WoS İndeksli Yayınlar Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8653
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Article Citation - WoS: 5Citation - Scopus: 5On the Composition of the Distributions X+-R and X+μ(indian Nat Sci Acad, 2005) Fisher, B.; Tas, K.Let 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 Fn (x) = F (x) * δn (x) and {δn (x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta function δ (x). The distribution (x+μ)+-r and ( l x lμ)+-r are evaluated for μ > 0, r = 1, 2, ..., and kμ ≠ 1, 2,... © Printed in India.Article Citation - WoS: 1Citation - Scopus: 1On the Non-Commutative Neutrix Product of the Distributions X<sup>-r</Sup>+ Ln<sup>p</Sup> X+ and X<sup>μ</Sup>+ln<sup>q< X+(Taylor & Francis Ltd, 2006) Tas, Kenan; Fisher, BrianLet f and g be distributions and g(n) = (g*delta(n))(x), where delta(n)(x ) is a certain sequence converging to the Dirac delta-function. The non-commutative neutrix product f o g of f and g is defined to be the neutrix limit of the sequence {fg(n) }, provided its limit h exists in the sense that [GRAPHICS] for all functions phi in D. It is proved that (x(+)(-r) ln(p) x(+)) o (x(+)(mu) ln(q) x(+)) = x(+)(-r+mu) ln(p+q) x(+) (x(-)(-r) ln(p) (x)-) o (x(-)(mu) ln(q) x(-)) = x(-)(-r+mu) ln(p+q) x(-) for mu < r - 1;mu not equal 0, +/- 1, +/- 2,..., r = 1,2,..., and p, q = 0, 1, 2,....Article Citation - WoS: 3Citation - Scopus: 2On the Non-Commutative Neutrix Product of the Distributions X<sup>λ</Sup>+ and X<sup>μ</Sup>+(Springer Heidelberg, 2006) Tas, K.; Fisher, B.Let f and g be distributions and let g(n) = (g * delta(n))(x), where delta(n)(x) is a certain sequence converging to the Dirac delta function. The non-commutative neutrix product f circle g of f and g is defined to be the limit of the sequence {fg(n)}, provided its limit h exists in the sense that [GRAPHICS] for all functions p in D. It is proved that (x(+)(lambda)ln(p)x(+)) circle (x(+)(mu)ln(q)x(+)) = x(+)(lambda+mu)ln(p+q)x(+), (x(-)(lambda)ln(p)x(-)) circle (x(-)mu ln(q)x(-)) = x(-)(lambda+mu)ln(p+q)x(-), for lambda + mu < -1; lambda,mu,lambda+mu not equal -1,-2,... and p,q = 0,1,2.....Article Citation - WoS: 3Citation - Scopus: 1On the Non-Commutative Neutrix Product of the Distributions X<sup>r</Sup> Ln<sup>p</Sup> | X | and X<sup>-s</Sup>(Taylor & Francis Ltd, 2005) Tas, K; Fisher, BThe non-commutative neutrix product of the distributions x(r) ln(P) \x\ and x(-s) is evaluated for r - s -2. -3,..../ p = 1, 2,....Article Citation - WoS: 9Citation - Scopus: 13The Convolution of Functions and Distributions(Academic Press inc Elsevier Science, 2005) Tas, K; Fisher, BThe non-commutative convolution f * g of two distributions f and g in V is defined to be the limit of the sequence {(f tau(n)) * g}, provided the limit exists, where {tau(n)} is a certain sequence of functions in D converging to 1. It is proved that vertical bar x vertical bar(lambda) * (sgnx vertical bar x vertical bar(mu)) = 2 sin(lambda pi/2)cos(mu pi/2)/sin[(lambda+mu)pi/2] B(lambda+1, mu+1) sgn x vertical bar x vertical bar(lambda+mu+1), for -1 < lambda + mu < 0 and lambda, mu not equal -1, -2,..., where B denotes the Beta function. (c) 2005 Elsevier Inc. All rights reserved.Article Citation - WoS: 5Citation - Scopus: 5On the Composition of the Distributions X<sup>-1</Sup> Ln|x| and X+<sup>r</Sup>(Taylor & Francis Ltd, 2005) Fisher, B; Tas, KLet 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 (x). The distribution (x(+)(r))(-1) ln \x(+)(r)\ is evaluated for r = 1, 2.....Article On the non-commutative neutrix product of the distributions x(+)(lambda) and x(+)(mu)(Springer Heidelberg, 2006) Fisher, Brian; Taş, KenanLet f and g be distributions and let g(n) = (g * delta(n))(x), where delta(n)(x) is a certain sequence converging to the Dirac delta function. The non-commutative neutrix product f circle g of f and g is defined to be the limit of the sequence {fg(n)}, provided its limit h exists in the sense that [GRAPHICS] for all functions p in D. It is proved that (x(+)(lambda)ln(p)x(+)) circle (x(+)(mu)ln(q)x(+)) = x(+)(lambda+mu)ln(p+q)x(+), (x(-)(lambda)ln(p)x(-)) circle (x(-)mu ln(q)x(-)) = x(-)(lambda+mu)ln(p+q)x(-), for lambda + mu < -1; lambda,mu,lambda+mu not equal -1,-2,... and p,q = 0,1,2.....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.Article Citation - WoS: 1Citation - Scopus: 2Some Results on the Non-Commutative Neutrix Product of Distributions(Taylor & Francis Ltd, 2009) Tas, Kenan; Fisher, BrianIt is proved that the non-commutative neutrix product of the distributions x-r and xslnq|x| exists and [image omitted] for r, q=1, 2, , s=0,1,2, and r-s1.