A discrete time method to the first variation of fractional order variational functionals

Pooseh, Shakoor; Almeida, Ricardo; Torres, Delfim F. M.

doi:10.2478/s11534-013-0250-0

Cited by 6 publications

(5 citation statements)

References 13 publications

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“…On the other hand, very well performed research has been conducted on the numerical side of FODEs. In this regard plenty of research articles addressing numerical and qualitative analysis have been presented in the past few years (for instance see [11][12][13][14][15][16] and the references therein). Here we remark that stability analysis is also an important aspect of qualitative analysis.…”

Section: Introductionmentioning

confidence: 99%

Study of a nonlinear multi-terms boundary value problem of fractional pantograph differential equations

et al. 2021

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In this research work, a class of multi-term fractional pantograph differential equations (FODEs) subject to antiperiodic boundary conditions (APBCs) is considered. The ensuing problem involves proportional type delay terms and constitutes a subclass of delay differential equations known as pantograph. On using fixed point theorems due to Banach and Schaefer, some sufficient conditions are developed for the existence and uniqueness of the solution to the problem under investigation. Furthermore, due to the significance of stability analysis from a numerical and optimization point of view Ulam type stability and its various forms are studied. Here we mention different forms of stability: Hyers–Ulam (HU), generalized Hyers–Ulam (GHU), Hyers–Ulam Rassias (HUR) and generalized Hyers–Ulam–Rassias (GHUR). After the demonstration of our results, some pertinent examples are given.

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Section: Introductionmentioning

confidence: 99%

Study of a nonlinear multi-terms boundary value problem of fractional pantograph differential equations

et al. 2021

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“…Therefore, it is imperative to develop appropriate numerical methods for such dynamic systems. The generalized ELE, condition of transversality and the FOVPs are discussed in [51][52][53][54]. The subject of FOVPs generates more reliable models of physical phenomena and has several applications in engineering and physics [55][56][57] such as automatic control, porous media, mechanical problems involved with dissipative systems, Brownian motions, vibrating string and so on.…”

Section: Introductionmentioning

confidence: 99%

An approximate wavelets solution to the class of variational problems with fractional order

Rayal

Verma

2020

J. Appl. Math. Comput.

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In the present work, a generalized fractional integral operational matrix is derived by using classical Legendre wavelets. Then, a numerical scheme based on this operational matrix and Lagrange multipliers is proposed for solving variational problems with fractional order. This approach has been applied on some illustrative examples. The results obtained for these examples demonstrate that the suggested technique is efficient for solving variational problems with fractional order and gives a very perfect agreement with the exact solution. The results are depicted in graphical maps and data tables. The integral square error, maximum absolute error, and order of convergence have been evaluated to analyze the precision of the suggested method. The present scheme provides better and comparable results with some other existing approaches available in the literature.

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“…The development and analysis of fractional calculus began in recent decades, when the fractional differential equation emerged as a tool for the description of phenomena in nature. Fractional differential equations (Giona and Roman, 1992;Kirchner et al, 2000;Magin, 2006;Li and Deng, 2007;Garrappa and Popolizio, 2011;Alipour et al, 2012;Baleanu et al, 2012;Machado et al, 2013;Rostamy et al, 2013) are used to model many phenomena in several fields (Pooseh et al, 2013;Dehghan et al, 2014;Lazo and Torres, 2014;Pinto and Carvalho, 2015;Raja and Chaudhary, 2015;Xu et al, 2015). Numerical techniques are widely used by scientists and engineers to solve fractional PDEs.…”

Section: Introductionmentioning

confidence: 99%

A highly accurate collocation algorithm for 1 + 1 and 2 + 1 fractional percolation equations

Bhrawy

2015

Journal of Vibration and Control

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This paper reports two successive spectral collocation methods, that enable easy and highly accuracy discretization, for 1 þ 1 and 2 þ 1 fractional percolation equations (FPEs). The first step depends mainly on the shifted Legendre GaussLobatto collocation method for spatial discretization. An expansion in a series of shifted Legendre polynomials for the approximate solution and its spatial derivatives occurring in the FPE is investigated. In addition, the Legendre-GaussLobatto quadrature rule is established to treat the boundary conditions. Thereby, the expansion coefficients are then determined by reducing the FPE with its boundary conditions to a system of ordinary differential equations for these coefficients. The second step is to propose the shifted Chebyshev Gauss-Radau collocation scheme, for temporal discretization, to reduce such a system to a system of algebraic equations, which is far easier to solve. The proposed collocation scheme, both in temporal and spatial discretizations, is successfully extended to the numerical solution of two-dimensional FPEs. An upper bound of the absolute error is obtained of the approximate solution for the twodimensional case. Convergence properties of the method are discussed through numerical examples. Several numerical examples with comparisons are reported to highlight the high accuracy of the present method over other numerical techniques.

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A discrete time method to the first variation of fractional order variational functionals

Cited by 6 publications

References 13 publications

Study of a nonlinear multi-terms boundary value problem of fractional pantograph differential equations

Study of a nonlinear multi-terms boundary value problem of fractional pantograph differential equations

An approximate wavelets solution to the class of variational problems with fractional order

A highly accurate collocation algorithm for 1 + 1 and 2 + 1 fractional percolation equations

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