinusoidal solutions. It is shown that the suggested method works well.

Numerical solution of full fractional Duffing equations with Cubic-Quintic-Heptic nonlinearities

In this article, based on the operational matrix of fractional order integration, we introduce a method for the numerical solution of fractional strongly nonlinear Duffing oscillators with cubic-quintic-heptic nonlinear restoring force and then use it in some cases. For this purpose, concerning the Caputo sense, we implement the block-pulse wavelets matrix of fractional order integration. To reach this aim, we analyse the errors. The approach has been examined by some numerical examples and changes in coefficients as well as in the derivative of the equation too. It is shown that this method works well for all the parameters and order of the fractional derivative. Results indicate the precision and computational performance of the suggested algorithm.

Numerical solution of fractional Mathieu equations by using block-pulse wavelets

In this paper, we introduce a method based on operational matrix of fractional order integration for the numerical solution of fractional Mathieu equation and then apply it in a number of cases. For this, we use the block-pulse wavelets matrix of fractional order integration with respect to the Caputo sense. The method was tested by some numerical examples and changes occurred in the coefficients as well as in the derivative of the equation. Results prove the accuracy and computational efficiency of the proposed algorithm.

Third-Kind Chebyshev Wavelet Method for the Solution of Fractional Order Riccati Differential Equations

In this paper, we derive the numerical solutions of the various fractional-order Riccati type differential equations using the third-kind Chebyshev wavelet operational matrix of fractional order integration (C3WOMFI) method. The operational matrix of fractional order integration method converts the fractional differential equations to a system of algebraic equations. The third-kind Chebyshev wavelet method provides sparse coefficient matrices, therefore the computational load involved for this method is not as severe and also the resulting method is faster. The numerical solutions agree with the exact solutions for non-fractional orders, and also the solutions for the fractional orders approach those of the integer orders as the fractional order coefficient [Formula: see text] approaches to 1.

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.

Numerical solution of fractional differential equation by wavelets and hybrid functions

In this paper, we introduce methods based on operational matrix of fractional order integration for solving a typical n-term non-homogeneous fractional differential equation (FDE). We use Block pulse wavelets matrix of fractional order integration where a fractional derivative is defined in the Caputo sense. Also we consider Hybrid of Block-pulse functions and shifted Legendre polynomials to approximate functions. By uses these methods we translate an FDE to an algebraic linear equations which can be solve. Methods has been tested by some numerical examples.

Numerical Studies for Fractional Pantograph Differential Equations Based on Piecewise Fractional-Order Taylor Function Approximations

Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, a reliable and efficient technique as a solution is regarded. In this paper, a new numerical method for solving fractional pantograph differential equations is presented. The fractional derivative is described in the Caputo sense. The method is based upon piecewise fractional-order Taylor function approximations. The piecewise fractional-order Taylor function is presented. An operational matrix of fractional order integration is derived and is utilized to reduce the under study problem to the solution of a system of algebraic equations. Using Newton’s iterative method, this system is solved and the solution of fractional pantograph differential equations is achieved. A bound of the error is given in the sense of Sobolev norms. Five examples are given and the numerical results are shown to demonstrate the efficiency of the proposed method.

Application of fractional-order Bernoulli functions for solving fractional Riccati differential equation

In this paper, a new numerical method for solving the fractional Riccati differential equation is presented. The fractional derivatives are described in the Caputo sense. The method is based upon fractional-order Bernoulli functions approximations. First, the fractional-order Bernoulli functions and their properties are presented. Then, an operational matrix of fractional order integration is derived and is utilized to reduce the under study problem to a system of algebraic equations. Error analysis included the residual error estimation and the upper bound of the absolute errors are introduced for this method. The technique and the error analysis are applied to some problems to demonstrate the validity and applicability of our method.

CAS Wavelet Function Method for Solving Abel Equations with Error Analysis

In this paper we use a computational method based on CAS wavelets for solving nonlinear fractional order Volterra integral equations. We solve particularly Abel equations. An operational matrix of fractional order integration for CAS wavelets is used. Block Pulse Functions (BPFs) and collocation method are employed to derive a general procedure for forming this matrix. The error analysis of proposed numerical scheme is studied theoretically. Finally, comparison of numerical results with exact solution are shown.

The Linear B-Spline Scaling Function Operational Matrix of Fractional Integration and Its Applications in Solving Fractional-Order Differential Equations

In the present paper, the linear B-spline scaling functions operational matrix of fractional-order integration is derived and is used to solve the fractional differential equations, including the linear and nonlinear Ricatti and composite fractional oscillation equations. The mentioned matrix is utilized to reduce the initial equations into a system of algebraic equations. Also, an error bound when using the linear B-spline scaling functions approximation is derived. Some examples are presented to demonstrate the validity and the applicability of the method. Moreover, compared with the known techniques, it is shown that the method is much more efficient and accurate.

Sine-cosine wavelet operational matrix of fractional order integration and its applications in solving the fractional order Riccati differential equations

In this paper, the sine-cosine wavelet method is presented for solving Riccati differential equations. The sine-cosine wavelet operational matrix of fractional integration is derived and utilized to transform the equations to system of algebraic equations. Also, the error analysis of the sine-cosine wavelet bases is given. The proposed method can be used to solve not only the classical Riccati differential equations but also the fractional ones. Some examples are included to demonstrate the validity and applicability of the technique.

The Construction of Operational Matrix of Fractional Integration Using the Fractional Chebyshev Polynomials

In this paper we present a computational method for solving generalized Abel integral equations. To this end, first, a new operational matrix of fractional order integration for fractional Chebyshev polynomials is derived. The error analysis of the proposed method is studied theoretically. Then, by applying this matrix, collocation and Galerkin methods, we reduce the original problem to a linear system of algebraic equations. Finally, comparison of numerical results with the exact solution shows the validity and efficiency of the presented method.

Solving nonlinear Volterra integro-differential equations of fractional order by using Euler wavelet method

In this paper, a wavelet numerical method for solving nonlinear Volterra integro-differential equations of fractional order is presented. The method is based upon Euler wavelet approximations. The Euler wavelet is first presented and an operational matrix of fractional-order integration is derived. By using the operational matrix, the nonlinear fractional integro-differential equations are reduced to a system of algebraic equations which is solved through known numerical algorithms. Also, various types of solutions, with smooth, non-smooth, and even singular behavior have been considered. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Numerical Simulation of One-Dimensional Fractional Nonsteady Heat Transfer Model Based on the Second Kind Chebyshev Wavelet

In the current study, a numerical technique for solving one-dimensional fractional nonsteady heat transfer model is presented. We construct the second kind Chebyshev wavelet and then derive the operational matrix of fractional-order integration. The operational matrix of fractional-order integration is utilized to reduce the original problem to a system of linear algebraic equations, and then the numerical solutions obtained by our method are compared with those obtained by CAS wavelet method. Lastly, illustrated examples are included to demonstrate the validity and applicability of the technique.

SCW method for solving the fractional integro-differential equations with a weakly singular kernel

In this paper, based on the second Chebyshev wavelets (SCW) operational matrix of fractional order integration, a numerical method for solving a class of fractional integro-differential equations with a weakly singular kernel is proposed. By using the operational matrix, the fractional integro-differential equations with weakly singular kernel are transformed into a system of algebraic equations. The upper bound of the error of the second Chebyshev wavelets expansion is investigated. Finally, some numerical examples are shown to illustrate the efficiency and accuracy of the approach.

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

In this paper, a new numerical method for solving fractional differential equations is presented. The fractional derivative is described in the Caputo sense. The method is based upon Bernoulli wavelet approximations. The Bernoulli wavelet is first presented. An operational matrix of fractional order integration is derived and is utilized to reduce the initial and boundary value problems to system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Wavelet operational matrix method for solving fractional differential equations with variable coefficients

In this paper, another operational matrix method based on Haar wavelet is proposed to solve the fractional differential equations with variable coefficients. The Haar wavelet operational matrix of fractional order integration is derived without using the block pulse functions considered in Li and Zhao (2010) [1]. The operational matrix of fractional order integration is utilized to reduce the initial equations to a system of algebraic equations. Some examples are included to demonstrate the validity and applicability of the method. Moreover, compared with the known technique, the methodology is shown to be much more efficient and accurate.