WoS İndeksli Yayınlar Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8653
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Article Further Results on the Neutrix Composition of Distributions Involving the Delta Function and the Function Cosh+<sup>-1</Sup> (x<sup>1/R<(de Gruyter Poland Sp Z O O, 2019) Tas, Kenan; Fisher, BrianThe neutrix composition F(f(x)) of a distribution F(x) and a locally summable function f(x) is said to exist and be equal to the distribution h(x) if the neutrix limit of the sequence {F-n(f(x))) is equal to h(x), where F-n(x) = F(x) * delta(n)(x) and {delta(n)(x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function delta(x). The function cosh(+)(-1)(x + 1) is defined by cosh(+)(-1)(x+ 1) = H(x) cosh(-1)(vertical bar x vertical bar + 1), where H(x) denotes Heaviside's function. It is then proved that the neutrix composition delta((s))[cosh(+)(-1)(x(1/r) + 1)] exists and delta((s))[cosh(+)(-1)(x(1/r) + 1] = Sigma(s-1)(k=0) Sigma(kr+r-1)(j=0) Sigma(j)(i=0) (-1)(kr+r+s-j-1)r/2(j+2) ((kr + r -1)(j)) ((j)(i)) [(j - 2i + 1)(s) - (j - 2i -1)(s)]delta((k))(x) for r, s = 1, 2, .... Further results are also proved. Our results improve, extend and generalize the main theorem of [Fisher B., Al-Sirehy F., Some results on the neutrix composition of distributions involving the delta function and the function cosh(+)(-1) (x + 1), Appl. Math. Sci. (Ruse), 2014, 8(153), 7629-7640].Article Citation - WoS: 12Citation - Scopus: 13On Defining the Distributions Δ<sup>r</Sup> and (δ′)<sup>r</Sup> by Conformable Derivatives(Springeropen, 2018) Abdeljawad, Thabet; Jarad, Fahd; Adjabi, Yassine; Baleanu, DumitruIn this paper, starting from a fixed delta-sequence, we use the generalized Taylor's formula based on conformable derivatives and the neutrix limit to find the powers of the Dirac delta function delta(r) and (delta')(r) for any r is an element of R.Erratum Citation - WoS: 5Retracted: on the Composition of the Distributions X+<sup>λ</Sup> and X+<sup>μ</Sup> (Retracted Article. See Vol. 330, Pg. 1494 2007)(Academic Press inc Elsevier Science, 2006) Tas, K; Fisher, BLet F be a distribution and let f be a locally summable function. The distribution F(f) is defined as the neutrix limit of the sequence {F-n(f)}, where F-n(x) = F(x) * delta(n)(x) and {delta(n)(x)) is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function delta(x). The distributions (x(+)(mu) )(+)(lambda) are evaluated for lambda < 0, mu > 0 and lambda, lambda mu not equal -1, -2.... (c) 2005 Elsevier Inc. All rights reserved.