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

Abstract

The derivation of the Embedded pair Diagonally Implicit Type Runge-Kutta Method (EDITRKM) for solving 3rd special order ordinary differential equations (ODEs) is introduced in the current study. The EDITRKM techniques are the name of the approach. This approach in the present study has two types: EDITRKM 4(3) for order 4 and 3 of the first pair and EDITRKM 5(4) for orders 5 and 4 of the second pair. To investigate the current study, a variety of tests for five various initial value problems (IVPs) with different step sizes h were implemented. Then, a comparation of the present study between the EDITRKM 4(3) and EDITRKM 5(4) for five different problems are made. The numerical techniques elucidated as the qualification regarding the efficiency and decimal logarithm for highest the time curve against logarithm of number of the function call estimate.

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.…”

“…Both the order 𝑝 method (𝐶, 𝐴, 𝑏, 𝑏 ′𝑇 , 𝑏 ′′𝑇 , 𝑏 ′′′𝑇 ) and the order 𝑞 method (𝐶, 𝐴 ̅ , 𝑏 The main idea is to obtain single cost error estimation for usage in step size approach values before generating the embedded pair of implicit EDIRKTO approaches. The methods are illustrated by increasing the significant pairs and local error estimates using the values step size ℎ, as follows [8],…”

“…Where 𝑇𝑜𝑙 refers to the required level of accuracy, and local truncation error (LTE) is performed at each stage. Therefore, the step will be admitted if 𝐿𝑇𝐸 ≤ 𝑇𝑜𝑙, uses the local extrapolation method, which indicates employing more precise calculations to boost the integration and ℎ can be improved by using Equation (8). If 𝐿𝑇𝐸 > 𝑇𝑜𝑙, the step will be denied and the step size ℎ will be cut in half.…”

“…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.…”