Two Embedded Pairs for Solve Directly Third-Order Ordinary Differential Equation by Using Runge-Kutta Type Method (RKTGD)

Abstract

In this paper, two pairs of embedded Runge-Kutta (RK) type techniques for straightforwardly tackling third-order ordinary differential equations (ODEs) of the form v″′ = f(x, v, v′) signified as RKTGD strategies were proposed and explored. Relying on the order conditions, the primary pair with mathematical order 4 and 3 was called RKTGD4(3), while different has order 5 and 4, and was named RKTGD5(4). The new strategies were determined so that the higher-order techniques were exact and the lower order techniques would bring about the best error estimates. At that point, variables step-size codes were created to support the methods and utilized in solving a lot of third-order problems. Comparisons were made between mathematical results and existing embedded RK pairs within the scientific literature, that require the problems to be reduced into a system of first-order ODEs, and the effectiveness of the new RKTGD pairs have appeared.