Development and Analysis of Embedded Explicit Runge-Kutta Methods for Directly Solving Special Fifth-Order Quasi-Linear Ordinary Differential Equations

This paper introduces two pairs of numerical embedded methods of Runge-Kutta (RK) type, known as ERKMF methods, for directly solving a class of quasi-linear ordinary differential equations (ODEs) of 5 th -order with the form (5) () = (, ()).The first pair, ERKMF6(5), is a 5 th -order embedded in 6th-order method that satisfies the condition first the same as last (FSAL), and the vector output is represented by the coefficient matrix's final row.The second pair, ERKMF7(6), is also an embedded method.The paper then applies these methods to solve specific 5 th -order problems using variable step-size codes and compares the results to those obtained using an existing embedded RK method.The ERKMF methods have shown to be advantageous and effective compared to existing methods, as demonstrated by the numerical findings.In conclusion, these new pairs of ERKMF methods provide a promising approach to directly solve quasi-linear ODEs of 5 th -order and could have significant implications for the field of numerical methods for differential equations.