University of TabrizComputational Methods for Differential Equations2345-398210320220701Local fractal Fourier transform and applications5956071304910.22034/cmde.2021.42554.1832ENAlirezaKhalili GolmankhanehDepartment of Physics
Islamic Azad University, Urmia Branch
Urmia, Iran.0000-0002-5008-0163Karmina KamalAliFaculty of Science, Department of Mathematics, University of Zakho, Iraq.0000-0002-3815-4457ResatYilmazerFaculty of Science, Department of Mathematics, Firat University, Elazig, Turkey.0000-0002-5059-3882Mohammed Khalid AwadKaabarDepartment of Mathematics and Statistics, Washington State University, Pullman, WA, USA.Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia.0000-0003-2260-0341Journal Article20201030In this manuscript, we review fractal calculus and the analogues of both local Fourier transform with its related properties and Fourier convolution theorem are proposed with proofs in fractal calculus. The fractal Dirac delta with its derivative and the fractal Fourier transform of the Dirac delta is also defined. In addition, some important applications of the local fractal Fourier transform are presented in this paper such as the fractal electric current in a simple circuit, the fractal second order ordinary differential equation, and the fractal Bernoulli-Euler beam equation. All discussed applications are closely related to the fact that, in fractal calculus, a useful local fractal derivative is a generalized local derivative in the standard calculus sense. In addition, a comparative analysis is also carried out to explain the benefits of this fractal calculus parameter on the basis of the additional alpha parameter, which is the dimension of the fractal set, such that when α = 1, we obtain the same results in the standard calculus. https://cmde.tabrizu.ac.ir/article_13049_dc2ffbc2729ed53148807a978137b821.pdf