A hybrid convergent method of tenth-order is presented in this work for directly solving fifth-order boundary value problems in ordinary differential equations. A unique direct block approach is obtained by combining multiple Finite Difference Formulas which are derived via the collocation technique. The proposed method is fully analyzed and the existence and uniqueness of the discrete solution is established. Different numerical examples are considered and the results are compared with those provided by existing works in the literature. The comparison shows the good performance of the present method over some cited works in the literature, confirming the competitiveness and superiority of the new numerical integrator.

This work proposes a hybrid block numerical method of tenth order for the direct solution of fifth-order initial value problems. The formulas that constitute the block method are derived from a continuous approximation obtained through interpolation and collocation techniques. In order to obtain better accuracy, sixth-order derivatives are incorporated to develop the formulas. The main characteristics of the method are analyzed, namely, the order, local truncation errors, zero-stability, consistency and convergence. The proposed strategy performs well, as shown by some numerical examples and the corresponding efficiency curves. Compared to existing numerical methods in the literature, the proposed method is competitive and the numerical approximations it provides are significantly close to the precise solutions.

\noindent The use of collocation process for constructing numerical methods for solving ordinary differential equations has been attractive for stiff and problems with highly oscillatory solutions. In this paper, a class of block hybrid collocation methods has been developed which can efficiently solve stiff and highly oscillatory differential equations in block solution form. The block hybrid collocation methods are derived based on collocation at the polynomial nodes which are very effective for solving highly oscillatory systems. The block solution methods arising from the continuous formulation are discussed for various examples with applications. The methods are self-starting and produce dense output within the integration interval. The convergence of the derived methods is determined theoretically and asymptotic error constants are calculated. Improved performance over some known standard methods is achieved for a broad class of problems with oscillating solutions. Preliminary numerical calculations using our new methods clearly show improved performance, efficiency and effectiveness compared to some methods with strong algebraic stability properties. Efficiency curves of the solutions plotted, show rapid convergence of the proposed second-derivative block hybrid collocation methods. (2010) Mathematics Subject Classification:34M10,35B05,35B35,65L05

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