Bigeometric Taylor Theorem and its Application to the Numerical Solution of Bigeometric Differential Equations

Abstract

Many studies in the field of Bigeometric Calculus are based on an approximation to the Bigeometric Taylor series, as the correct version is not known. The Bigeometric Taylor Series introduced in this research, is derived and proven explicitly. As an application of the Bigeometric Taylor Series, the Bigeometric Runge-Kutta method is derived in analogy to the classical Runge-Kutta method. The stability, as well as the convergence analysis is given explicitly for Bigeometric Runge-Kutta method. Application of the Bigeometric Runge-Kutta method to problems with known closed form solutions show the advantage of this method for a certain family of problems compared to the classical Runge-Kutta Method. Keywords: Bigeometric calculus, Runge-Kutta, differential equations, numerical approximation, dynamical systems,electirical circuits.

ÖZ: Bigeometrik alanında yapılan birçok çalı¸smada Bigeometrik Taylor serisi do˘gru analiz edilmeden kullanılmı¸stır. Bu çalı¸smada Bigeometrik Taylor Serisinin ispatı açık olarak verilmi¸stir. Bigeometrik Taylor Serisinin bir uygulaması olarak, Bigeometric Runge- Kutta yöntemi nümerik analizde bilinen Runge-Kutta yöntemi baz alınarak çıkarılmı¸stır. Ayrıca Bigeometric Runge-Kutta yöntemi için yakınsak ve kararlılık testleri de analiz edilmi¸stir. Yöntem dinamik sistemler, bioloji ve elektrik devrelerinde uygulanmı¸s ve Bigeometrik Runge Kutta ile elde edilen sonuçlar nümerik analizde bilinen Runge- Kutta yöntemi ile kar¸sıla¸stırılmı¸stır. Anahtar Kelimeler: Çarpımsal analiz„ Runge-Kutta, diferansiyel denklemler, numerik yakınsama, dinamik sistemler, elektrik devreleri.