Differential Equation Solver Simulator for Runge-Kutta Methods

Öz

Many of problems in engineering and science is modeled by differential equations mathematically, therefore their solutions have an important role. Many methods have been developed for analytical or numerical solutions of differential equations. In proportion to the development of technology, the numerical solution methods are utilized widely. In particular, the main objectives in real time applications are to reach the correct solution as soon as possible with minimal processing and maximum precision. In the performed study, a simulator that contains Runge-Kutta based 48 methods was developed for numerical solution of differential equations. In the user friendly simulator which can be used also for educational purposes, the solution of defined differential equation under the specified initial condition with given step size or according to the number of points requested within the specified range can be obtained by the selected method. Solutions can be presented to the user both numerical (step values, computation time) and graphically; also the subject explanations about the methods/solutions can be given. Furthermore, the comparative solutions (performance analysis) can be implemented by the simulator. So, the users can realize the numerical solutions of differential equations with different methods by the simulator; the students learn the methods in this field visually with the aid of subject explanation and can implement step by step; the designers can choose the most appropriate method easily, effectively and accurately for their systems by the comparative analysis.

Anahtar Kelimeler

• Ordinary differential equation; Runge-Kutta methods; Simulator

RUNGE-KUTTA YÖNTEMLERİ İÇİN DİFERANSİYEL DENKLEM ÇÖZÜM SİMÜLATÖRÜ

Öz

Mühendislik ve fen bilimlerine ait problemlerinin birçoğu diferansiyel denklemlerle matematiksel olarak modellenmekte, dolayısıyla çözümleri önemli yer tutmaktadır. Diferansiyel denklemlerin analitik veya sayısal çözümleri için çok sayıda yöntemler geliştirilmiştir. Teknolojinin gelişmesiyle orantılı olarak sayısal çözüm yöntemlerinden yoğun bir şekilde faydalanılmaktadır. Özellikle gerçek zamanlı uygulamalarda en az işlemle, en kısa sürede, en yüksek hassasiyetle, en doğru çözüme ulaşmak başlıca hedeflerdendir. Gerçekleştirilen çalışmada, diferansiyel denklemlerin sayısal çözümü için Runge-Kutta tabanlı 48 yöntemi içeren bir simülatör geliştirilmiştir. Kullanıcı dostu ve eğitim amaçlı da kullanılabilecek simülatörde tanımlanan diferansiyel denklemin, belirtilen başlangıç koşulu altında, verilen adım boyutu veya nokta sayısına göre, istenen aralıktaki çözümü seçilen yöntemle elde edilebilmektedir. Çözümler kullanıcıya hem sayısal (adım değerleri, hesaplama süresi vb.) hem de grafiksel olarak sunulabilmekte; bunun yanında yöntemlerle/çözümlerle ilgili konu anlatımları da verilebilmektedir. Ayrıca simülatörle karşılaştırmalı çözümler (performans analizleri) de gerçekleştirilebilmekte. Böylece simülatör ile kullanıcılar farklı yöntemlerle diferansiyel denklemlerin sayısal çözümlerini gerçekleştirebilmekte; öğrenciler bu alandaki yöntemleri konu anlatımı destekli görsel olarak öğrenip adım adım uygulayabilmekte; tasarımcılar da karşılaştırmalı analizlerle sistemleri için en uygun yöntemi kolay, etkin ve doğru bir şekilde seçebilmektedirler.

Anahtar Kelimeler

• Adi diferansiyel denklem; Runge-Kutta yöntemleri; Simülatör

Kaynakça

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