MİTTAG LEFFLER ÇEKİRDEĞİ İLE LOKAL TÜREVLER İÇİN ÇİFT KATLI LAPLACE DÖNÜŞÜMÜ TEORİSİ VE UYGULAMALARI

Yıl 2025, Cilt: 25 Sayı: 4, 774 - 784, 04.08.2025

Burak Özküçük , Erdal Baş

https://doi.org/10.35414/akufemubid.1569312

Öz

Bu makale, Mittag-Leffler fonksiyonunu içeren altı parametreli uyumlu türevin genel formu olan M-türevi çift katlı Laplace dönüşümünü tanımlamayı amaçlamaktadır. Birkaç teoremle ifade edilir ve bize M-türevi kısmi diferansiyel denklemleri çözmek için kullanışlı ve güvenilir bir yöntem verecektir. Ayrıca, verilen bu tanım ve teoremlerin kesirli kısmi diferansiyel denklemlere uygulanması gösterilmiştir. Matematiksel, mühendislik ve fiziksel modellerle eşleşebilecek M-türevlerini içeren kısmi diferansiyel denklemlere çözümler bulmak, bu dönüşümün kullanılmasıyla gerçekleştirilebilir.

Anahtar Kelimeler

M-türev , Laplace dönüşümü , İki katlı M- Laplace dönüşümü , Mattag Leffler fonksiyonu

THEORY AND APPLICATIONS OF THE DOUBLE LAPLACE TRANSFORM FOR LOCAL DERİVATİVES WİTH THE MITTAG LEFFLER KERNEL

Öz

This article aims to define the 𝑀-derivative double Laplace transform, which is the general form of the six-parameter conformable derivative involving the Mittag-Leffler function. It is expressed with several theorems and will give us a useful and dependable method for solving fractional 𝑀-derivative partial differenial equations. Furthermore, the application of these given definitions and theorems to fractional partial differential equations is shown. Finding solutions to partial differential equations containing 𝑀-derivatives that can match mathematical, engineering, and physical mode ls may be accomplished with the use of this transformation.

Anahtar Kelimeler

M-derivative , Laplace transform , Double M-Derivative , Truncated Mittag-Leffler function

Kaynakça

• Abdeljawad, T., 2015. On Conformable fractional calulus, Journal of Computational Applied Mathematics, 279, 57-66. https://doi.org/10.48550/arXiv.1402.6892
• Bas, E., Acay, B., and Abdeljawad, T., 2020. Non-Local fractional calculus from different viewpoint generated by truncated M-derivative, Journal of Computational and Applied Mathematics, 366, 112410. https://doi.org/10.1016/j.cam.2019.112410
• Bas, E., Acay, B., 2020. The direct spectral problem via local derivative including truncated Mittag-Leffler function, Applied Mathematıcs and Computation, 366, 124787. https://doi.org/10.1016/j.amc.2019.124787
• Jarad, F., Uğurlu, E., Abdeljawad, T. and Baleanu, D., 2017. On a new class of fractional operators, Advances in Difference Equations, 2017;247. https://doi.org/10.1186s13662-017-130-z
• Jarad, F., Uğurlu, E., Abdeljawad, T., D., 2020. Generalized fractional derivatives and Laplace transform, American Institute of Mathematical Sciences, 13, 3, 709-722. https://dx.doi.org/10.3934/dcdss.2020039.
• Katugampola, U.N., 2017. A new approach to generalized fractional derivatives, mBulletın of Mathematıcal Analysıs and Applıcatıons, 16, 4, 1-15. https://dx.doi.org/10.48550/arXiv.1106.0965
• Khalil R., Al Horani, M., Yousef, A., Sababheh, M., 2014. A new definition of derivative, Journal of Computational and Applied Mathematics, 264, 65-70. https://doi.org/10.1016/j.cam.2014.01.002
• Kilbas, A., A., Srivastava, H., M., and Trujillo, J., J., 2006. Theory and Applications of Fractional Differential Equations, Jan van Mill (editor), Elsevıer B. V. Amsterdam, Netherlands, 283-326.
• Kurt, A.,2018. Comformable Laplace Dönüşümleri ve Uygulamaları, Doctoral Thesis, Institute of Science, Selçuk Üniversiy, 93.
• Ozkan, O., Kurt, A., 2018. On Conformable double Laplace transform, Opt quant Electron, 50, 103. https://doi.org/10.1007/s11082-018-1372-9
• Sousa, J., V., D., C., Oliviera, E., C., D., 2017. A new Truncated M-fractional derivate type unifying some fractional derivative types with classical properties, International Journal of Analysis and Applications, 16, 1, 83-96. https://doi.org/10.28924/2291-8639