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espanolEste documento contiene una comparacion detallada entre los metodos de solucion numerica de las ecuaciones diferenciales ordinarias, que comienzan con el metodo de la serie Taylor de orden 2, indicando que esta serie dificulta los calculos para derivadas de orden superior de funciones de varias variables, de modo que los metodos de orden Runge Kutta 2 se implementan, lo que logra el proposito requerido evitando los dificiles calculos de derivadas de orden superior. En este documento, se expondran diferentes variantes de los metodos Runge-Kutta de orden 2 a partir de una introduccion y demostracion de la conexion de estos con la serie Taylor de orden 2, estos metodos son: el metodo de Heun, el metodo de punto medio y El metodo de Ralston. Se observara a partir de la solucion de ecuaciones diferenciales de prueba su respectivo error con respecto a la solucion analitica, obteniendo un indice de error dictado por el error cuadratico medio EMC. A traves de este documento conoceremos la mejor aproximacion numerica a la solucion analitica de los diferentes PVI (problemas de valor inicial) planteados, tambien fijando un patron de solucion para ciertos problemas, es decir, se estipulara el metodo apropiado para cada tipo de problema. Se observo que metodo de Ralston presento mayor exactitud seguido por el metodo del punto medio y el de Heun, en los demas PVI se observa que el metodo del punto medio proporciona la mejor solucion numerica puesto que tiene un EMC muy bajo y dificil de alcanzar por los demas metodos. EnglishThis manuscript contains a detailed comparison between numerical solution methods of ordinary differential equations, which start from the Taylor series method of order 2, stating that this series hinders calculations for higher order derivatives of functions of several variables, so that the Runge Kutta methods of order 2 are implemented, which achieve the required purpose avoiding the cumbersome calculations of higher order derivatives. In this document, different variants of the Runge-Kutta methods of order 2 will be exposed from an introduction and demonstration of the connection of these with the Taylor series of order 2, these methods are: the method of Heun, the method of midpoint and the Ralston method. It will be observed from the solution of test differential equations its respective error with respect to the analytical solution, obtaining an error index dictated by the mean square error EMC. Through this document we will know the best numerical approximation to the analytical solution of the different PVI (initialvalue problems) raised, also fixing a solution pattern for certain problems, that is, the appropriate method for each type of problem will be stipulated.It was observed that the Ralston method presented greater accuracy followed by the midpoint method and the Heun method, in the other PVI it is observed that the midpoint method yields the best numerical solution since it has a very low EMC and difficult to reach by the other methods.

scientia et technica

Numerical comparison by different methods (second order Runge Kutta methods, Heun method, fixed point method and Ralston method) to differential equations with initial condition