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.

The Implementations of the Embedded Diagonally Implicit Type Runge-Kutta Method (EDITRKM) For Special Third Order of the Ordinary Differential Equations

“…In the current study, the numerical results between the EDITRKM 4(3) and EDITRKM 5(4) have a good comparison as shown in Figure (1). The current study was based on Runge-Kutta Method that has been analyzed earlier by [12,13,18], however, the research in hand expanded and improved the method from explicit to implicit and from directly to diagonally.…”

“…Moreover, [9][10] developed a solution of the special third order for the ODEs directly by RK technique. Finally, Senu [11] and Fawzi et al [12] constructed the embedded RK technique to solve third order for the ODEs. The explicit embedded pair Runge-Kutta (RK) method that known as TFRKF6 ( 5) is improved to compute the numerical solution of the initial value problems of first-order for oscillatory approximations.…”

Derivation of Embedded Diagonally Implicit Methods for Directly Solving Fourth-order ODEs

“…Figure 1 illustrates the numerical comparison between the EDIRKTO4(3) and EDIRKTO5(4) outcomes from the current study. The current study is based on the Runge-Kutta Method, which has been previously studied by [4,5,8]. However, the research at hand broadened and enhanced the method from explicit to implicit and from direct to diagonal.…”

“…In this study, we discuss a class of quasi-linear, fourth-order (ODEs) and their numerical integration in the following form: 𝑞 (4) (𝑡) = 𝑓(𝑡, 𝑞(𝑡), 𝑞′(𝑡)), 𝑡 ≥ 𝑡 0 , (1) with initial conditions 𝜎 𝑖 (𝜏) = 𝜁 𝑖 , 𝑖 = 0, 1, … , 3.…”

“…However, it would be more efficient if numerical methods could be used to solve the problem accurately and quickly. In [4][5][6][7][8][9][10][11][12] contain such works. Multistep strategies for solving ODEs require initial values.…”