The Runge‐Kutta method with Arcs of equal length and its applications

Abstract The Runge‐Kutta method, a popular numerical procedure for solving a differential equation, is based on Taylor series expansions of the solution. It provides the solution if the function is single‐valued. However, it is difficult to obtain the solution in the case of a multivalued function. In this paper, a coordinate system that follows the shape of the solution is introduced for developing a numerical solution of a differential equation with a multivalued functional solution. For a step describing accurately the shape of the function, the arc length of the solution function is used. the Runge‐Kutta method with arcs of equal length is proposed in which the arc length is kept identical. From the error formula for the Runge‐Kutta method, the condition is shown under which the solution by the proposed method is more accurate than the one by the conventional Runge‐Kutta method. Next, a numerical method for a differential equation with a multivalued solution function is given by combining the new Runge‐Kutta method and the coordinate transformation. This transformation is applied not only to numerical values but also to the differential equation itself. This process can eliminate the points at which the function on the right‐hand side of the differential equation becomes singular and provide solutions even when the solution function is multivalued.