Abstract
This manuscript presents a step by step guide on derivation and analysis of a new numerical method to solve initial value problem of fourth order ordinary differential equations. The method adopted hybrid techniques using power series as the basic function. Collocation of the fourth derivatives was done at both grid and off-grid points. The interpolation of the approximate function is also taken at the first four points. The complete derivation of the new technique is introduced and shown here, as well as the full analysis of the method. The discrete schemes and its first, second, and third derivatives were combined together and solved simultaneously to obtain the required 32 family of block integrators. The block integrators are then applied to solve problem. The method was tested on a linear system of equations of fourth order ordinary differential equation in order to check the practicability and reliability of the proposed method. The results are displaced in tables; it converges faster and uses smaller time for its computations. The basic properties of the method were examined, the method has order of accuracy p=10, the method is zero stable, consistence, convergence and absolutely stable. In future study, we will investigate the feasibility, convergence, and accuracy of the method by on some standard complex boundary value problems of fourth order ordinary differential equations. The extension of this new numerical method will be illustrated and comparison will also be made with some existing methods.
Highlights
This work proposes the derivation and analysis of a new numerical method for direct approximation of the differential equations of the form.y '''' = f ( x, y, y ', y '', y '''), y ( x0 ) = τ0, y '( x0 ) = τ1, y ''( x0 ) = τ 2, y '''( x0 ) = τ3 (1)where R× Rm × Rm → Rm and y, y0, y ', y '', y '''∈R are given real constants.Several authors have devoted a lot of attention to the development of various methods for solving (1) directly without reducing it to system of first order
We are motivated by the work of [1] to develop an hybrid block method with four-step and four-hybrid points with characteristic of high order for solving directly fourth order ordinary differential equations
This work is motivated by the need to develop a new numerical method which can handle system of equations of initial value problems of fourth order ordinary differential equations
Summary
This work proposes the derivation and analysis of a new numerical method for direct approximation of the differential equations of the form. Several authors have devoted a lot of attention to the development of various methods for solving (1) directly without reducing it to system of first order. Numerous numerical methods based on the use of different polynomial functions and non polynomials have been adopted including. We are motivated by the work of [1] to develop an hybrid block method with four-step and four-hybrid points with characteristic of high order for solving directly fourth order ordinary differential equations. This work is motivated by the need to develop a new numerical method which can handle system of equations of initial value problems of fourth order ordinary differential equations.
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