A 4-point block method for solving higher order ordinary differential equations directly

An alternative block method for solving fifth-order initial value problems (IVPs) is proposed with an adaptive strategy of implementing variable step size. The derived method is designed to compute four solutions simultaneously without reducing the problem to a system of first-order IVPs. To validate the proposed method, the consistency and zero stability are also discussed. The improved performance of the developed method is demonstrated by comparing it with the existing methods and the results showed that the 4-point block method is suitable for solving fifth-order IVPs

A 4-Point Block Method For Solving Higher Order Ordinary Differential Equations Directly

An alternative block method for solving fifth-order initial value problems (IVPs) is proposed with an adaptive strategy of implementing variable step size. The derived method is designed to compute four solutions simultaneously without reducing the problem to a system of first-order IVPs. To validate the proposed method, the consistency and zero stability are also discussed. The improved performance of the developed method is demonstrated by comparing it with the existing methods and the results showed that the 4-point block method is suitable for solving fifth-order IVPs

Accurate Four-Step Hybrid Block Method for Solving Higher-Order Initial Value Problems

يركز هذا البحث على التعامل مع طريقة عددية ذاتية البدء والتي يمكن استخدامها لايجاد التكامل بشكل مباشر لمسائل القيم الابتدائية للمعادلات التفاضلية الاعتيادية ذات الرتب العالية. هذه الطريقة مشتقة من تقريب متسلسلة القوى مع المعادلات الناتجة التي تم تقديرها في الشبكة المحددة ونقاط خارج الشبكة.تم تطبيق الطريقة باستخدام طريقة االبلوك  كمكامل عددي لمسألة القيمة الأبتدائية ذات الرتب الأعلى. تم التحقق من الخصائص الأساسية لطريقة البلوك والتحقق من أدائها ثم تنفيذها ببعض االتجارب الاختبارية للتحقق من دقة الطريقة وتقاربها.This paper focuses on developing a self-starting numerical approach that can be used for direct integration of higher-order initial value problems of Ordinary Differential Equations. The method is derived from power series approximation with the resulting equations discretized at the selected grid and off-grid points. The method is applied in a block-by-block approach as a numerical integrator of higher-order initial value problems. The basic properties of the block method are investigated to authenticate its performance and then implemented with some tested experiments to validate the accuracy and convergence of the method

Enactment of implicit two-step Obrechkoff-type block method on unsteady sedimentation analysis of spherical particles in Newtonian fluid media

Purpose: The analysis of the characteristics of particles motion is considered in this article, where a model which studies a Newtonian fluid media with specific interest on the analysis of unsteady sedimentation of particles is considered. The numerical solution of this first order differential equation model using an Obrechkoff-type block method is presented. Methodology: The algorithm for the conventional Nyström -type multistep scheme is considered with specific parameter choices in order to obtain the main k-step Obrechkoff-type block method and the required additional method. The unknown coefficients of these methods are obtained by using the concept of Taylor series expansion to obtain the required schemes for the block method which were combined as simultaneous integrators for the solution of the differential equation model.Findings: The block method gave highly accurate results as compared with the exact solution of the model. Furthermore, at selected values of the physical properties of nanoparticles, the solutions using the two-step Obrechkoff-type block method was compared with past literatures and the results were seen to be in agreement. The influence of the physical parameters on terminal velocity is also discussed

MATHEMATICA COMPUTER PROGRAMMING CODES OF EXPONENTIALLY FITTTED CONCURRENT MILNE'S DEVICE FOR SOLVING SPECIAL PROBLEMS

Over the years, scientific computing has contributed immensely to computational mathematics. Mathematica computer programming codes is known to provide easy computation and quick results. This research article is specifically built to generate Mathematica computer programming codes of exponentially fitted concurrent Milne’s device (EFCMD) for solving special problems. Exponentially fitted concurrent Miln device is formulated via collocation/interpolation with power series as the approximate solution. Analyzing the EFCMD will produce the main local truncation error (MLTE) after showing the order, thereby bringing forth the bounds of convergence. Numerical results display that EFCMD do better than existing methods in terms of the maximum errors in the least studied bound of convergence as a result of varying/designing a suitable pace size, ascertain bound of convergence and error control