The main contribution of this work is the development of direct explicit methods of Runge-Kutta (RK) type for solving class of seventh-order ordinary differential equations (ODEs) to improve computational efficiency. For this purpose, we have generalized RK, RKN, RKD, RKT, RKFD and RKM methods for solving class of first-, second-, third-, fourth-, and fifth-order ODEs. Using Taylor expansion approach, we have derived the algebraic equations of the order conditions for the proposed RKM integrators up to the tenth-order. Based on these order conditions, two RKM methods of fifth- and sixth-order with four- and five-stage are derived. The zero stability of the methods is proven. Stability polynomial of the methods for linear special seventh-order ODE is given. Numerical results have clearly shown the advantage and the efficiency of the new methods and agree well with analytical solutions due to the fact the proposed integrators are zero stable, more efficient and accurate integrators.
Mechee, Mohammed S. and Mshachal, Jawad K.
"Derivation of Direct Explicit Integrators of RK Type for Solving Class of Seventh-Order Ordinary Differential Equations,"
Karbala International Journal of Modern Science: Vol. 5
, Article 8.
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