A class of explicit Runge–Kutta type methods with the involvement of fourth derivative, denoted as two-derivative Runge–Kutta type (TDRKT) methods, are proposed and investigated for solving a special class of third-order ordinary differential equations in the form u ‴ ( x ) = f ( x , u ( x ) ) . In this paper, two stages with algebraic order four and three stages with algebraic order five are presented. The derivation of TDRKT methods involves single third derivative and multiple evaluations of fourth derivative for every step. Stability property of the methods are analysed. Accuracy and efficiency of the new methods are exhibited through numerical experiments.

The improved block hybrid collocation method (KBHK(B)) with two off-step points and four collocation points is proposed for the direct solution of general third order ordinary differential equations. Modification is done by adding first derivative of third order function into the general form of block hybrid collocation method to yield KBHK(B). Both main and additional methods are derived via interpolation (power series function) and collocation of the basic polynomial. An improved block method is derived to provide the approximation at five points concurrently. Zero stability, consistency, convergence and absolute stability region of KBHK(B) are investigated. Some numerical examples with exact solution are tested to illustrate the efficiency of the method. KBHK(B) is compared by other block hibrid collocation method in term of global error and step number. It is shown that KBHK(B) generates minimum global error with minimum step number.

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