Abstract
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
Highlights
Third-order problems considered in this work are of the form:y = f (x, y, y, y ), x ∈ [a, b] (1)with appropriate initial conditions and boundary conditions respetively
The problem in (1) is assumed well posed and the numerical solution is the interest in this work
The collocation technique is employed. This method is found to be flexible and more efficient in that since it approximates the solution of (1) at several intra-points, its block formulations consists of several linear multistep methods which are required for direct solution of (1), such that it overcomes the overlapping of pieces of solutions and it is self starting
Summary
Third-order problems considered in this work are of the form:. with appropriate initial conditions and boundary conditions respetively. The problem in (1) is assumed well posed and the numerical solution is the interest in this work. Numerical methods for the solution of (1) with initial conditions or boundary conditions and for third order singularly perturbed boundary value problems are numerous. This method is found to be flexible and more efficient in that since it approximates the solution of (1) at several intra-points, its block formulations consists of several linear multistep methods which are required for direct solution of (1), such that it overcomes the overlapping of pieces of solutions and it is self starting. In this work we seek the numerical solution of (1) directly with three different three-step continuous hybrid block methods with three off-step points. These methods are developed using the collocation approach
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