Matematik Bölümü Yayın Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/413
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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: 158Citation - Scopus: 181Hamiltonian Formulation of Systems With Linear Velocities Within Riemann-Liouville Fractional Derivatives(Academic Press inc Elsevier Science, 2005) Muslih, SI; Baleanu, D; Avkar, T.The link between the treatments of constrained systems with fractional derivatives by using both Hamiltonian and Lagrangian formulations is studied. It is shown that both treatments for systems with linear velocities are equivalent. (c) 2004 Elsevier Inc. All rights reserved.Article Citation - WoS: 2On Oscillatory Solutions of Certain Forced Emden-Fowler Like Equations(Academic Press inc Elsevier Science, 2008) Mustafa, Octavian G.We give a constructive proof of existence to oscillatory solutions for the differential equations x ''(t) + a(t)vertical bar x(t)vertical bar lambda sign[x(t)] = e(t), where t >= t(0) >= 1 and lambda > 1, that decay to 0 when t -> infinity as 0(t(-mu)) for mu > 0 as close as desired to the "critical quantity" mu* = 2/lambda-1 For this class of equations, we have lim(t ->+infinity) E(t) = 0, where E(t) 0 and E ''(t) e(t) throughout [t(0) + infinity). We also establish that for any mu > mu* and any negative-valued E(t) = 0(t(-mu)) as t ->+infinity the differential equation has a negative-valued solution decaying to 0 at +infinity as o(t(-mu)). In this way, we are not in the reach of any of the developments from the recent paper [C.H Ou, J.S.W. Wong, Forced oscillation of nth-order functional differential equations, J. Math. Anal. Appl. 262 (2001) 722-732]. (C) 2008 Elsevier Inc. All rights reserved.Article Citation - WoS: 393Citation - Scopus: 432Anomalous Diffusion Expressed Through Fractional Order Differential Operators in the Bloch-Torrey Equation(Academic Press inc Elsevier Science, 2008) Abdullah, Osama; Baleanu, Dumitru; Zhou, Xiaohong Joe; Magin, Richard L.Diffusion weighted MRI is used clinically to detect and characterize neurodegenerative, malignant and ischemic diseases. The correlation between developing pathology and localized diffusion relies on d iffusi on -weighted pulse sequences to probe biophysical models of molecular diffusion-typically exp[-(bD)]-where D is the apparent diffusion coefficient (turn (2)/s) and b depends on the specific gradient pulse sequence parameters. Several recent studies have investigated the so-called anomalous diffusion stretched exponential model-exp[-(bD)(alpha)], where alpha is a measure of tissue complexity that can be derived from fractal models of tissue structure. In this paper we propose an alternative derivation for the stretched exponential model using fractional order space and time derivatives. First, we consider the case where the spatial Laplacian in the Bloch-Torrey equation is generalized to incorporate a fractional order Brownian model of diffusivity. Second, we consider the case where the time derivative in the Bloch-Torrey equation is replaced by a Riemann-Liouville fractional order time derivative expressed in the Caputo form. Both cases revert to the classical results for integer order operations. Fractional order dynamics derived for the first case were observed to fit the signal attenuation in diffusion-weighted images obtained from Sephadex gels, human articular cartilage and human brain. Future developments of this approach may be useful for classifying anomalous diffusion in tissues with developing pathology. (c) 2007 Elsevier Inc. All rights reserved.Article Decoupling Structure of the Principal Sigma Model-Maxwell Interactions(Academic Press inc Elsevier Science, 2008) Yilmaz, Nejat T.The principal sigma model and Abelian gauge fields coupling is studied. By expressing the first-order formulation of the gauge field equations an implicit on-shell scalar-gauge field decoupling structure is revealed. It is also shown that due to this decoupling structure the scalars of the theory belong to the pure sigma model and the gauge fields sector consists of a number of coupled Maxwell theories with currents partially induced by the scalars. (C) 2008 Elsevier Inc. All rights reserved.Article Citation - WoS: 168Citation - Scopus: 192The Hamilton Formalism With Fractional Derivatives(Academic Press inc Elsevier Science, 2007) Nawafleh, Khaled I.; Hijjawi, Raed S.; Muslih, Sami I.; Baleanu, Dumitru; Rabei, Eqab M.Recently the traditional calculus of variations has been extended to be applicable for systems containing fractional derivatives. In this paper the passage from the Lagrangian containing fractional derivatives to the Hamiltonian is achieved. The Hamilton's equations of motion are obtained in a similar manner to the usual mechanics. In addition, the classical fields with fractional derivatives are investigated using Hamiltonian formalism. Two discrete problems and one continuous are considered to demonstrate the application of the formalism, the results are obtained to be in exact agreement with Agrawal's formalism. (c) 2006 Elsevier Inc. All rights reserved.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.