On Some Comparison of Adam’s Methods with Multistep Methods and Their Application to Solve Initial-Value Problems for First-Order ODEs
DOI:
https://doi.org/10.22105/kmisj.v1i2.55Keywords:
Initial-value problem, ordinary differential equations (ODEs), Adams- Moulton and Adams-Bashforth method, stability and degree, Simson and trapezoid methods, methods of Runge-KuttaAbstract
Among in the class of Numerical Methods for solving initial-value problem one of the popular methods is the Adams-Moulton and Adams-Bashforth, which make up the Adamsfamily. Many experts believe that Multistep methods are obtained from the generalization of Adams methods. Historically it happened that first the methods of Adams appeared. And after the emergence of Adams methods specialists constructed methods that is a special case of the Adams methods. Noted that Adams method intersects with the Runge-Kutta methods at one point, which is called Euler’s method. Adams methods and Runge-Kutta methods are the intersects at the multiple points in the application them to calculation of definite integrals. As is known the fourth order Runge-Kutta method, which was constructed by Runge, coincides with Simpson’s method in the application them to calculation of the definite integral. Here, have compared Adam’s methods with Multistep Methods in the application of them to solve initial-value problem for the Ordinary Differential Equations the first order. By using specific examples it is shown, how one can obtain Adams methods from the Runge-Kutta methods and vice versa.
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