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Journal articles on the topic 'Fractional Fourier series expansion'

Author: Grafiati

Published: 5 June 2025

Last updated: 1 August 2025

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1

Chii-Huei, Yu. "Fractional Fourier series Expansion of Two Types of Fractional Trigonometric Functions." International Journal of Electrical and Electronics Research 10, no. 3 (2022): 4–9. https://doi.org/10.5281/zenodo.7043902.

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<strong>Abstract:</strong> In this paper, we find the fractional Fourier series expansion of two type of fractional trigonometric functions based on Jumarie’s modified Riemann-Liouville (R-L) fractional calculus. A new multiplication of fractional analytic functions plays an important role in this paper. The main methods we used are fractional Euler’s formula and the fractional power series expansion of complex fractional analytic function. On the other hand, two examples are provided to illustrate our results. <strong>Keywords:</strong> fractional Fourier series expansion, fractio

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2

Yang, Yong-Ju, and Shun-Qin Wang. "Local Fractional Fourier Series Method for Solving Nonlinear Equations with Local Fractional Operators." Mathematical Problems in Engineering 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/481905.

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We apply the local fractional Fourier series method for solving nonlinear equation with local fractional operators. This method is the coupling of the local fractional Fourier series expansion method with other methods, such as the Yang-Laplace transformation method and the local fractional power series method, which effectively separates the variables of partial differential equation. Some testing nonlinear equations and equation systems are given to demonstrate the accuracy and applicability of the proposed approach.

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Chii-Huei, Yu. "Fractional Fourier Series Expansions of Two Types of Fractional Trigonometric Functions." International Journal of Electrical and Electronics Research 12, no. 4 (2024): 1–5. https://doi.org/10.5281/zenodo.13955269.

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<strong>Abstract:</strong> In this paper, we obtain the fractional Fourier series expansions of two types of fractional trigonometric functions. A new multiplication of fractional analytic functions plays an important role in this article. In fact, our results are generalizations of ordinary calculus results. <strong>Keywords:</strong> fractional Fourier series expansions, fractional trigonometric functions, new multiplication, fractional analytic functions. <strong>Title:</strong> Fractional Fourier Series Expansions of Two Types of Fractional Trigonometric Functions <strong>Author:</strong>

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Chii-Huei, Yu. "Study of Fractional Fourier Series Expansions of Two Types of Matrix Fractional Functions." International Journal of Mathematics and Physical Sciences Research 12, no. 2 (2024): 13–17. https://doi.org/10.5281/zenodo.14039321.

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<strong>Abstract:</strong> In this paper, based on a new multiplication of fractional analytic functions, we obtain the fractional Fourier series expansions of two types of matrix fractional functions. In fact, our results are generalizations of ordinary calculus results. <strong>Keywords:</strong> New multiplication, fractional analytic functions, fractional Fourier series expansions, matrix fractional functions. <strong>Title:</strong> Study of Fractional Fourier Series Expansions of Two Types of Matrix Fractional Functions <strong>Author:</strong> Chii-Huei Yu <strong>International Journal

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Soo-Chang Pei, Min-Hung Yeh, and Tzyy-Liang Luo. "Fractional Fourier series expansion for finite signals and dual extension to discrete-time fractional Fourier transform." IEEE Transactions on Signal Processing 47, no. 10 (1999): 2883–88. http://dx.doi.org/10.1109/78.790671.

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Chii-Huei, Yu. "Evaluating Fractional Fourier Series of Two Types of Matrix Fractional Functions Based on a New Multiplication of Fractional Analytic Functions." International Journal of Interdisciplinary Research and Innovations 12, no. 4 (2024): 40–43. https://doi.org/10.5281/zenodo.14177803.

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<strong>Abstract:</strong> In this paper, based on a new multiplication of fractional analytic functions, we use some techniques to obtain the fractional Fourier series expansions of two types of matrix fractional functions. Matrix fractional Euler’s formula and matrix fractional DeMoivre’s formula play important roles in this article. In fact, our results are generalizations of the results in ordinary calculus.  <strong>Keywords:</strong> New multiplication, fractional analytic functions, fractional Fourier series expansions, matrix fractional functions. <strong>Title:</stron

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Chii-Huei, Yu. "Evaluating Fractional Fourier Series Expansions of Two Types of Matrix Fractional Functions." International Journal of Novel Research in Physics Chemistry & Mathematics 11, no. 3 (2024): 28–32. https://doi.org/10.5281/zenodo.14064960.

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<strong>Abstract:</strong> In this paper, based on a new multiplication of fractional analytic functions, we find the fractional Fourier series expansions of two types of matrix fractional functions. Matrix fractional Euler’s formula and matrix fractional DeMoivre’s formula play important roles in this article. In fact, our results are generalizations of ordinary calculus results.     <strong>Keywords:</strong> New multiplication, fractional analytic functions, fractional Fourier series expansions, matrix fractional functions, matrix fractional Euler’s for

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8

Candan, Çag̃atay, and Haldun M. Ozaktas. "Sampling and series expansion theorems for fractional Fourier and other transforms." Signal Processing 83, no. 11 (2003): 2455–57. http://dx.doi.org/10.1016/s0165-1684(03)00196-8.

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9

Coëtmellec, Sébastien, Marc Brunel, Denis Lebrun, and Jean-Bernard Lecourt. "Fractional-order Fourier series expansion for the analysis of chirped pulses." Optics Communications 249, no. 1-3 (2005): 145–52. http://dx.doi.org/10.1016/j.optcom.2005.01.004.

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BRUNEL, MARC, and SEBASTIEN COETMELLEC. "ANALYSIS OF SPIDER INTERFEROGRAMS WITH THE FRACTIONAL-ORDER FOURIER SERIES EXPANSION." Journal of Nonlinear Optical Physics & Materials 15, no. 04 (2006): 501–11. http://dx.doi.org/10.1142/s0218863506003475.

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We propose the use of fractional-order Fourier series expansion for the characterization of ultrashort pulses. The analysis and reconstruction of simulated SPIDER interferograms is demonstrated with this decomposition process. We consider the case of ultrashort pulses (about 100 fs) that have been subject to second- and third-order dispersion. Our technique allows the simultaneous determination of the quadratic and cubic spectral phase coefficients of the pulses. The technique can be further used to analyze, filter or reconstruct noisy SPIDER interferograms.

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Abd El-Latief, A. M. "Application of Generalized Fractional Thermoelasticity Theory with Two Relaxation Times to an Electromagnetothermoelastic Thick Plate." Advances in Materials Science and Engineering 2016 (2016): 1–9. http://dx.doi.org/10.1155/2016/5821604.

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The fractional mathematical model of Maxwell’s equations in an electromagnetic field and the fractional generalized thermoelastic theory associated with two relaxation times are applied to a 1D problem for a thick plate. Laplace transform is used. The solution in Laplace transform domain has been obtained using a direct method and its inversion is calculated numerically using a method based on Fourier series expansion technique. Finally, the effects of the two fractional parameters (thermo and magneto) on variable fields distributions are made. Numerical results are represented graphically.

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Hamza, F., A. M. Abd El-Latief, and W. Khatan. "Thermomechanical Fractional Model of Two Immiscible TEMHD." Advances in Materials Science and Engineering 2015 (2015): 1–16. http://dx.doi.org/10.1155/2015/391454.

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We introduce a mathematical model of unsteady thermoelectric MHD flow and heat transfer of two immiscible fractional second-grade fluids, with thermal fractional parametersαiand mechanical fractional parametersβi,i=1,2. The Laplace transform with respect to time is used to obtain the solution in the transformed domain. The inversion of Laplace transform is obtained by using numerical method based on a Fourier-series expansion. The numerical results for temperature, velocity, and the stress distributions are represented graphically for different values ofαiandβi. The graphs describe the fractio

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Ahmad, Anwar, and Dumitru Baleanu. "On two backward problems with Dzherbashian-Nersesian operator." AIMS Mathematics 8, no. 1 (2022): 887–904. http://dx.doi.org/10.3934/math.2023043.

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<abstract><p>We investigate the initial-boundary value problems for a fourth-order differential equation within the powerful fractional Dzherbashian-Nersesian operator (FDNO). Boundary conditions considered in this manuscript are of the Samarskii-Ionkin type. The solutions obtained here are based on a series expansion using Riesz basis in a space corresponding to a non-self-adjoint spectral problem. Conditional to some regularity, consistency, alongside orthogonality dependence, the existence and uniqueness of the obtained solutions are exhibited by using Fourier method. Acquired r

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Brzeziński, Dariusz W. "Review of numerical methods for NumILPT with computational accuracy assessment for fractional calculus." Applied Mathematics and Nonlinear Sciences 3, no. 2 (2018): 487–502. http://dx.doi.org/10.2478/amns.2018.2.00038.

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AbstractIn the paper we present results of accuracy evaluation of numerous numerical algorithms for the numerical approximation of the Inverse Laplace Transform. The selected algorithms represent diverse lines of approach to this problem and include methods by Stehfest, Abate and Whitt, Vlach and Singhai, De Hoog, Talbot, Zakian and a one in which the FFT is applied for the Fourier series convergence acceleration. We use C++ and Python languages with arbitrary precision mathematical libraries to address some crucial issues of numerical implementation. The test set includes Laplace transforms c

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Kheybari, Samad, Farzaneh Alizadeh, Mohammad Taghi Darvishi, and Kamyar Hosseini. "A Fourier Series Technique for Approximate Solutions of Modified Anomalous Time-Fractional Sub-Diffusion Equations." Fractal and Fractional 8, no. 12 (2024): 718. https://doi.org/10.3390/fractalfract8120718.

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This article presents an innovative approximating technique for addressing modified anomalous time-fractional sub-diffusion equations (MAFSDEs) of the Caputo type. These equations generalize classical diffusion equations, which involve fractional derivatives with respect to time, capturing the non-local and history-dependent behavior typical in sub-diffusion processes. In such a model, the particle transports slower than in a standard diffusion, often due to obstacles or memory effects in the medium. The core of the proposed technique involves transforming the original problem into a family of

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Li, Hongwei, Xiaonan Wu, and Jiwei Zhang. "Numerical Solution of the Time-Fractional Sub-Diffusion Equation on an Unbounded Domain in Two-Dimensional Space." East Asian Journal on Applied Mathematics 7, no. 3 (2017): 439–54. http://dx.doi.org/10.4208/eajam.031116.080317a.

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AbstractThe numerical solution of the time-fractional sub-diffusion equation on an unbounded domain in two-dimensional space is considered, where a circular artificial boundary is introduced to divide the unbounded domain into a bounded computational domain and an unbounded exterior domain. The local artificial boundary conditions for the fractional sub-diffusion equation are designed on the circular artificial boundary by a joint Laplace transform and Fourier series expansion, and some auxiliary variables are introduced to circumvent high-order derivatives in the artificial boundary condition

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17

Li, Shu-Nan, and Bing-Yang Cao. "On Entropic Framework Based on Standard and Fractional Phonon Boltzmann Transport Equations." Entropy 21, no. 2 (2019): 204. http://dx.doi.org/10.3390/e21020204.

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Generalized expressions of the entropy and related concepts in non-Fourier heat conduction have attracted increasing attention in recent years. Based on standard and fractional phonon Boltzmann transport equations (BTEs), we study entropic functionals including entropy density, entropy flux and entropy production rate. Using the relaxation time approximation and power series expansion, macroscopic approximations are derived for these entropic concepts. For the standard BTE, our results can recover the entropic frameworks of classical irreversible thermodynamics (CIT) and extended irreversible

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18

Islam, Mohsin. "Fractional Order Two Temperature Generalized Thermoelasticity in a Symmetric Spherical Shell." Science & Technology Journal 8, no. 1 (2020): 91–104. http://dx.doi.org/10.22232/stj.2020.08.01.12.

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This paper deals with the determination of thermoelastic stresses, strain and conductive temperature in a spherically symmetric spherical shell in the context of the fractional order two temperature generalized thermoelasticity theory (2TT). The two temperature three-phase-lag thermoelastic model (2T3P) and two temperature Green Naghdi model III (2TGN-III) are combined into a unified formulation. There is no temperature at the outer boundary and thermal load is applied at the inner boundary. The basic equations have been written in the form of a vector-matrix differential equation in the Lapla

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Mondal, Sudip, Sadek Hossain Mallik, and M. Kanoria. "Fractional Order Two-Temperature Dual-Phase-Lag Thermoelasticity with Variable Thermal Conductivity." International Scholarly Research Notices 2014 (October 29, 2014): 1–13. http://dx.doi.org/10.1155/2014/646049.

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A new theory of two-temperature generalized thermoelasticity is constructed in the context of a new consideration of dual-phase-lag heat conduction with fractional orders. The theory is then adopted to study thermoelastic interaction in an isotropic homogenous semi-infinite generalized thermoelastic solids with variable thermal conductivity whose boundary is subjected to thermal and mechanical loading. The basic equations of the problem have been written in the form of a vector-matrix differential equation in the Laplace transform domain, which is then solved by using a state space approach. T

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Zhang, Sumei, Hongquan Yong, and Haiyang Xiao. "Option Pricing with Fractional Stochastic Volatilities and Jumps." Fractal and Fractional 7, no. 9 (2023): 680. http://dx.doi.org/10.3390/fractalfract7090680.

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Empirical studies suggest that asset price fluctuations exhibit “long memory”, “volatility smile”, “volatility clustering” and asset prices present “jump”. To fit the above empirical characteristics of the market, this paper proposes a fractional stochastic volatility jump-diffusion model by combining two fractional stochastic volatilities with mixed-exponential jumps. The characteristic function of the log-return is expressed in terms of the solution of two-dimensional fractional Riccati equations of which closed-form solution does not exist. To obtain the explicit characteristic function, we

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21

Singh, Biswajit, Smita Pal (Sarkar), and Krishnendu Barman. "Memory-dependent derivative under generalized three-phase-lag thermoelasticity model with a heat source." Multidiscipline Modeling in Materials and Structures 16, no. 6 (2020): 1337–56. http://dx.doi.org/10.1108/mmms-10-2019-0182.

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PurposeThis study aims to attempt to construct a new mathematical model of the generalized thermoelasiticity theory based on the memory-dependent derivative (MDD) considering three-phase-lag effects. The governing equations of the problem associated with kernel function and time delay are illustrated in the form of vector matrix differential equations. Implementing Laplace and Fourier transform tools, the problem is sorted out analytically by an eigenvalue approach method. The inversion of Laplace and Fourier transforms are executed, incorporating series expansion procedures. Displacement comp

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22

Lovejoy, Shaun. "Fractional relaxation noises, motions and the fractional energy balance equation." Nonlinear Processes in Geophysics 29, no. 1 (2022): 93–121. http://dx.doi.org/10.5194/npg-29-93-2022.

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Abstract. We consider the statistical properties of solutions of the stochastic fractional relaxation equation and its fractionally integrated extensions that are models for the Earth's energy balance. In these equations, the highest-order derivative term is fractional, and it models the energy storage processes that are scaling over a wide range. When driven stochastically, the system is a fractional Langevin equation (FLE) that has been considered in the context of random walks where it yields highly nonstationary behaviour. An important difference with the usual applications is that we inst

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23

Lukomsky, V. P., and I. S. Gandzha. "Fractional Fourier approximations for potential gravity waves on deep water." Nonlinear Processes in Geophysics 10, no. 6 (2003): 599–614. http://dx.doi.org/10.5194/npg-10-599-2003.

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Abstract. In the framework of the canonical model of hydrodynamics, where fluid is assumed to be ideal and incompressible, waves are potential, two-dimensional, and symmetric, the authors have recently reported the existence of a new type of gravity waves on deep water besides well studied Stokes waves (Lukomsky et al., 2002b). The distinctive feature of these waves is that horizontal water velocities in the wave crests exceed the speed of the crests themselves. Such waves were found to describe irregular flows with stagnation point inside the flow domain and discontinuous streamlines near the

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Bia, Pietro, Luciano Mescia, and Diego Caratelli. "Fractional Calculus-Based Modeling of Electromagnetic Field Propagation in Arbitrary Biological Tissue." Mathematical Problems in Engineering 2016 (2016): 1–11. http://dx.doi.org/10.1155/2016/5676903.

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The interaction of electromagnetic fields and biological tissues has become a topic of increasing interest for new research activities in bioelectrics, a new interdisciplinary field combining knowledge of electromagnetic theory, modeling, and simulations, physics, material science, cell biology, and medicine. In particular, the feasibility of pulsed electromagnetic fields in RF and mm-wave frequency range has been investigated with the objective to discover new noninvasive techniques in healthcare. The aim of this contribution is to illustrate a novel Finite-Difference Time-Domain (FDTD) schem

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Dadsena, Ravi, Deboleena Sadhukhan, and Ramakrishnan Swaminathan. "Differentiation of Mild Cognitive Impairment Conditions in MR Images using Fractional order Jacobi Fourier Moment Features." Current Directions in Biomedical Engineering 7, no. 2 (2021): 724–27. http://dx.doi.org/10.1515/cdbme-2021-2185.

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Abstract Mild Cognitive Impairment (MCI) is the asymptomatic, preclinical transitional stage among aging and Alzheimer’s Disease (AD). Detection of MCI can ensure the timely intervention required to manage the disease’s severity. Morphological alterations of Lateral Ventricle (LV) is considered as a significant biomarker for disease diagnosis. This research aims to analyze the shape alterations of the LV region using Fractional Order Jacobi Fourier Moment (FOJFM) features, which are categorized by their generic nature and capabilities to perform time-frequency analysis. T1-weighted transaxial

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Aban, Ulbolsyn, and Batirkhan Turmetov. "ON THE SOLVABILITY OF DIRECT AND INVERSE PROBLEMS FOR A CLASS OF DEGENERATE PARABOLIC EQUATIONS WITH INVOLUTION." Q A Iasaýı atyndaǵy Halyqaralyq qazaq-túrіk ýnıversıtetіnіń habarlary (fızıka matematıka ınformatıka serııasy) 32, no. 1 (2025): 15–30. https://doi.org/10.47526/2025-1/2524-0080.19.

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In this paper, for degenerate diffusion equations with involution, the solvability of the direct and inverse problems for determining the right-hand side is studied. The equation with a fractional derivative in the Caputo sense is considered. The elliptic part of the studied equation involves a nonlocal analogue of the Laplace operator with a coefficient depending on the time variable. By studying these problems with respect to the time variable, we obtain a one-dimensional degenerate equation with a fractional Caputo derivative. The solution of this equation is expressed by a special function

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GOREV, Vyacheslav, Alexander GUSEV, Valerii KORNIIENKO, Yana SHEDLOVSKA, and Ivan LAKTIONOV. "POLYNOMIAL SOLUTIONS FOR THE KOLMOGOROV–WIENER PREDICTION OF MODELED SMOOTHED HEAVY-TAIL PROCESS." Information Technology: Computer Science, Software Engineering and Cyber Security, no. 1 (June 12, 2024): 28–34. http://dx.doi.org/10.32782/it/2024-1-4.

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Nowadays telecommunication traffic in systems with data packet transfer is considered as a heavy-tail random process. In a couple of rather simple models traffic is considered to be stationary one. In our recent papers we generated modeled heavy-tail data, which is based on the smoothing of the fractional Gaussian noise. In particular, the applicability if the continuous Kolmogorov–Wiener filter to the prediction of such data was investigated, the corresponding Wiener–Hopf integral equation was solved on the basis of the truncated Walsh function expansion. However, a question occurs – may anot

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Mittal, Gaurav, and Vinayak Kulkarni. "Dual-phase-lag thermoelastic problem in finite cylindrical domain with relaxation time." Multidiscipline Modeling in Materials and Structures 14, no. 5 (2018): 837–56. http://dx.doi.org/10.1108/mmms-03-2018-0041.

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Purpose The purpose of this paper is to frame a dual-phase-lag model using the fractional theory of thermoelasticity with relaxation time. The generalized Fourier law of heat conduction based upon Tzou model that includes temperature gradient, the thermal displacement and two different translations of heat flux vector and temperature gradient has been used to formulate the heat conduction model. The microstructural interactions and corresponding thermal changes have been studied due to the involvement of relaxation time and delay time translations. This results in achieving the finite speed of

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29

Lamba, N. K., and K. C. Deshmukh. "MEMORY DEPENDENT RESPONSE IN AN INFINITELY LONG THERMOELASTIC SOLID CIRCULAR CYLINDER." PNRPU Mechanics Bulletin, no. 1 (December 15, 2024): 5–12. http://dx.doi.org/10.15593/perm.mech/2024.1.01.

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Memory-dependent derivatives (MDD) have physical meaning, and compared to fractional derivatives, they are more suitable and convenient for temporal remodeling. In this study, the temperature and stress distributions in an infinitely extended generalized thermally elastic solid circular cylinder have been investigated by utilizing the concept of a memory-dependent heat conduction model. The homogeneous, isotropic, infinitely long solid circular cylinder is considered to have a lateral surface that is free of traction and is subjected to a known surrounding temperature. In the domain of the int

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Bezvesilnaya, E., and V. Ilchenko. "Method of measuring precession details on a coordinate measuring machine." System technologies 2, no. 157 (2025): 128–34. https://doi.org/10.34185/1562-9945-2-157-2025-13.

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The article focuses on the development and implementation of an effective methodology for measuring high-precision parts on coordinate measuring machines (CMM). The proposed approach addresses the challenges associated with complex geometry measurements under variable environmental conditions by combining advanced mathematical modeling techniques with adaptive error compensation algorithms. The mathematical foundation is based on the application of tensor formalism in Riemannian space, which allows for more precise model-ing of geometric errors using Christoffel symbols and covariant derivativ

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TARASOV, VASILY E. "FRACTIONAL DERIVATIVE AS FRACTIONAL POWER OF DERIVATIVE." International Journal of Mathematics 18, no. 03 (2007): 281–99. http://dx.doi.org/10.1142/s0129167x07004102.

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Definitions of fractional derivatives as fractional powers of derivative operators are suggested. The Taylor series and Fourier series are used to define fractional power of selfadjoint derivative operator. The Fourier integrals and Weyl quantization procedure are applied to derive the definition of fractional derivative operator. Fractional generalization of concept of stability is considered.

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Wang, Chen, Xianming Hou, Qingyan Wu, Pei Dang, and Zunwei Fu. "Fractional Fourier Series on the Torus and Applications." Fractal and Fractional 8, no. 8 (2024): 494. http://dx.doi.org/10.3390/fractalfract8080494.

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This paper introduces the fractional Fourier series on the fractional torus and proceeds to investigate several fundamental aspects. Our study includes topics such as fractional convolution, fractional approximation, fractional Fourier inversion, and the Poisson summation formula. We also explore the relationship between the decay of its fractional Fourier coefficients and the smoothness of a function. Additionally, we establish the convergence of the fractional Féjer means and Bochner–Riesz means. Finally, we demonstrate the practical applications of the fractional Fourier series, particularl

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Chii-Huei, Yu. "Application of Fractional Fourier series in Evaluating Fractional Integrals." International Journal of Computer Science and Information Technology Research 10, no. 3 (2022): 38–44. https://doi.org/10.5281/zenodo.7049853.

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<strong>Abstract:</strong> In this paper, we use fractional Fourier series to solve three types of fractional integrals based on Jumarie’s modified Riemann-Liouville (R-L) fractional calculus. Fractional Euler’s formula, fractional DeMoivre’s formula and a new multiplication of fractional analytic functions play important roles in this paper. In fact, our results are generalization of the traditional calculus results. On the other hand, three examples are given to illustrate our results.  <strong>Keywords:</strong> fractional Fourier series, fractional integrals, fractio

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Yu, Chiihuei. "Fractional fourier series and its applications." Journal of Physics: Conference Series 1976, no. 1 (2021): 012080. http://dx.doi.org/10.1088/1742-6596/1976/1/012080.

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Zhu, H., M. Ding, and Y. Li. "Gibbs phenomenon for fractional Fourier series." IET Signal Processing 5, no. 8 (2011): 728. http://dx.doi.org/10.1049/iet-spr.2010.0348.

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Chii-Huei, Yu. "A Study on Fractional Fourier Series of Four Types of Matrix Fractional Functions." International Journal of Computer Science and Information Technology Research 12, no. 4 (2024): 34–38. https://doi.org/10.5281/zenodo.14177896.

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<strong>Abstract: </strong>In this paper, based on a new multiplication of fractional analytic functions, we find the fractional Fourier series of four types of matrix fractional functions. Matrix fractional Euler’s formula and matrix fractional DeMoivre’s formula play important roles in this article. In fact, our results are generalizations of ordinary calculus results.  <strong>Keywords:</strong> New multiplication, fractional analytic functions, fractional Fourier series, matrix fractional functions. <strong>Title:</strong> A Study on Fractional Fourier Series of Four Types

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Li, Zheng-Biao, and Wei-Hong Zhu. "Fractional series expansion method for fractional differential equations." International Journal of Numerical Methods for Heat & Fluid Flow 25, no. 7 (2015): 1525–30. http://dx.doi.org/10.1108/hff-05-2014-0160.

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Purpose – The purpose of this paper is to suggest a new analytical technique called the fractional series expansion method for solving linear fractional differential equations (FDEs). Design/methodology/approach – This method is based on the idea of Kantorovich method, convergent series, and the modified Riemann-Liouville derivative. Findings – This work suggests a new analytical technique. The FDEs are described in Jumarie’s sense. Originality/value – It finds a new method for solving linear FDEs. The solution procedure is elucidated by two examples.

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Montiel, M. Eugenia, Alberto S. Aguado, and Ed Zaluska. "Fourier Series Expansion of Irregular Curves." Fractals 05, no. 01 (1997): 105–19. http://dx.doi.org/10.1142/s0218348x97000115.

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Fourier theory provides an important approach to shape analyses; many methods for the analysis and synthesis of shapes use a description based on the expansion of a curve in Fourier series. Most of these methods have centered on modeling regular shapes, although irregular shapes defined by fractal functions have also been considered by using spectral synthesis. In this paper we propose a novel representation of irregular shapes based on Fourier analysis. We formulate a parametric description of irregular curves by using a geometric composition defined via Fourier expansion. This description al

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Radchenko, V. M. "Fourier Series Expansion of Stochastic Measures." Theory of Probability & Its Applications 63, no. 2 (2018): 318–26. http://dx.doi.org/10.1137/s0040585x97t989064.

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Nakhman, A. D. "Generalized Fractional Integrals of the Fourier Series." Vestnik Tambovskogo gosudarstvennogo tehnicheskogo universiteta 28, no. 3 (2022): 496–506. http://dx.doi.org/10.17277/vestnik.2022.03.pp.496-506.

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Hu, Ming-Sheng, Ravi P. Agarwal, and Xiao-Jun Yang. "Local Fractional Fourier Series with Application to Wave Equation in Fractal Vibrating String." Abstract and Applied Analysis 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/567401.

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We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the Mittag-Leffler function.

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Bouchenak, Ahmed, Khalil Roshdi, and Alhorani Mohammed. "Fractional Fourier Series with Separation of Variables Technique and Its Application on Fractional differential Equations." WSEAS TRANSACTIONS ON MATHEMATICS 20 (September 13, 2021): 461–69. http://dx.doi.org/10.37394/23206.2021.20.48.

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When using some classical methods, such us separation of variables; it is impossible to find a general solution for some differential equations. Therefore, we suggest adding conformable fractional Fourier series to get a new technique to solve fractional Benjamin Bana Mahony and Heat Equations. Furtheremore, we give new numerical approximation for functions using mathematica coding called conformable fractional Fourier series approximation

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Shokoohinia, Mohammad Reza, and Mohammad Mehdi Fateh. "Robust dynamic sliding mode control of robot manipulators using the Fourier series expansion." Transactions of the Institute of Measurement and Control 41, no. 9 (2018): 2488–95. http://dx.doi.org/10.1177/0142331218802357.

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This paper presents a robust dynamic sliding mode control for electrically driven robot manipulators. The control law computes the motor voltages based on the voltage control strategy. Uncertainties are estimated using the Fourier series expansion and the truncation error is compensated. The Fourier coefficients are tuned based on the stability analysis. The contribution of this paper is designing a robust controller using a novel adaptive Fourier series expansion. In comparison with previous related works based on the Fourier series expansion, the superiority of this paper is presenting an ad

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Pashaei, Ronak, Mohammad Sadegh Asgari, and Amir Pishkoo. "Conformable Derivatives in Laplace Equation and Fractional Fourier Series Solution." International Annals of Science 9, no. 1 (2019): 1–7. http://dx.doi.org/10.21467/ias.9.1.1-7.

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In this paper the solution of conformable Laplace equation, \frac{\partial^{\alpha}u(x,y)}{\partial x^{\alpha}}+ \frac{\partial^{\alpha}u(x,y)}{\partial y^{\alpha}}=0, where 1 < α ≤ 2 has been deduced by using fractional fourier series and separation of variables method. For special cases α =2 (Laplace's equation), α=1.9, and α=1.8 conformable fractional fourier coefficients have been calculated. To calculate coefficients, integrals are of type "conformable fractional integral".

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Zainal, Nor Hafizah, and Adem Kılıçman. "Solving Fractional Partial Differential Equations with Corrected Fourier Series Method." Abstract and Applied Analysis 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/958931.

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The corrected Fourier series (CFS) is proposed for solving partial differential equations (PDEs) with fractional time derivative on a finite domain. In the previous work, we have been solving partial differential equations by using corrected Fourier series. The fractional derivatives are described in Riemann sense. Some numerical examples are presented to show the solutions.

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Chen, Zhi-Yong, Carlo Cattani, and Wei-Ping Zhong. "Signal Processing for Nondifferentiable Data Defined on Cantor Sets: A Local Fractional Fourier Series Approach." Advances in Mathematical Physics 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/561434.

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From the signal processing point of view, the nondifferentiable data defined on the Cantor sets are investigated in this paper. The local fractional Fourier series is used to process the signals, which are the local fractional continuous functions. Our results can be observed as significant extensions of the previously known results for the Fourier series in the framework of the local fractional calculus. Some examples are given to illustrate the efficiency and implementation of the present method.

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Zhao, Yang, Dumitru Baleanu, Mihaela Cristina Baleanu, De-Fu Cheng, and Xiao-Jun Yang. "Mappings for Special Functions on Cantor Sets and Special Integral Transforms via Local Fractional Operators." Abstract and Applied Analysis 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/316978.

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The mappings for some special functions on Cantor sets are investigated. Meanwhile, we apply the local fractional Fourier series, Fourier transforms, and Laplace transforms to solve three local fractional differential equations, and the corresponding nondifferentiable solutions were presented.

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Samad, Muhammad Adnan, Yuanqing Xia, Nader Al-Rashidi, Saima Siddiqui, Muhammad Younus Bhat, and Huda M. Alshanbari. "Generalized Sampling Theory in the Quaternion Domain: A Fractional Fourier Approach." Fractal and Fractional 8, no. 12 (2024): 748. https://doi.org/10.3390/fractalfract8120748.

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The field of quaternions has made a substantial impact on signal processing research, with numerous studies exploring their applications. Building on this foundation, this article extends the study of sampling theory using the quaternion fractional Fourier Transform (QFRFT). We first propose a generalized sampling expansion (GSE) for fractional bandlimited signals via the QFRFT, extending the classical Papoulis expansion. Next, we design fractional quaternion Fourier filters to reconstruct both the signals and their derivatives, based on the GSE and QFRFT properties. We illustrate the practica

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Biermé, Hermine, and Hans-Peter Scheffler. "Fourier Series Approximation of Linear Fractional Stable Motion." Journal of Fourier Analysis and Applications 14, no. 2 (2008): 180–202. http://dx.doi.org/10.1007/s00041-008-9011-7.

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Yangklan, Pussadee, Vichian Laohakosol, and Sukrawan Mavecha. "Integer factorization and finite Fourier series expansion." Lithuanian Mathematical Journal 61, no. 2 (2021): 274–84. http://dx.doi.org/10.1007/s10986-020-09506-5.

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