Derivation of Embedded Diagonally Implicit Methods for Directly Solving Fourth-order ODEs - Saleh, Omar Mohammed; Fawzi , Firas Adel; Hussain, Kasim Abas; Nizam G. Ghawadri, (Ibn AL-Haitham Journal For Pure and Applied Sciences, 2024)
EDIRKTO, an Implicit Type Runge-Kutta  Method of Diagonally Embedded pairs, is a novel approach presented in the paper that may be used to solve 4th-order ordinary differential equations of the form . There are two pairs of EDIRKTO, with three stages each: EDIRKTO4(3) and EDIRKTO5(4). The derivation techniques of the method indicate that the higher-order pair is more accurate, while the lower-order pair provides superior error estimates. Next, using these pairs as a basis, we developed variable step codes and applied them to a series of -order ODE problems. The numerical outcomes demonstrated how much more effective their approach is in reducing the quantity of function evaluations needed to resolve fourth-order ODE issues.