High Speed Algorithm for Computation of Fractional Differentiation and Integration

2011 ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Standard Method ◽  
High Speed ◽  
Computational Method ◽  
Integration Time ◽  
Fractional Differentiation ◽  
History Of ◽  
Weighted Integrals ◽  
Fractional Differential

A high speed algorithm for computing fractional differentiations and fractional integrations in fractional differential equations is proposed. In this algorithm the stored data is not the history of the function to be differentiated or integrated but the history of the weighted integrals of the function. It is shown that, by the computational method based on the new algorithm, the integration time only increases in proportion to n log n, different from n2 by a standard method, for n steps of integrations of a differential integration.

scholarly journals A high-speed algorithm for computation of fractional differentiation and fractional integration

2013 ◽  
Vol 371 (1990) ◽  
pp. 20120152 ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
High Speed ◽  
Fractional Integration ◽  
Computing Time ◽  
Fractional Differentiation ◽  
Time Step ◽  
Weighted Integrals ◽  
Fractional Differential

A high-speed algorithm for computing fractional differentiations and fractional integrations in fractional differential equations is proposed. In this algorithm, the stored data are not the function to be differentiated or integrated but the weighted integrals of the function. The intervals of integration for the memory can be increased without loss of accuracy as the computing time-step n increases. The computing cost varies as , as opposed to n 2 of standard algorithms.

scholarly journals A Collocation Method Based on Discrete Spline Quasi-Interpolatory Operators for the Solution of Time Fractional Differential Equations

2021 ◽  
Vol 5 (1) ◽  
pp. 5
Author(s):  
Enza Pellegrino ◽  
Laura Pezza ◽  
Francesca Pitolli
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Linear Time ◽  
Collocation Methods ◽  
Numerical Tests ◽  
History Of ◽  
Fractional Differential ◽  
Time Fractional Derivative ◽  
Time Fractional Differential Equations

In many applications, real phenomena are modeled by differential problems having a time fractional derivative that depends on the history of the unknown function. For the numerical solution of time fractional differential equations, we propose a new method that combines spline quasi-interpolatory operators and collocation methods. We show that the method is convergent and reproduces polynomials of suitable degree. The numerical tests demonstrate the validity and applicability of the proposed method when used to solve linear time fractional differential equations.

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.

scholarly journals Computational method based on Bernstein operational matrices for multi-order fractional differential equations

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 591-601 ◽  
Author(s):  
Davood Rostamy ◽  
Hossein Jafari ◽  
Mohsen Alipour ◽  
Chaudry Khalique
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Operational Matrix ◽  
Adomian Decomposition Method ◽  
Computational Method ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Transform Method ◽  
Adomian Decomposition ◽  
Fractional Differential

In this paper, the Bernstein operational matrices are used to obtain solutions of multi-order fractional differential equations. In this regard we present a theorem which can reduce the nonlinear fractional differential equations to a system of algebraic equations. The fractional derivative considered here is in the Caputo sense. Finally, we give several examples by using the proposed method. These results are then compared with the results obtained by using Adomian decomposition method, differential transform method and the generalized block pulse operational matrix method. We conclude that our results compare well with the results of other methods and the efficiency and accuracy of the proposed method is very good.

scholarly journals APPLICATIONS OF BI-FRAMELET SYSTEMS FOR SOLVING FRACTIONAL ORDER DIFFERENTIAL EQUATIONS

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040051 ◽  
Author(s):  
MUTAZ MOHAMMAD ◽  
CARLO CATTANI
Keyword(s):  
Numerical Analysis ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Sparse Matrices ◽  
Computational Method ◽  
Extension Principle ◽  
Vanishing Moments ◽  
Fractional Order Differential Equations ◽  
Fractional Differential

Framelets and their attractive features in many disciplines have attracted a great interest in the recent years. This paper intends to show the advantages of using bi-framelet systems in the context of numerical fractional differential equations (FDEs). We present a computational method based on the quasi-affine bi-framelets with high vanishing moments constructed using the generalized (mixed) oblique extension principle. We use this system for solving some types of FDEs by solving a series of important examples of FDEs related to many mathematical applications. The quasi-affine bi-framelet-based methods for numerical FDEs show the advantages of using sparse matrices and its accuracy in numerical analysis.

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