Noyes-Whitney Dinamik Denkleminin Çözümlerinin Oransal Kesirli Türeve Göre İncelenmesi

Yıl 2024, Cilt: 36 Sayı: 1, 35 - 41, 28.03.2024

Tuba Gülşen , Melek Dönmez

Öz

Katı maddenin bir çözücü içinde çözünmesinin dinamiği temel olarak Noyes-Whitney denklemi ile tanımlanır. Karmaşık süreçleri hafıza etkileri ve yerel olmayan davranışlarla simüle etmek amacıyla kesirli analiz güçlü bir temel sunar. Noyes-Whitney denklemini kesirli türevler kullanarak çözerek hafızanın ve yerel olmamanın çözünme kinetiği üzerindeki etkilerini araştırıyoruz. Matematiksel analiz yoluyla, Noyes-Whitney denkleminin orantılı kesirli türevle davranışını açıklığa kavuşturarak kimya mühendisliği ve farmasötik uygulamalardaki çözünme süreçlerine ilişkin bilgiler sağlıyoruz. Bu çalışmada oransal kesirli türevin özelliklerini ve teorilerini zaman ölçeğinde verdikten sonra oransal kesirli Noyes-Whitney dinamik denklemini başlangıç koşulunun varlığında ve oransal kesirli türev üzerinden çeşitli zaman ölçeklerinde birkaç örnek vererek çözüyoruz.

Anahtar Kelimeler

Noyes-Whitney dinamik denklemi, kesirli hesap, oransal kesirli türev

Investigating Solutions of the Noyes-Whitney Dynamic Equation via Proportional Fractional Derivative

Öz

The dynamics of solid material dissolving in a solvent are fundamentally described by the Noyes-Whitney equation. For the purpose of simulating intricate processes with memory effects and non-local behaviors, fractional calculus offers a strong foundation. We explore the effects of memory and non-locality on dissolution kinetics by solving the Noyes-Whitney equation using fractional derivatives. By means of mathematical analysis, we provide insights into the dissolving processes in chemical engineering and pharmaceutical applications by clarifying the behavior of the Noyes-Whitney equation with proportional fractional derivative. In this study, after discussing the characteristics and theories of the proportional fractional derivative on a time scale, we solve the proportional fractional Noyes-Whitney dynamic equation in the presence of the initial condition and give several examples on various time scales via the proportional fractional derivative.

Anahtar Kelimeler

Noyes-Whitney dynamic equation, fractional calculus, proportional fractional derivative

Kaynakça

• Abdeljewad T. On conformable fractional calculus. J Comput Appl Math 2015; 279: 57–66.
• Agarwal R, Bohner M, O'Regan D, Peterson A. Dynamic equations on time scales: a survey. J Comput Appl Math 2002; 141(1-2): 1-26.
• Alkan A. Improving homotopy analysis method with an optimal parameter for time-fractional Burgers equation. Karamanoğlu Mehmetbey Univ J Engin Natural Sci 2022; 4(2), 117-134.
• Anderson DR, Georgiev SG. Conformable Dynamic Equations on Time Scales. Chapman and Hall/CRC, 2020.
• Anderson DR, Ulness DJ. Newly defined conformable derivatives. Adv Dyn Syst Appl 2015; 10(2): 109-137.
• Aulbach B, Hilger S. A unified approach to continuous and discrete Dynamics. Qual Theory Differ Equ (Szeged, 1988), 37–56, Colloq Math Soc János Bolyai, 53 North-Holland, Amsterdam, 1990.
• Avcı HH, Anaç H. The New Conformable methods to solve conformable time-fractional generalized Burgers equation with proportional delay. Erciyes Univ Inst Sci Tech J Sci 2023; 39(2), 315-329.
• Bekiryazıcı Z, Merdan M, Kesemen T, Khaniyev T. Mathematical modeling of biochemical reactions under random effects. TJMCS 2016; 5, 8-18.
• Bektaş U, Anaç H. Q-homotopy Shehu analysis transform method of time-fractional coupled Burgers equations. Eskişehir Technical Univ J Sci Tech A-Appl Sci Eng 2023; 24(3), 177-191.
• Benkhettou N, Brito da Cruz AMC, Torres DFM. A fractional calculus on arbitrary time scales: Fractional differentiation and fractional integration. Signal Process 2015; 107: 230– 237.
• Benkhettou N, Hassani S, Torres, DFM. A conformable fractional calculus on arbitrary time scales. J King Saud Univ Sci 2016; 28(1): 93-98.
• Bohner M, Peterson A. Dynamic equations on time scales, An introduction with applications. Boston, MA: Birkhauser, 2001.
• Bohner M, Peterson A. Advances in Dynamic Equations on Time Scales. Boston: Birkhauser, 2004.
• Bohner M, Svetlin G. Multivariable dynamic calculus on time scales. Cham: Springer, 2016.
• Dokoumetzidis A, Papadopoulou V, Macheras P. Analysis of dissolution data using modified versions of Noyes–Whitney equation and the Weibull function. Pharm Res 2006; 23: 256-261.
• Erol AS, Anaç H, Olgun A. Numerical solutions of conformable time-fractional Swift-Hohenberg equation with proportional delay by the novel methods. Karamanoglu Mehmetbey Univ J Engin Natural Sci 2023; 5(1), 1-24.
• Gulsen T, Yilmaz E, Goktas S. Conformable fractional Dirac system on time scales. J Inequal Appl 2017; 2017(1): 161.
• Gülşen T, Acar M. Self-Adjoint Sturm-Liouville Dynamic Problem via Proportional Derivative. JIST 2023; 13(4): 2945-2957.
• Gülşen T, Yilmaz E, Kemaloğlu H. Conformable fractional Sturm-Liouville equation and some existence results on time scales. Turk J Math 2018; 42(3): 1348-1360.
• Hilger S. Analysis on measure chains a unified approach to continuous and discrete calculus. Results Math 1990; 18(1).
• Kartal A, Anaç H, Olgun A. Numerical solution of conformable time fractional generalized Burgers equation with proportional delay by new methods. Black Sea Sci J 2023; 13(2), 310-335.
• Kartal A, Anaç H, Olgun A. The new numerical solutions of conformable time fractional generalized Burgers equation with proportional delay. Gümüşhane Univ J Sci 2023; 13(4), 927-938.
• Katsanevakis S, Maravelias CD. Modelling fish growth: multi‐model inference as a better alternative to a priori using von Bertalanffy equation. Fish fish 2008; 9(2): 178-187.
• Katugampola U. A new fractional derivative with classical properties, arXiv:1410.6535v2, 2014.
• Khalil R, Horani MAl, Yousef A, Sababheh M. A new definition of fractional derivative. J Comput Appl Math 2014; 264: 57–66.
• Lindberg NO, Lundstedt, T. The relationship between the dissolution rate and the particle size of prednimustine: a disagreement with the noyes-whitney equation. Drug Dev Ind Pharm 1994; 20(16): 2547-2550.
• Ortigueira MD, Machado JT. What is a fractional derivative? J Comput Phys 2015; 293: 4-13.
• Öner BK, Anaç H. The Numerical Solutions of the Conformable Time-Fractional Noyes Field Model via a New Hybrid Method. IKJM 2023; 5(2), 76-91.
• Segi Rahmat MR. A new definition of conformable fractional derivative on arbitrary time scales. Adv Differ Equ 2019; 2019(1): 1-16.
• Shi Y, Wan A, Shi Y, Zhang Y, Chen Y. Experimental and mathematical studies on the drug release properties of aspirin loaded chitosan nanoparticles. BioMed Res Int 2014; 2014: 8 pages.
• Tomar BS, Tirumkudulu MS, Yu W, Berchielli A, Manthena S, Doshi P. A model to predict drug release from a single-layer osmotic controlled-release tablet. J Drug Deliv Sci Technol 2023; 80, 104138.
• Van der Zwaan I, Frenning G. A new modelling approach for dissolution of polydisperse powders. Int J Pharma 2023; 633, 122626.
• Wimalasiri VW, Dunuweera SP, Dunuweera AN, Rajapakse RMG. Research article Noyes-Whitney dissolution model-based pH-sensitive slow release of paclitaxel (taxol) from human hair-derived keratin microparticle carriers. BioMed Res Int 2021; 2021, Article ID 6657482: 8 pages.
• Yilmaz E, Gulsen T, Panakhov ES. Existence Results for a Conformable Type Dirac System on Time Scales in Quantum Physics. Appl Comput Math Int J 2022; 21(3): 279-291.