Fractional boundary value problems: Analysis and numerical methods

Ford, Neville J.; Morgado, Maria Luísa

doi:10.2478/s13540-011-0034-4

Cited by 82 publications

(55 citation statements)

References 8 publications

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“…[4,7,8]. It would be of interest to find out whether the conditions of our Nagumo-type result, Theorem 3.1, can be modified to prove a corresponding uniqueness theorem for this class of problems too.…”

Section: W(t)mentioning

confidence: 99%

The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus

Diethelm

2012

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Dedicated to the memory of my teacher, Professor Dr. Helmut BraßWe generalize the classical mean value theorem of differential calculus by allowing the use of a Caputo-type fractional derivative instead of the commonly used first-order derivative. Similarly, we generalize the classical mean value theorem for integrals by allowing the corresponding fractional integral, viz. the Riemann-Liouville operator, instead of a classical (firstorder) integral. As an application of the former result we then prove a uniqueness theorem for initial value problems involving Caputo-type fractional differential operators. This theorem generalizes the classical Nagumo theorem for first-order differential equations.MSC 2010 : Primary 26A33; Secondary 34A08, 34A12 Key Words and Phrases: Caputo derivative, mean value theorem of differential calculus, mean value theorem of integral calculus, fractional differential equation, Nagumo's theorem, uniqueness of solutions

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Section: W(t)mentioning

confidence: 99%

The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus

Diethelm

2012

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“…[2,5,7,10,11]. A rigorous theory of FDE has been started quite recently, see for example -the books [4,9,11] and to mention only few papers related to ordinary FDE as [3,6,8]. Separately, there are many recent works related c 2012 Diogenes Co., Sofia pp.…”

Section: Introductionmentioning

confidence: 99%

On the oscillation of fractional differential equations

Grace

Agarwal

Wong

et al. 2012

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117

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In this paper we initiate the oscillation theory for fractional differential equations. Oscillation criteria are obtained for a class of nonlinear fractional differential equations of the formwhere D q a denotes the Riemann-Liouville differential operator of order q, 0 < q ≤ 1. The results are also stated when the Riemann-Liouville differential operator is replaced by Caputo's differential operator.MSC 2010 : Primary 34A08: Secondary 34C10, 26A33

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“…For some recent works on the topic, see ( [1, 2, 3, 4, 5, 7, c 2012 Diogenes Co., Sofia pp. 362-382 , DOI: 10.2478/s13540-012-0027-y 8,9,10,11,12,13,14,15,16,17,18,21,22,23,24,27,31,35,36,37,38]) and references therein.…”

Section: Introductionmentioning

confidence: 99%

Fractional differential inclusions with fractional separated boundary conditions

Ahmad

Ntouyas

2012

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This paper studies a new class of boundary value problems of nonlinear fractional differential inclusions of order q ∈ (1, 2] with fractional separated boundary conditions. New existence results are obtained for this class of problems by using some standard fixed point theorems. A possible generalization for the inclusion problem with fractional separated integral boundary conditions is also discussed. Some illustrative examples are presented.MSC 2010 : Primary 34A60: Secondary 26A33, 34A08

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Fractional boundary value problems: Analysis and numerical methods

Cited by 82 publications

References 8 publications

The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus

The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus

On the oscillation of fractional differential equations

Fractional differential inclusions with fractional separated boundary conditions

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