One of the main problems in the theory of numerical methods is the construction of the methods with the high accuracy and having extended stability regions. Depending from the object of investigation, it usually requires the necessary of some additional requirements on the using methods as a decreasing ofvolume computational work, the using of the certain properties of the solutions of the investigated problem, the selection of the step size, the determination of the values of the considering problem at intermediate points, etc. Therefore, the method which has the widely application must satisfy certain additional conditions. As a rule, the methods have constructed by the above mentioned requirements do not have high accuracy. Therefore, the methods specially constructed to solve specific problems, are more effective. By using the above presentationhere we are consider to the construction of a special method and its application to solving of the initial value problem for ODE of ...
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