Further results on the neutrix composition of distributions involving the delta function and the function cosh-1+(x1/r + 1)

• Autorzy: Fisher Brian , Tas Kenan

Identyfikatory

DOI

10.1515/dema-2019-0021

• Języki publikacji: EN

Abstrakty

EN

The 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 {Fn(f(x))} is equal to h(x), 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 function cosh-1+(x + 1) is defined by cosh-1+(x + 1) =H(x) cosh-1(|x| + 1), where H(x) denotes Heaviside’s function. It is then proved that the neutrix composition δ(s)[cosh-1+(x1/r+1)] exists and (…), 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].

Słowa kluczowe

EN

distribution; Dirac delta function; composition of distributions; neutrix; neutrix limit

PL

dystrybucja; funkcja delta; funkcja impulsowa; delta Diraca; neutrix

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2019
• Tom: Vol. 52, nr 1
• Strony: 249--255
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Fisher Brian

• Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK

autor

Tas Kenan

• Department of Mathematics, Cankaya University, Ankara, Turkey

Bibliografia

• [1] Temple G., The theory of generalized functions, In: Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences, 1955, 228(1173), 175–190
• [2] Antosik P., Composition of distributions, Technical Report no.9, University of Wisconsin, Milwaukee, 1988-1989, 1–30
• [3] Van der Corput J. G., Introduction to the neutrix calculus, J. Analyse Math., 1959, 7, 291–398
• [4] Fisher B., On defining the change of variable in distributions, Rostock. Math. Kolloq., 1985, 28, 75–86
• [5] Jones, D. S., Hadamard’s Finite Part., Mathematical Methods in the Applied Sciences, 1996, 19(13), 1017–1052
• [6] Gel’fand I. M., Shilov G. E., Generalized functions, Vol 1, Academic Press, London and New York, 1964
• [7] Fisher B., The composition and neutrix composition of distributions, In: Tas K., Tenreiro Machado J. A., Baleanu D. (Eds.), Mathematical Methods in Engineering, Springer, The Netherlands, 2007, 59–69
• [8] 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
• [9] Fisher B., Kılıçman A., On the composition and neutrix composition of the delta function and powers of the inverse hyperbolic sine function, Integral Transforms Spec. Funct., 2010, 21(12), 935–944
• [10] Fisher B., Özça ̄g E., Al-Sirehy F., On the composition and neutrix composition of the delta function and the function cosh−1(|x|1/r + 1), Internat. J. Anal. Appl., 2017, 13(2), 161–169
• [11] Fisher B., The delta function and the composition of distributions, Dem. Math., 2002, 35(1), 117–123
• [12] Fisher B., Kılıçman A., On the composition and neutrix composition of the delta function with the hyperbolic tangent and its inverse functions, Journal Appl. Math., 2011, Article ID 846736, http://dx.doi.org/10.1155/2011/846736
• [13] Fisher B., Özçaḡ E., Some results on the neutrix composition of the delta function, Filomat, 2012, 26(6), 1247–1256
• [14] Kraiweeradechachai T., Orankitjaroen S., Fisher B., Özçaḡ E., Further results on the neutrix composition of the delta function, East-West J. Math., 2009, 11(2), 151–164

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).