This study is therefore aimed at developing classes of efficient numerical integration schemes, for direct solution of second-order Partial Differential Equations (PDEs) with the aid of the method of lines. The power series polynomials were used as basis functions for trial solutions in the derivation of the proposed schemes via collocation and interpolation techniques at some appropriately chosen grid and off-grid points the derivedschemes are consistent, zero-stable and convergent. the proposed methods perform better in terms of accuracy than some existing methods in the literature.
A Block of hybrid method with three off-step points based is presented in this work for direct approximation of solution of third-order Initial and Boundary Value Problems (IVPs and BVPs). This off-step points are formulated such that they exist only on a single step at a time. Hence, these points are shifted to three positions respectively in order to obtain three different integrators for computational analysis. These analysis includes; order of the methods, consistency, stability and convergence, global error, number of functions evaluation and CPU time. The superiority of these methods over existing methods is established numerically on different test problems in literature
Fifth-order Korteweg-de Vries (KdV) equations, arise in modeling waves phenomena such as the propagation of shallow water waves over a flat surface, gravity-capillary waves and sound waves in plasmas. In this work, a one-step block hybrid linear multistep method was derived using the collocation technique, to solve fifth-order KdV models via the Method of Line (MoL). The consistency, stability and convergence of the method were established. The efficiency of the method can be seen from comparison of the exact solutions of problems and other methods cited from literature.
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