A computational matrix method for solving systems of high order fractional differential equations

Khader, M. M.; El-Danaf, Talaat S.; Hendy, Ahmed S.

doi:10.1016/j.apm.2012.08.009

Cited by 62 publications

(24 citation statements)

References 26 publications

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“…A fundamental solution of a fractional order distributed parameter system was presented in [31]. The numerical solutions of such systems were proposed in the literature by means of finite difference methods [32], spectral collocation methods and others [33,34]. However, it is worth pointing out that up to now, there is no concern about ILC for fractional order distributed parameter systems.…”

Section: Introductionmentioning

confidence: 99%

ILC with Initial State Learning for Fractional Order Linear Distributed Parameter Systems

Lan

Cui

2018

Algorithms

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This paper presents a second order P-type iterative learning control (ILC) scheme with initial state learning for a class of fractional order linear distributed parameter systems. First, by analyzing the control and learning processes, a discrete system for P-type ILC is established, and the ILC design problem is then converted to a stability problem for such a discrete system. Next, a sufficient condition for the convergence of the control input and the tracking errors is obtained by introducing a new norm and using the generalized Gronwall inequality, which is less conservative than the existing one. Finally, the validity of the proposed method is verified by a numerical example.

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Section: Introductionmentioning

confidence: 99%

ILC with Initial State Learning for Fractional Order Linear Distributed Parameter Systems

Lan

Cui

2018

Algorithms

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“…The equations in partial derivatives of a fractional order were subdivided into two big classes: with a fractional derivative on space and with a fractional derivative on time. Now there are many papers on numerical methods of the solution of such equations [19][20][21]. In this paper, we depend on results of work [22] in which explicit and purely implicit numerical methods for the solution of the equation with two sided space fractional derivative were investigated.…”

Section: Introductionmentioning

confidence: 99%

A Fractional Analog of Crank–nicholson Method for the Two Sided Space Fractional Partial Equation With Functional Delay

Пименов

Hendy

2016

UMJ

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“…The iterative methods is utilized widely and found the solution of the equations in the area of the initial and boundary value problems. We already have a number of famous computational techniques dealing with the solution of high order fractional differential equations, such as weighted essentially non-oscillatory scheme 18 , Chebyshev collocation method 19,20 , Green functions method 21 and blockby-block approach 22 . Moreover, as an excellent modeling tool, fractional integro-differential equations issue have attracted much attention recently 23,24 .…”

Section: Introductionmentioning

confidence: 99%

Using ANNs Approach for Solving Fractional Order Volterra Integro-differential Equations

Jafarian¹,

Rostami²,

Golmankhaneh³

et al. 2017

IJCIS

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Indeed, interesting properties of artificial neural networks approach made this non-parametric model a powerful tool in solving various complicated mathematical problems. The current research attempts to produce an approximate polynomial solution for special type of fractional order Volterra integrodifferential equations. The present technique combines the neural networks approach with the power series method to introduce an efficient iterative technique. To do this, a multi-layer feed-forward neural architecture is depicted for constructing a power series of arbitrary degree. Combining the initial conditions with the resulted series gives us a suitable trial solution. Substituting this solution instead of the unknown function and employing the least mean square rule, converts the origin problem to an approximated unconstrained optimization problem. Subsequently, the resulting nonlinear minimization problem is solved iteratively using the neural networks approach. For this aim, a suitable three-layer feed-forward neural architecture is formed and trained using a back-propagation supervised learning algorithm which is based on the gradient descent rule. In other words, discretizing the differential domain with a classical rule produces some training rules. By importing these to designed architecture as input signals, the indicated learning algorithm can minimize the defined criterion function to achieve the solution series coefficients. Ultimately, the analysis is accompanied by two numerical examples to illustrate the ability of the method. Also, some comparisons are made between the present iterative approach and another traditional technique. The obtained results reveal that our method is very effective, and in these examples leads to the better approximations.

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A computational matrix method for solving systems of high order fractional differential equations

Cited by 62 publications

References 26 publications

ILC with Initial State Learning for Fractional Order Linear Distributed Parameter Systems

ILC with Initial State Learning for Fractional Order Linear Distributed Parameter Systems

A Fractional Analog of Crank–nicholson Method for the Two Sided Space Fractional Partial Equation With Functional Delay

Using ANNs Approach for Solving Fractional Order Volterra Integro-differential Equations

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