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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013, Article ID 379856, 2 pageshttp://dx.doi.org/10.1155/2013/379856

EditorialRecent Developments in Integral Transforms, Special Functions,and Their Extensions to Distributions Theory

Adem KJlJçman,1 Mustafa Bayram,2 and Hassan Eltayeb3

1 Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia,43400 Serdang, Selangor, Malaysia

2 Department of Mathematical Engineering, Chemical and Metallurgical Faculty, Yildiz Technical University,Esenler, 34210 Istanbul, Turkey

3 Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Correspondence should be addressed to Adem Kılıcman; [email protected]

Received 2 December 2013; Accepted 2 December 2013

Copyright © 2013 Adem Kılıcman et al.This is an open access article distributed under the Creative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The distributions are also known as generalized functionswhich generalize the idea of classical functions and allow usto extend the concept of derivative not only to all continuousfunctions but also to the discontinuous functions in theclassical sense. The theory of distributions has applicationsin various fields especially in science and engineering wherethere are many noncontinuous phenomena which mightnaturally lead to differential equations whose solutions aredistributions. For example, such as the delta distribution isnot mathematical function in classical sense. However it isvery useful in several applications, therefore the distributionscan help us to develop an operational calculus in orderto investigate linear ordinary differential equations as wellas partial differential equations with constant and variablecoefficients through their fundamental solutions.

By considering some regular operations which are validfor ordinary functions such as addition, multiplication byscalar scan is extended into distributions. Other operationscan only be defined for certain restricted subclasses; theseare also known as irregular operations. However, generalizedfunctions are useful tools to extend the concept of deriva-tives and further to use them to formulate the generalizedsolutions of partial differential equations. They are alsovery important in physics and engineering where manynoncontinuous problems naturally might lead to differentialequations whose solutions are distributions, such as theDiracdelta distribution. This special issue is focused on someintegrals transform, special functions, and their applications

with fractional orders. In some papers, further relationshipbetween linear and nonlinear partial differential equationsand generalized functions was also considered.

We are pleased to announce the completion of this specialissue onRecentDevelopments in Integral Transforms, SpecialFunctions, andTheir Extensions toDistributionsTheory.Thisspecial issue was opened in late August of 2012 and closed inlate May of 2013.

In this special issue, a total of 33 articles were publishedand they cover a wide range of special functions changes fromtheoretical sides of special functions to their applications insolutions of certain differential equations by mostly focusingon some special functions in several areas. There werealso some submissions on the different types of integraltransforms and their applications including some works ongeneralized functions. For example, the modeling of thermaldistributions around a barrier at the interface of coatingand substrate was studied by A. Sahin; A. Secer studied thenumerical solution and simulation of second-order parabolicPDEs with Sinc-Galerkin method which covers to solvesecond-order PDEs by numerical method. Similarly, someintegrals involving q-Laguerre polynomials and applicationswere studied by J. Cao.

An efficient pseudospectral method was reported to solvea class of nonlinear optimal control problems, which werepresented by E. Tohidi et al. By M. Inc et al., numerical solu-tions of the second-order one-dimensional telegraph equa-tion were studied which was based on the reproducing kernel

2 Abstract and Applied Analysis

Hilbert space method; in an interesting work by H.-L. Wuand J.-C. Lan, it was reported that Lipschitz estimates canbe applied to fractional multilinear singular integrals onvariable exponent Lebesgue spaces. In a paper by D. Kumaret al., an efficient approach for fractional harry dym equationby using Sumudu transform was presented. The stabilityof trigonometric functional equations in distributions andhyperfunctions was studied by J. Chung and J. Chang.

Further, by A. Atangana and A. Secer, the fractional orderderivatives and table of fractional derivatives of some specialfunctions were reported.

Since it would be very lengthy to list all the contributions,thus, we suggest that the readers read the full special issue tosee the further details of each study.

Acknowledgments

The editors would like to express their gratitude to all theauthors for their contribution and collaboration and to themany reviewers who were involved in the reviewing processand for their valuable comments, suggestions, and timelyreports. Further we also acknowledge the assistance and helpwhich were provided by the editorial board members of thejournal during the preparation of this special issue as well asduring the smooth publishing process.

Adem KılıcmanMustafa BayramHassan Eltayeb

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