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

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

Hatice Muti , Gaye Yeşim Taflan

https://doi.org/10.17714/gumusfenbil.1567137

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

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

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

Kaynakça

• Alkan, A., & Anaç, H. (2024). The novel numerical solutions for time-fractional Fornberg-Whitham equation by using fractional natural transform decomposition method. AIMS Mathematics, 9(9), 25333-25359.
• Al-Omari, S. K. Q. (2013). On the application of natural transforms. Int. J. Pure Appl. Math, 85(4), 729-744.
• Arifoğlu, U. (2013). Elektrik-Elektronik Devrelerinin Analizi, Alfa Basım Yayım.
• Bateman, H., Erdélyi, A., Magnus, W., & Oberhettinger, F. (1954). Tables of integral transforms (Vol. 1, p. 197). New York: McGraw-Hill.
• Belgacem, F. B. M., & Silambarasan, R. (2012). Theory of natural transform. Math. Engg. Sci. Aeros, 3, 99-124.
• Butcher, J.C. (2008). Numerical Methods for Ordinary Differential Equations, John Wiley &Sons.
• Chindhe, A. D., & Kiwne, S. (2017). Application of Natural transform in Cryptography. Journal of New Theory, (16), 59-67.
• Debnath, L., & Bhatta, D. (2016). Integral transforms and their applications. Chapman and Hall/CRC.
• Doetsch, G. (2013). Theorie und Anwendung der Laplace-transformation (Vol. 67). Springer-Verlag.
• Gardner, M. F., & Barnes, J. L. (1942). Transients in linear systems studied by the Laplace transformation (Vol. 1). J. Wiley & Sons, Incorporated.
• Heun K. (1900) Neue Methoden zur approximativen Integration der Differentialgleichungen einer unabh¨angigen Ver¨anderlichen. Z. Math. Phys., 45, 23–38.
• Higazy, M., Aggarwal, S., & Hamed, Y. S. (2020). Determination of Number of Infected Cells and Concentration of Viral Particles in Plasma during HIV‐1 Infections Using Shehu Transformation. Journal of Mathematics, 2020(1), 6624794.
• Huˇta A. (1956) Une am´elioration de la m´ethode de Runge–Kutta–Nystr¨om pour la ´esolution num´erique des ´equations diff´erentielles du premier ordre. Acta Fac. Nat. Univ. Comenian. Math., 1, 201–224.
• Huˇta A. (1957) Contribution `a la formule de sixi`eme ordre dans la m´ethode de Runge–Kutta–Nystr¨om. Acta Fac. Nat. Univ. Comenian. Math., 2, 21–24. Iserles A., Munthe-Kaas H. Z
• Irwin, J. D., Nelms R.M. (2015). Temel Mühendislik Devre Analizi, Editor: Timur Aydemir, Çevirenler: Hasan Dağ, Sedat Sünter, Timur Aydemir, Halis Altun, Nobel Akademik Yayıncılık.
• Iyengar, S.R., & Jain R.K. (2009). Numerical Methods, New Age International.
• Jadhav, C. P., Dale, T. B., & Boadh, R. (2022). Solution of fractional differential equations for LC, RC and LR circuits using sumudu transform method. NeuroQuantology, 20(17), 994.
• Khalouta, A. (2023). A New Exponential Type Kernel Integral Transform: Khalouta Transform and Its Application. Vol. LVII.
• Khan, Z. H., & Khan, W. A. (2008). N-transform-properties and applications. NUST journal of engineering sciences, 1(1), 127-133.
• Khidir, A. A., Alfaifi, W. I., Alsharari, S. L., Alanazi, I. M., & Alyasi, L. S. (2023). Study on Determining the Blood Glucose Concentration During an Intravenous Injection Using Volterra Integral Equations. Journal of Advances in Mathematics and Computer Science, 38(10), 172-184.
• Kiliçman, A., & Omran, M. (2017). On double Natural transform and its applications. J. Nonlinear Sci. Appl, 10(4), 1744-1754.
• Köklü, K. (2020). Resolvent, natural, and Sumudu transformations: solution of logarithmic Kernel integral equations with natural transform. Mathematical Problems in Engineering, 2020(1), 9746318.
• Kumar, A., Bansal, S., & Aggarwal, S. (2021). Determination of the blood glucose concentration of a patient during continuous intravenous injection using Anuj transform. Neuroquantology, 19(12), 303.
• Kumar, A., Bansal, S., & Aggarwal, S. (2021). A new novel integral transform “Anuj transform” with application. Design Engineering, 9, 12741-12751.
• Kutta W. (1901) Beitrag zur n¨aherungsweisen Integration totaler Differentialgleichungen. Z. Math. Phys., 46, 435–453.
• Lado, F. (1971). Numerical fourier transforms in one, two, and three dimensions for liquid state calculations. Journal of Computational Physics, 8(3), 417-433.
• Maitama, S., & Zhao, W. (2019). New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations. arXiv preprint arXiv:1904.11370.
• Mahgoub, M. M. A., & Mohand, M. (2019). The new integral transform “Sawi Transform”. Advances in Theoretical and Applied Mathematics, 14(1), 81-87.
• Nyström, E. Johannes. (1925). Über die numerische integration von differentialgleichungen. Helsingfors
• Peker, H. A., & Çuha, F. A. (2023). Exact solutions of some basic cardiovascular models by Kashuri Fundo transform. Journal of New Theory, (43), 63-72.
• Rawashdeh, M. S., & Maitama, S. (2014). Solving coupled system of nonlinear PDE’s using the natural decomposition method. Int. J. Pure Appl. Math, 92(5), 757-776.
• Rawashdeh, M. S., & Maitama, S. (2015). Solving nonlinear ordinary differential equations using the NDM. J. Appl. Anal. Comput, 5(1), 77-88.
• Rawashdeh, M. S., & Maitama, S. (2016). Solving PDEs using the natural decomposition method. Nonlinear studies, 23(1).
• Runge C. (1895) Uber die numerische Aufl osung von Differentialgleichungen. Math. Ann., 46, 167–178. Silambarasan, R. & Belgacem, F. B. M. (2011). Applications of the Natural transform to Maxwell’s Equations, PIERS Suzhou, China,, Sept 12-16, pp 899–902.
• Upadhyaya, L. M. (2019). Introducing the Upadhyaya integral transform. Bulletin of Pure and Applied Sciences, 38E (1), 471-510.
• Upadhyaya, L. M., Shehata, A., & Kamal, A. (2021). An update on the Upadhyaya transform Bulletin of Pure & Applied Sciences-Mathematics, 2021, Vol 40E, Issue 1, p26.
• Vashi, J., & Timol, M. G. (2016). Laplace and Sumudu transforms and their application. Int. J. Innov. Sci., Eng. Technol, 3(8), 538-542.
• Watugala, G. (1998). Sumudu transform-a new integral transform to solve differential equations and control engineering problems. Mathematical engineering in industry, 6(4), 319-329.