Analytical and numerical solutions of some differential equations: a comparative study using Natural transform and Runge-Kutta method

Yıl 2025, Cilt: 15 Sayı: 1, 245 - 259, 15.03.2025

Hatice Muti , Gaye Yeşim Taflan

Öz

This paper focuses on Natural transform and the Runge-Kutta numerical method. These techniques have been used to analyze blood glucose concentrations and electrical circuits. These examples have been selected to demonstrate the applicability of Natural transform in many different areas. It was aimed to solve electrical circuits, which are generally solved by Laplace transform in engineering literature, with Natural transform and to obtain a comparative solution with Runge-Kutta method, which is a numerical method. These engineering problems defined with differential equations were analyzed using Natural transform. Firstly, the differential equations were written using the Natural transform, then the new equations were solved and applying the inverse transform, the results of the equations were obtained. The fourth-order Runge-Kutta numerical method was the second method employed in this study. The results found with this methods were presented in tables and compared graphically. The results applying Natural transform and Runge–Kutta method are equivalent to exact solution.

Anahtar Kelimeler

Glucose concentration, Natural transform, RLC circuit, Runge-Kutta

Bazı diferansiyel denklemlerin analitik ve numerik çözümleri: Natural dönüşüm ile Runge-Kutta yöntemi kullanılarak karşılaştırmalı bir çalışma

Öz

Bu çalışma, Naturel dönüşüm ve Runge-Kutta sayısal yöntemi üzerine odaklanmaktadır. Bu teknikler kan şekeri konsantrasyonlarının analizinde ve devre analizlerinde kullanılmıştır. Örnekler, Naturel dönüşümün birçok farklı alanda uygulanabilirliğini göstermek için seçilmiştir. Mühendislik literatüründe genellikle Laplace dönüşümü ile çözülen elektrik devrelerinin Naturel dönüşüm ile çözülmesi ve sayısal bir yöntem olan Runge-Kutta yöntemi ile karşılaştırmalı bir çözüm elde edilmesi amaçlanmıştır. Diferansiyel denklemlerle tanımlanmış bu mühendislik problemleri Natural dönüşümü ile analiz edilmiştir. İlk olarak diferansiyel denklemler Naturel dönüşüm kullanılarak yazılmış, daha sonra yeni denklemler çözülmüş ve ters dönüşüm uygulanarak denklemlerin sonuçları elde edilmiştir. Bu çalışmada kullanılan ikinci yöntem, dördüncü dereceden Runge-Kutta sayısal yöntemidir. Bu metotlarla elde edilen sonuçlar tablolar halinde sunulmuş ve grafiksel olarak karşılaştırılmıştır. Naturel dönüşüm ve Runge-Kutta yöntemi uygulanarak elde edilen sonuçların tam çözüme eşdeğer olduğu görülmüştür.

Anahtar Kelimeler

Glikoz konsantrasyonu, Naturel dönüşüm, RLC devresi, Runge-Kutta

Kaynakça

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