Numerical solution of fractional differential equations using the generalized block pulse operational matrix
01 Aug 2011-Computers & Mathematics With Applications (COMPUTERS AND MATHEMATICS WITH APPLICATIONS)-Vol. 62, Iss: 3, pp 1046-1054
TL;DR: A way to solve the fractional differential equations using the Riemann-Liouville fractional integral for repeated fractional integration and the generalized block pulse operational matrices of differentiation are proposed.
Abstract: The Riemann-Liouville fractional integral for repeated fractional integration is expanded in block pulse functions to yield the block pulse operational matrices for the fractional order integration. Also, the generalized block pulse operational matrices of differentiation are derived. Based on the above results we propose a way to solve the fractional differential equations. The method is computationally attractive and applications are demonstrated through illustrative examples.
Citations
More filters
TL;DR: The main characteristic behind this approach in this paper is that it derive two kinds of operational matrixes of Bernstein polynomials, which can be viewed as the system of linear equations after dispersing the variable.
126 citations
Cites methods from "Numerical solution of fractional di..."
...The most commonly used ones are Adomian decomposition method (ADM) [16,17], Variational iteration method (VIM) [18], generalized differential transform method (GDTM) [19,20], generalized block pulse operational matrix method [21] and wavelet method [22,23]....
[...]
TL;DR: The second kind Chebyshev wavelet method is presented for solving linear and nonlinear fractional differential equations and the operational matrix of fractional order integration is utilized to reduce the fractions of differential equations to system of algebraic equations.
119 citations
Cites methods from "Numerical solution of fractional di..."
...These methods include eigenvector expansion [5], adomian decomposition method (ADM) [6–8], fractional differential transform method (FDTM) [9–12] and generalized block pulse operational matrix method [13]....
[...]
TL;DR: An efficient and accurate computational method based on the Legendre wavelets (LWs) is proposed for solving a class of fractional optimal control problems (FOCPs) and reveals that the proposed method is very accurate and efficient.
93 citations
TL;DR: In this paper, the authors generalized the wavelet collocation method to fractional differential equations using cubic B-spline wavelet and analyzed expressions of fractional derivatives in Caputo sense.
84 citations
References
More filters
TL;DR: In this article, a method for tuning the PI λ D μ controller is proposed to fulfill five different design specifications, including gain crossover frequency, phase margin, and iso-damping property of the system.
881 citations
TL;DR: The main aim is to generalize the Legendre operational matrix to the fractional calculus and reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem.
Abstract: Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. For that reason we need a reliable and efficient technique for the solution of fractional differential equations. This paper deals with the numerical solution of a class of fractional differential equations. The fractional derivatives are described in the Caputo sense. Our main aim is to generalize the Legendre operational matrix to the fractional calculus. In this approach, a truncated Legendre series together with the Legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations. The main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem. The method is applied to solve two types of fractional differential equations, linear and nonlinear. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.
704 citations
TL;DR: A new generalized Taylor’s formula of the kind f ( x) = ∑ j = 0 n a j ( α ) ( x - a ) j α + R n α ( x ) , where a j ∈ R, x > a, 0 α ⩽ 1, is established.
678 citations
TL;DR: In this paper, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations (FPDE) with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives, and the results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.
Abstract: In this article, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations. On the basis of the homotopy analysis method, a scheme is developed to obtain the approximate solution of the fractional KdV, K(2,2), Burgers, BBM-Burgers, cubic Boussinesq, coupled KdV, and Boussinesq-like B(m,n) equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. 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 of the studied models 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. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010
554 citations
TL;DR: In this article, the authors implemented a well known transformation technique, Differential Transform Method (DTM), to the area of fractional differential equations and the results obtained are in good agreement with the existing ones in open literature.
Abstract: In this study, we implement a well known transformation technique, Differential Transform Method (DTM), to the area of fractional differential equations. Theorems that never existed before are introduced with their proofs. Also numerical examples are carried out for various types of problems, including the Bagley–Torvik, Ricatti and composite fractional oscillation equations for the application of the method. The results obtained are in good agreement with the existing ones in open literature and it is shown that the technique introduced here is robust, accurate and easy to apply.
423 citations