ISSN 2348-1196 (print) International Journal of Computer Science and Information Technology Research ISSN 2348-120X (online) Vol. 10, Issue 3, pp: (38-44), Month: July - September 2022, Available at: www.researchpublish.com
Application of Fractional Fourier series in Evaluating Fractional Integrals Chii-Huei Yu School of Mathematics and Statistics, Zhaoqing University, Guangdong, China DOI: https://doi.org/10.5281/zenodo.7049853
Published Date: 05-September-2022
Abstract: 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. Keyword: fractional Fourier series, fractional integrals, fractional Euler’s formula, fractional DeMoivre’s formula, new multiplication, fractional analytic functions.
I. INTRODUCTION In recent years, fractional calculus has attracted more and more attention due to its wide application in science and engineering [1-11]. However, the definition of fractional derivative is not unique, there are many useful definitions, including Riemann-Liouville (R-L) fractional derivative, Caputo fractional derivative, Grunwald-Letnikov (G-L) fractional derivative, Jumarie’s modified R-L fractional derivative [12-15]. Jumarie modified the definition of R-L fractional derivative with a new formula, and we obtained that the modified fractional derivative of a constant function is zero. Thus, it is easier to connect fractional calculus with traditional calculus by using this definition. In this paper, based on Jumarie type of R-L fractional calculus, we use the fractional Fourier series to solve the following three types of 𝛼-fractional integrals: ( 0𝐼𝑇𝛼𝛼 ) [
( 0𝐼𝑇𝛼𝛼 ) [
(𝑟 ∙ 𝑐𝑜𝑠𝛼 (𝑡 𝛼 ) + 𝑐)⨂[𝑟 2 ∙ 𝑐𝑜𝑠𝛼 (2𝑡 𝛼 ) + (𝑎 + 𝑏)𝑟 ∙ 𝑐𝑜𝑠𝛼 (𝑡 𝛼 ) + 𝑎𝑏] + 𝑟 ∙ 𝑠𝑖𝑛𝛼 (𝑡 𝛼 )⨂[𝑟 2 ∙ 𝑠𝑖𝑛𝛼 (2𝑡 𝛼 ) + (𝑎 + 𝑏)𝑟 ∙ 𝑠𝑖𝑛𝛼 (𝑡 𝛼 )] ], ⨂[2𝑎𝑏𝑟 2 ∙ 𝑐𝑜𝑠𝛼 (2𝑡 𝛼 ) + [2(𝑎 + 𝑏)𝑟 3 + 2𝑎𝑏(𝑎 + 𝑏)𝑟] ∙ 𝑐𝑜𝑠𝛼 (𝑡 𝛼 ) + 𝑟 4 + (𝑎 + 𝑏)2 𝑟 2 + 𝑎2 𝑏 2 ]⨂ −1 (1) (𝑟 ∙ 𝑐𝑜𝑠𝛼 (𝑡 𝛼 ) + 𝑐)⨂[𝑟 2 ∙ 𝑐𝑜𝑠𝛼 (2𝑡 𝛼 ) + (𝑎 + 𝑏)𝑟 ∙ 𝑐𝑜𝑠𝛼 (𝑡 𝛼 ) + 𝑎𝑏] + 𝑟 ∙ 𝑠𝑖𝑛𝛼 (𝑡 𝛼 )⨂[𝑟 2 ∙ 𝑠𝑖𝑛𝛼 (2𝑡 𝛼 ) + (𝑎 + 𝑏)𝑟 ∙ 𝑠𝑖𝑛𝛼 (𝑡 𝛼 )] ], ⨂[2𝑎𝑏𝑟 2 ∙ 𝑐𝑜𝑠𝛼 (2𝑡 𝛼 ) + [2(𝑎 + 𝑏)𝑟 3 + 2𝑎𝑏(𝑎 + 𝑏)𝑟] ∙ 𝑐𝑜𝑠𝛼 (𝑡 𝛼 ) + 𝑟 4 + (𝑎 + 𝑏)2 𝑟 2 + 𝑎2 𝑏 2 ]⨂ −1 ⨂𝑐𝑜𝑠𝛼 (𝑘𝑡 𝛼 )
(2) and ( 0𝐼𝑇𝛼𝛼 ) [
−(𝑟 ∙ 𝑐𝑜𝑠𝛼 (𝑡 𝛼 ) + 𝑐)⨂[𝑟 2 ∙ 𝑠𝑖𝑛𝛼 (2𝑡 𝛼 ) + (𝑎 + 𝑏)𝑟 ∙ 𝑠𝑖𝑛𝛼 (𝑡 𝛼 )] + 𝑟 ∙ 𝑠𝑖𝑛𝛼 (𝑡 𝛼 )⨂[𝑟 2 ∙ 𝑐𝑜𝑠𝛼 (2𝑡 𝛼 ) + (𝑎 + 𝑏)𝑟 ∙ 𝑐𝑜𝑠𝛼 (𝑡 𝛼 ) + 𝑎𝑏] ]. ⨂[2𝑎𝑏𝑟 2 ∙ 𝑐𝑜𝑠𝛼 (2𝑡 𝛼 ) + [2(𝑎 + 𝑏)𝑟 3 + 2𝑎𝑏(𝑎 + 𝑏)𝑟] ∙ 𝑐𝑜𝑠𝛼 (𝑡 𝛼 ) + 𝑟 4 + (𝑎 + 𝑏)2 𝑟 2 + 𝑎2 𝑏 2 ]⨂ −1 ⨂𝑠𝑖𝑛𝛼 (𝑘𝑡 𝛼 )
(3) Where 0 < α ≤ 1, 𝑎, 𝑏, 𝑐, 𝑟 are real numbers, 𝑘 is any positive integer, and 𝑎 ≠ 0, 𝑏 ≠ 0, 𝑎 ≠ 𝑏, |𝑟| < |𝑎|, |𝑟| < |𝑏|. The major methods used in this article are fractional Euler’s formula, fractional DeMoivre’s formula, and a new multiplication of fractional analytic functions. In fact, the results we obtained are natural generalization of classical calculus results. In addition, some examples are provided to illustrate our results.
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