Kesir mertebeden diferensiyel denklemlerin sayısal çözümleri
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Dosyalar
Tarih
2024
Yazarlar
Dergi Başlığı
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Cilt Başlığı
Yayıncı
Aksaray Üniversitesi Fen Bilimleri Enstitüsü
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
Bu çalışmada kesir mertebeden diferensiyel denklemlerin sayısal çözümleri elde edildi. Kesir mertebeden diferensiyel denklemler doğada birçok problemin modellenmesi için son yıllarda birçok çalışmada kullanılmaktadır. Kesirli bir diferensiyel denklemin tam çözümünü bulmak, tam sayı mertebeden diferensiyel denklemlerdeki kadar kolay olmamakla beraber, bazen tam çözümlere ulaşmak mümkün olamamaktadır. Bu durum sayısal çözüm yöntemlerinin kesir mertebeden türevli denklemlere modifiye edilmesi ihtiyacını ortaya çıkarmıştır. Bu çalışmada sayısal çözüm yöntemi olarak Adams-Bashforth-Moulton yöntemi kullanıldı. Yöntemin formulizasyonu için lineer interpolasyon polinomları kullanıldı. Yöntemde ardışık iterasyonlar kullanıldığı ve sayısal sonuçların hassas ondalıklı kısımları göz ardı edilemeyeceği için uygun algoritma dillerinde kodlar yazıldı. Elde edilen çözümlerin değerlendirilebilmesi için denklemlerin tam çözümleri Laplace dönüşümleri ile elde edildi ve sonuçlar karşılaştırıldı. Elde edilen bulgular Adams-Bashforth-Moulton yönteminin kesir mertebeden diferensiyel denklemlerin ya da diferensiyel denklem sistemlerinin çözümlerinde oldukça düşük hata oranlarında sonuç verdiğini gösterdi. Adım genişliğinin yeterince küçük seçildiğinde hata oranlarının da istenildiği düzeyde düşük olacağı görüldü. Uygulamalarda kesir mertebenin farklı değerleri için çözümler elde edildi. Bu sonuçlara göre, tam çözümü bulunamayan ya da hesaplama açısından zorluklar içeren denklem ya da denklem sistemlerinin sayısal çözümlerinde Adams-Bashforth-Moulton yönteminin güvenle kullanılabileceği görüldü.
In this study, numerical solutions of fractional-order differential equations were obtained. Fractional-order differential equations have been widely used in recent years to model many problems in nature. Finding the exact solution of a fractional differential equation is not as straightforward as in integer-order differential equations, and sometimes it is not possible to reach exact solutions. This situation has led to the need to modify numerical solution methods for fractional-order derivative equations. In this study, the Adams-Bashforth-Moulton method was used as the numerical solution method. Linear interpolation polynomials were used for the formulation of the method. Since consecutive iterations were used in the method and the accurate decimal parts of the numerical results could not be ignored, codes were written in suitable algorithmic languages. In order to evaluate the obtained solutions, the exact solutions of the equations were obtained using Laplace transforms, and the results were compared. The findings showed that the Adams-Bashforth-Moulton method provided results with very low error rates in the solutions of fractional-order differential equations or systems of differential equations. It was observed that when the step size was chosen small enough, the error rates would also be sufficiently low. Solutions were obtained for different values of the fractional order in applications. Based on these results, it was observed that the Adams-Bashforth-Moulton method could be safely used in the numerical solutions of equations or systems of equations that do not have exact solutions or pose computational difficulties.
In this study, numerical solutions of fractional-order differential equations were obtained. Fractional-order differential equations have been widely used in recent years to model many problems in nature. Finding the exact solution of a fractional differential equation is not as straightforward as in integer-order differential equations, and sometimes it is not possible to reach exact solutions. This situation has led to the need to modify numerical solution methods for fractional-order derivative equations. In this study, the Adams-Bashforth-Moulton method was used as the numerical solution method. Linear interpolation polynomials were used for the formulation of the method. Since consecutive iterations were used in the method and the accurate decimal parts of the numerical results could not be ignored, codes were written in suitable algorithmic languages. In order to evaluate the obtained solutions, the exact solutions of the equations were obtained using Laplace transforms, and the results were compared. The findings showed that the Adams-Bashforth-Moulton method provided results with very low error rates in the solutions of fractional-order differential equations or systems of differential equations. It was observed that when the step size was chosen small enough, the error rates would also be sufficiently low. Solutions were obtained for different values of the fractional order in applications. Based on these results, it was observed that the Adams-Bashforth-Moulton method could be safely used in the numerical solutions of equations or systems of equations that do not have exact solutions or pose computational difficulties.
Açıklama
Anahtar Kelimeler
Adams-Bashforth-Moulton Yöntemi, Kesirli Adams-Bashforth Moulton Yöntemi, Sayısal Çözüm, Kesirli Türev, Laplace Dönüşümü, Fractional Adams-Bashforth Moulton Method, Numerical Solution
