13: Discovering Three Fail-Proof Numerical Methods in Java

Creator QR Code JIS X 0510 in Java 13: Discovering Three Fail-Proof Numerical Methods

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When using the smaller step size, the Euler method and the improved Euler method are only off by about 035% at x = 99 Much better

If you don t want to use either of the Euler methods, you re in luck There s another numerical method that you can use to solve differential equations: the Runge-Kutta method This method gives excellent results that are even more accurate than the Euler or improved Euler methods

In the Runge-Kutta method, the recurrence relation is a weighted average of terms: y n +1 = y n + h 7c1 + c 2 + c 3 + c 4A 6

Carl David Tolm Runge was a German mathematician He was born in Havana, Cuba, on August 30, 1856, where his father was the Danish consul The family later moved to Bremen, Germany Runge died on January 3, 1927 Martin Wilhelm Kutta was also a German mathematician He was born in Pitschen, Germany (which today is part of Poland), on November 3, 1867 He went to the University of Breslau and continued to work in Munich He also spent a year at the University of Cambridge He became a professor in Stuttgart, Germany, in 1911 Kutta died on December 25, 1944 In 1901, Runge and Kutta co-developed the Runge-Kutta method, which is the powerful method that s used to solve ordinary differential equations numerically

where: c1 = f(xn yn) c 2 = f c x n + h, y n + c1 h m 2 2 c3 = f c x n + h, y n + c2 h m 2 2 c4 = f(xn + h, yn + c3h) This method can easily be adapted to the code you use earlier in the chapter, because f(x, y) doesn t depend on y it s just f(x) In this case, the recurrence relation becomes: y n + 1 = y n + h = f _ x n i + f c x n + h m + f c x n + h m + f _ x n + hi G 6 2 2