scholarly journals Numerical Solutions for Systems of Fractional and Classical Integro-Differential Equations via Finite Integration Method Based on Shifted Chebyshev Polynomials

2021 ◽  
Vol 5 (3) ◽  
pp. 103
Author(s):  
Ampol Duangpan ◽  
Ratinan Boonklurb ◽  
Matinee Juytai
Keyword(s):  
Differential Equations ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Integration Method ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Computational Cost ◽  
Stiff System ◽  
Numerical Convergence ◽  
Shifted Chebyshev Polynomial

In this paper, the finite integration method and the operational matrix of fractional integration are implemented based on the shifted Chebyshev polynomial. They are utilized to devise two numerical procedures for solving the systems of fractional and classical integro-differential equations. The fractional derivatives are described in the Caputo sense. The devised procedure can be successfully applied to solve the stiff system of ODEs. To demonstrate the efficiency, accuracy and numerical convergence order of these procedures, several experimental examples are given. As a consequence, the numerical computations illustrate that our presented procedures achieve significant improvement in terms of accuracy with less computational cost.

Numerical Simulation of Fractional Delay Differential Equations Using the Operational Matrix of Fractional Integration for Fractional-Order Taylor Basis

2021 ◽  
Vol 6 (1) ◽  
pp. 10
Author(s):  
İbrahim Avcı 
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Absolute Error ◽  
Algebraic Equations ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

In this paper, we consider numerical solutions for a general form of fractional delay differential equations (FDDEs) with fractional derivatives defined in the Caputo sense. A fractional integration operational matrix, created using a fractional Taylor basis, is applied to solve these FDDEs. The main characteristic of this approach is, by utilizing the operational matrix of fractional integration, to reduce the given differential equation to a set of algebraic equations with unknown coefficients. This equation system can be solved efficiently using a computer algorithm. A bound on the error for the best approximation and fractional integration are also given. Several examples are given to illustrate the validity and applicability of the technique. The efficiency of the presented method is revealed by comparing results with some existing solutions, the findings of some other approaches from the literature and by plotting absolute error figures.

scholarly journals Numerical Solutions for Multi-Term Fractional Order Differential Equations with Fractional Taylor Operational Matrix of Fractional Integration

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 96 ◽  
Author(s):  
İbrahim Avcı ◽  
Nazim I. Mahmudov
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Main Idea ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Fractional Order Differential Equations ◽  
Fractional Differential ◽  
The Given

In this article, we propose a numerical method based on the fractional Taylor vector for solving multi-term fractional differential equations. The main idea of this method is to reduce the given problems to a set of algebraic equations by utilizing the fractional Taylor operational matrix of fractional integration. This system of equations can be solved efficiently. Some numerical examples are given to demonstrate the accuracy and applicability. The results show that the presented method is efficient and applicable.

scholarly journals Numerical solutions of multi-order fractional differential equations by Boubaker polynomials

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 226-230 ◽  
Author(s):  
A. Bolandtalat ◽  
E. Babolian ◽  
H. Jafari
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Algebraic Equations ◽  
Boubaker Polynomials ◽  
Fractional Differential ◽  
The Given

AbstractIn this paper, we have applied a numerical method based on Boubaker polynomials to obtain approximate numerical solutions of multi-order fractional differential equations. We obtain an operational matrix of fractional integration based on Boubaker polynomials. Using this operational matrix, the given problem is converted into a set of algebraic equations. Illustrative examples are are given to demonstrate the efficiency and simplicity of this technique.

A Numerical Method for Caputo Differential Equations and Application of High-Speed Algorithm

2019 ◽  
Vol 14 (9) ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
High Speed ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Computing Time ◽  
Integer Order ◽  
Truncation Errors ◽  
Fractional Differential

In this paper, a numerical algorithm to solve Caputo differential equations is proposed. The proposed algorithm utilizes the R2 algorithm for fractional integration based on the fact that the Caputo derivative of a function f(t) is defined as the Riemann–Liouville integral of the derivative f(ν)(t). The discretized equations are integer order differential equations, in which the contribution of f(ν)(t) from the past behaves as a time-dependent inhomogeneous term. Therefore, numerical techniques for integer order differential equations can be used to solve these equations. The accuracy of this algorithm is examined by solving linear and nonlinear Caputo differential equations. When large time-steps are necessary to solve fractional differential equations, the high-speed algorithm (HSA) proposed by the present authors (Fukunaga, M., and Shimizu, N., 2013, “A High Speed Algorithm for Computation of Fractional Differentiation and Integration,” Philos. Trans. R. Soc., A, 371(1990), p. 20120152) is employed to reduce the computing time. The introduction of this algorithm does not degrade the accuracy of numerical solutions, if the parameters of HSA are appropriately chosen. Furthermore, it reduces the truncation errors in calculating fractional derivatives by the conventional trapezoidal rule. Thus, the proposed algorithm for Caputo differential equations together with the HSA enables fractional differential equations to be solved with high accuracy and high speed.

Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions

2010 ◽  
Vol 65 (11) ◽  
pp. 935-949 ◽  
Author(s):  
Mehdi Dehghan ◽  
Jalil Manafian ◽  
Abbas Saadatmandi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Analysis Method ◽  
Partial Differential ◽  
Integer Order ◽  
Homotopy Analysis

In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

scholarly journals Numerical Investigation of Fractional-Order Differential Equations via φ -Haar-Wavelet Method

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
F. M. Alharbi ◽  
A. M. Zidan ◽  
Muhammad Naeem ◽  
Rasool Shah ◽  
Kamsing Nonlaopon
Keyword(s):  
Differential Equations ◽  
Fractional Integral ◽  
Numerical Technique ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Nonlinear Problems ◽  
Haar Wavelet ◽  
Mathematical Tool ◽  
Suggested Technique ◽  
Operational Matrix Of Integration

In this paper, we propose a novel and efficient numerical technique for solving linear and nonlinear fractional differential equations (FDEs) with the φ -Caputo fractional derivative. Our approach is based on a new operational matrix of integration, namely, the φ -Haar-wavelet operational matrix of fractional integration. In this paper, we derived an explicit formula for the φ -fractional integral of the Haar-wavelet by utilizing the φ -fractional integral operator. We also extended our method to nonlinear φ -FDEs. The nonlinear problems are first linearized by applying the technique of quasilinearization, and then, the proposed method is applied to get a numerical solution of the linearized problems. The current technique is an effective and simple mathematical tool for solving nonlinear φ -FDEs. In the context of error analysis, an exact upper bound of the error for the suggested technique is given, which shows convergence of the proposed method. Finally, some numerical examples that demonstrate the efficiency of our technique are discussed.

Wavelet Galerkin method for fourth-order multi-dimensional elliptic partial differential equations

2018 ◽  
Vol 16 (05) ◽  
pp. 1850045 ◽  
Author(s):  
B. V. Rathish Kumar ◽  
Gopal Priyadarshi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Computational Cost ◽  
Stability Estimates ◽  
Partial Differential ◽  
Wavelet Galerkin Method

We describe a wavelet Galerkin method for numerical solutions of fourth-order linear and nonlinear partial differential equations (PDEs) in 2D and 3D based on the use of Daubechies compactly supported wavelets. Two-term connection coefficients have been used to compute higher-order derivatives accurately and economically. Localization and orthogonality properties of wavelets make the global matrix sparse. In particular, these properties reduce the computational cost significantly. Linear system of equations obtained from discretized equations have been solved using GMRES iterative solver. Quasi-linearization technique has been effectively used to handle nonlinear terms arising in nonlinear biharmonic equation. To reduce the computational cost of our method, we have proposed an efficient compression algorithm. Error and stability estimates have been derived. Accuracy of the proposed method is demonstrated through various examples.

scholarly journals A Spectral Deferred Correction Method for Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Jia Xin ◽  
Jianfei Huang ◽  
Weijia Zhao ◽  
Jiang Zhu
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Caputo Derivative ◽  
Numerical Solutions ◽  
Computational Cost ◽  
Correction Method ◽  
Initial Value ◽  
Deferred Correction ◽  
Fractional Differential ◽  
Spectral Deferred Correction

A spectral deferred correction method is presented for the initial value problems of fractional differential equations (FDEs) with Caputo derivative. This method is constructed based on the residual function and the error equation deduced from Volterra integral equations equivalent to the FDEs. The proposed method allows that one can use a relatively few nodes to obtain the high accuracy numerical solutions of FDEs without the penalty of a huge computational cost due to the nonlocality of Caputo derivative. Finally, preliminary numerical experiments are given to verify the efficiency and accuracy of this method.

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