High Order Numerical Methods for Fractional Terminal Value Problems

Ford, Neville J.; Morgado, Maria Luísa; Rebelo, Magda

doi:10.1515/cmam-2013-0022

Cited by 27 publications

(14 citation statements)

References 20 publications

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“…Obviously, this investigation can only cover a limited number of alternatives to the Adams method and might be extended to other schemes. A different replacement for the Adams scheme, namely a collocation method using a combination of classical piecewise polynomials and generalized polynomials (with non-integer exponents) has been shown to also provide relatively good results in [95].…”

Section: Terminal Value Problemsmentioning

confidence: 99%

Trends, directions for further research, and some open problems of fractional calculus

et al. 2022

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The area of fractional calculus (FC) has been fast developing and is presently being applied in all scientific fields. Therefore, it is of key relevance to assess the present state of development and to foresee, if possible, the future evolution, or, at least, the challenges identified in the scope of advanced research works. This paper gives a vision about the directions for further research as well as some open problems of FC. A number of topics in mathematics, numerical algorithms and physics are analyzed, giving a systematic perspective for future research.

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Section: Terminal Value Problemsmentioning

confidence: 99%

Trends, directions for further research, and some open problems of fractional calculus

et al. 2022

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“…or [10,13]. For the numerical solution of such problems, one may apply a so-called shooting method [8,9,11,13,14], i.e. one starts with a first guess x 0,1 for x(0), (numerically) solves the initial value problem consisting of the differential equation given in (1.2) and the initial condition x(0) = x 0,1 , and in this way obtains a first approximate solution x * 1 for x(T ) = x * .…”

Section: Motivationmentioning

confidence: 99%

Upper and lower estimates for the separation of solutions to fractional differential equations

Diethelm¹

2022

Fract Calc Appl Anal

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Given a fractional differential equation of order $$\alpha \in (0,1]$$ α ∈ ( 0 , 1 ] with Caputo derivatives, we investigate in a quantitative sense how the associated solutions depend on their respective initial conditions. Specifically, we look at two solutions $$x_1$$ x 1 and $$x_2$$ x 2 , say, of the same differential equation, both of which are assumed to be defined on a common interval [0, T], and provide upper and lower bounds for the difference $$x_1(t) - x_2(t)$$ x 1 ( t ) - x 2 ( t ) for all $$t \in [0,T]$$ t ∈ [ 0 , T ] that are stronger than the bounds previously described in the literature.

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“…We shall address additional questions related to problems of this sort, and the quest for numerical methods for their solution (in this context, see [11] for first results), in a forthcoming separate paper.…”

Section: G(t S)| · |Y(s) −ỹ(S)| Dsmentioning

confidence: 99%

An extension of the well-posedness concept for fractional differential equations of Caputo’s type

Diethelm¹

2014

Applicable Analysis

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It is well known that, under standard assumptions, initial value problems for fractional ordinary differential equations involving Caputo-type derivatives are well posed in the sense that a unique solution exists and that this solution continuously depends on the given function, the initial value and the order of the derivative. Here we extend this well-posedness concept to the extent that we also allow the location of the starting point of the differential operator to be changed, and we prove that the solution depends on this parameter in a continuous way too if the usual assumptions are satisfied. Similarly, the solution to the corresponding terminal value problems depends on the location of the starting point and of the terminal point in a continuous way too.

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High Order Numerical Methods for Fractional Terminal Value Problems

Cited by 27 publications

References 20 publications

Trends, directions for further research, and some open problems of fractional calculus

Trends, directions for further research, and some open problems of fractional calculus

Upper and lower estimates for the separation of solutions to fractional differential equations

An extension of the well-posedness concept for fractional differential equations of Caputo’s type

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