Linear Fractional Integro-differential Equations Research Articles

Erratum to: Fast iterative refinement method for mixed systems of integral and fractional integro-differential equations

The authors’ new method of iterative refinement has been successfully applied to the linear fractional Fredholm integro-differential equation. In this work, we adapt our method to obtain approximate analytical solutions for linear Volterra equations of both the first and the second kind of fractional type. A detailed convergence analysis is presented for each kind. The authors also apply their method to the challenging first kind linear Fredholm integral equation. Three different cases are considered to test the efficacy of our method. These include mixed systems of various forms of linear integral and fractional integro-differential equations. We compare our method with six well-established methods: Picard’s successive approximations, an accelerated form of the latter, the Adomian decomposition, the variational iteration, the homotopy perturbation and the homotopy analysis. The results show the versatility of the new method in solving a wide variety of equations, its high convergence speed and accuracy and its capability of directly dealing with equations of the first kind. We also prove that Picard’s method for linear equations is a special case of our method.

Solving multi-dimensional fractional integro-differential equations with the initial and boundary conditions by using multi-dimensional Laplace Transform method

In this paper, the one-dimensional Laplace transform method is evolved to solve linear one-dimensional fractional Volterra integro-differential equations with initial conditions and extend this study by taking multi-dimensional Laplace transform method of the linear fractional multidimensional Volterra integro-differential equations to find solution of the initial and boundary value problems. It is noticed that the suggested methods is suitable to find solution for problems. The results of the noticed methods are supporter, easily and active.

Solution of Fuzzy Fractional Integro-Differential Equations Using A Domian Decomposition Method

In this paper, we present Adomian decomposition method to solve linear fuzzy fractional integro-differential equation with fuzzy initial conditions. Results are compared with the results obtained using Fuzzy Laplace transform method.

Analysis of Fractional Schrödinger Equation Occurring in Quantum Mechanics

In this paper, we present Adomian decomposition method to solve linear fuzzy fractional integro-differential equation with fuzzy initial conditions. Results are compared with the results obtained using Fuzzy Laplace transform method.

Chebyshev-Legendre Spectral Approach in Fractional Integro-Differential Equations

Objectives: In the current research we implement Chebyshev-Legendre spectral approach for a class of linear fractional integro-differential equations (FIDE) including those of Fredholm and Volterra types via initial conditions or two-point boundary value conditions. Methods: An equations derived that characterizes the approximation solution by using the shifted Legendre and shifted Chebyshev collocation points. By providing various numerical instances, the precision and efficiency of the developed algorithms were evaluated. We show that for a class of equations, we obtain the exact solutions. Results: Some particular linear solvers were presented for the linear and nonlinear FIDE with initial conditions by applying downshifted Legendre polynomials. The fractional derivatives were defined in the Caputo sense. Furthermore, a novel method developed by executing downshifted Legendre procedure in union with the downshifted Chebyshev collocation way for the solution of FIDE. Based on our research, the presented research is the first investigation regarding the Chebyshev-Legendre spectral approach for determining FIDE with initial conditions. Conclusion: In the current paper, A numerical algorithm was proposed to determine the overall equations, utilizing Gauss-collocation features and estimated the solution directly applying the downshifted Legendre polynomials.

Legendre Polynomial and Iterative Decomposition Methods for Linear Fractional Integro-Differential Equations

Legendre Polynomial and Iterative Decomposition Methods for Linear Fractional Integro-Differential Equations

SOLVING FRACTIONAL INTEGRO DIFFERENTIAL EQUATIONS BY HOMOTOPY ANALYSIS TRANSFORM METHOD

In this paper, we introduce an analytical method, which so called the homotopy analysis transform method (HATM) which is a combination of HAM and Laplace decomposition method (LDM). This scheme is simple to apply linear and nonlinear fractional integro-differential equation and having less computational work in comparison of other exiting methods. The fractional derivatives are described in the Caputo sense. The most useful advantage of this method is to solve the fractional integro-differential equation without using Adomian polynomials and He’s polynomials for the computation of nonlinear terms. AMS Subject Classification: 00A71, 45A05, 34A12, 45E10, 7Q10

Iterative refinement for a system of linear integro-differential equations of fractional type

A novel technique based on iterative refinement is developed to approximate the analytical solution of a system of linear fractional integro-differential equations. While the study focuses mainly on Fredholm-type equations, adaptation to the Volterra-type is also presented. A comparison is made with the method of successive approximations on the basis of convergence speed and accuracy. Several numerical examples are given to demonstrate the efficacy of our algorithm. The authors also present formulations of some error bounds.

Legendre Wavelet and He's Homotopy Perturbation Methods for Linear Fractional Integro-Differential Equations

paper, the Legendre wavelet method (LWM) and He's Homotopy perturbation method (HPM) are applied to approximate solution for linear fractional integro-differential equation with initial condition. A comparison between these methods takes place. Numerical examples are presented to illustrste the efficiency and accuracy of the proposed methods.

Approximate Solution of the System of Linear Fractional Integro-Differential Equations of Volterra Using B- Spline Method

Approximate Solution of the System of Linear Fractional Integro-Differential Equations of Volterra Using B- Spline Method

Legendre wavelets method for solving fractional integro-differential equations

In this paper, based on the constructed Legendre wavelets operational matrix of integration of fractional order, a numerical method for solving linear and nonlinear fractional integro-differential equations is proposed. By using the operational matrix, the linear and nonlinear fractional integro-differential equations are reduced to a system of algebraic equations which are solved through known numerical algorithms. The upper bound of the error of the Legendre wavelets expansion is investigated in Theorem 5.1. Finally, four numerical examples are shown to illustrate the efficiency and accuracy of the approach.

Numerical Approximation of Fractional Integro-differential Equations by an Iterative Decomposition Method

In this paper, an attempt is made to approximate the solution of linear and nonlinear fractional integro-differential equations, by applying an iterative decomposition method. The approximate solution of each problem is presented as a rapidly convergent series of easily computable terms. Results obtained are compared favourably with known results to illustrate the accuracy and efficiency of the method.

Numerical Solution of Fractional Integro-Differential Equations by Least Squares Method and Shifted Chebyshev Polynomial

We investigate the numerical solution of linear fractional integro-differential equations by least squares method with aid of shifted Chebyshev polynomial. Some numerical examples are presented to illustrate the theoretical results.

Spectral collocation method for linear fractional integro-differential equations

In this paper, we propose and analyze a spectral Jacobi-collocation method for the numerical solution of general linear fractional integro-differential equations. The fractional derivatives are described in the Caputo sense. First, we use some function and variable transformations to change the equation into a Volterra integral equation defined on the standard interval [-1,1]. Then the Jacobi–Gauss points are used as collocation nodes and the Jacobi–Gauss quadrature formula is used to approximate the integral equation. Later, the convergence order of the proposed method is investigated in the infinity norm. Finally, some numerical results are given to demonstrate the effectiveness of the proposed method.

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.

Numerical approach for solving fractional Fredholm integro-differential equation

In this article, we present a new method which is based on the Taylor Matrix Method to give approximate solution of the linear fractional Fredholm integro-differential equations. This method is based on first taking the truncated Taylor expansions of the functions in the linear fractional differential part and Fredholm integral part then, substituting their matrix forms into the equation. We solve this matrix equation with the assistance of Maple 13. In addition, illustrative examples are presented to demonstrate the effectiveness of the proposed method.

On the Solution of Linear and Nonlinear Fractional Integro-Differential Equations in Nano-Transistors

On the Solution of Linear and Nonlinear Fractional Integro-Differential Equations in Nano-Transistors

Comparison of Adomian decomposition and Homotopy perturbation methods for higher-order linear fractional integrodifferential equations

In this paper, we will compare between Adomian decomposition method (ADM) and Homotopy perturbation method (HPM) for obtaining the numerical solutions of higher-order linear fractional integro-differential equations with boundary conditions. Numerical examples are presented to illustrate the efficiency and accuracy of the proposed methods.

Approximate solution of fractional integro-differential equations by Taylor expansion method

In this paper, Taylor expansion approach is presented for solving (approximately) a class of linear fractional integro-differential equations including those of Fredholm and of Volterra types. By means of the m th-order Taylor expansion of the unknown function at an arbitrary point, the linear fractional integro-differential equation can be converted approximately to a system of equations for the unknown function itself and its m derivatives under initial conditions. This method gives a simple and closed form solution for a linear fractional integro-differential equation. In addition, illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method.

A Legendre collocation method for fractional integro-differential equations

A numerical method for solving the linear and non-linear fractional integro-differential equations of Volterra type is presented. The fractional derivative is described in the Caputo sense. The method is based upon Legendre approximations. The properties of Legendre polynomials together with the Gaussian integration method are utilized to reduce the fractional integro-differential equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique and a comparison is made with existing results.