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.

The derivation of 5th order diagonal implicit type Runge Kutta methods (DITRKM5) for solving 3rd special order ordinary differential equations (ODEs) is introduced in the present study. The DITRKM5 techniques are the name of the approach. This approach has three equivalent non-zero diagonal elements. 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 comparison was made with the methods indicated in the other literature of the implicit RK techniques. The numerical techniques are elucidated as the qualification regarding the efficiency and number of function evaluations compared with another literature of the implicit RK approaches from the result of the computations. In addition, the stability polynomial for DITRK method is derived and analyzed.