scholarly journals Approximating non linear higher order ODEs by a three point block algorithm

PLoS ONE ◽  
2021 ◽  
Vol 16 (2) ◽  
pp. e0246904
Author(s):  
Ahmad Fadly Nurullah Rasedee ◽  
Mohammad Hasan Abdul Sathar ◽  
Khairil Iskandar Othman ◽  
Siti Raihana Hamzah ◽  
Norizarina Ishak
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Numerical Algorithm ◽  
Computational Cost ◽  
Real Life ◽  
Higher Order ◽  
Multistep Method ◽  
Stability And Convergence ◽  
Block Algorithm ◽  
Life Problems

Differential equations are commonly used to model various types of real life applications. The complexity of these models may often hinder the ability to acquire an analytical solution. To overcome this drawback, numerical methods were introduced to approximate the solutions. Initially when developing a numerical algorithm, researchers focused on the key aspect which is accuracy of the method. As numerical methods becomes more and more robust, accuracy alone is not sufficient hence begins the pursuit of efficiency which warrants the need for reducing computational cost. The current research proposes a numerical algorithm for solving initial value higher order ordinary differential equations (ODEs). The proposed algorithm is derived as a three point block multistep method, developed in an Adams type formulae (3PBCS) and will be used to solve various types of ODEs and systems of ODEs. Type of ODEs that are selected varies from linear to nonlinear, artificial and real life problems. Results will illustrate the accuracy and efficiency of the proposed three point block method. Order, stability and convergence of the method are also presented in the study.

scholarly journals Formulative Visualization of Numerical Methods for Solving Non-Linear Ordinary Differential Equations

2021 ◽  
Vol 2 (2) ◽  
pp. 79-88
Author(s):  
Jeevan Kafle ◽  
Bhogendra Kumar Thakur ◽  
Grishma Acharya
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Real Life ◽  
Euler Method ◽  
Linear Ordinary Differential Equations ◽  
Life Problems ◽  
Backward Euler ◽  
Non Linear ◽  
Runge Kutta

Many physical problems in the real world are frequently modeled by ordinary differential equations (ODEs). Real-life problems are usually non-linear, numerical methods are therefore needed to approximate their solution. We consider different numerical methods viz., Explicit (Forward) and Implicit (Backward) Euler method, Classical second-order Runge-Kutta (RK2) method (Heun’s method or Improved Euler method), Third-order Runge-Kutta (RK3) method, Fourth-order Runge-Kutta (RK4) method, and Butcher fifth-order Runge-Kutta (BRK5) method which are popular classical iteration methods of approximating solutions of ODEs. Moreover, an intuitive explanation of those methods is also be presented, comparing among them and also with exact solutions with necessary visualizations. Finally, we analyze the error and accuracy of these methods with the help of suitable mathematical programming software.

scholarly journals Solution for nonlinear Duffing oscillator using variable order variable stepsize block method

MATEMATIKA ◽  
2017 ◽  
Vol 33 (2) ◽  
pp. 165 ◽  
Author(s):  
Ahmad Fadly Nurullah Rasedee ◽  
Mohamad Hasan Abdul Sathar ◽  
Norizarina Ishak ◽  
Nur Shuhada Kamarudin ◽  
Muhamad Azrin Nazri ◽  
...  
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Duffing Oscillator ◽  
Computational Cost ◽  
Real Life ◽  
Multistep Method ◽  
Variable Order ◽  
Block Method ◽  
Step Size ◽  
Variable Stepsize

Real life phenomena found in various fields such as engineering, physics, biology and communication theory can be modeled as nonlinear higher order ordinary differential equations, particularly the Duffing oscillator. Analytical solutions for these differential equations can be time consuming whereas, conventional numerical solutions may lack accuracy. This research propose a block multistep method integrated with a variable order step size (VOS) algorithm for solving these Duffing oscillators directly. The proposed VOS Block method provides an alternative numerical solution by reducing computational cost (time) but without loss of accuracy. Numerical simulations are compared with known exact solutions for proof of accuracy and against current numerical methods for proof of efficiency (steps taken).

scholarly journals Higher Order Haar Wavelet Method for Solving Differential Equations

2020 ◽  
Author(s):  
Jüri Majak ◽  
Mart Ratas ◽  
Kristo Karjust ◽  
Boris Shvartsman
Keyword(s):  
Differential Equations ◽  
Computational Cost ◽  
Haar Wavelet ◽  
Higher Order ◽  
Wavelet Method ◽  
Haar Wavelet Method ◽  
Convergence Results ◽  
Key Characteristics ◽  
Parameter Values ◽  
Excellent Accuracy

The study is focused on the development, adaption and evaluation of the higher order Haar wavelet method (HOHWM) for solving differential equations. Accuracy and computational complexity are two measurable key characteristics of any numerical method. The HOHWM introduced recently by authors as an improvement of the widely used Haar wavelet method (HWM) has shown excellent accuracy and convergence results in the case of all model problems studied. The practical value of the proposed HOHWM approach is that it allows reduction of the computational cost by several magnitudes as compared to HWM, depending on the mesh and the method parameter values used.

Problem-Based Learning (PBL)

2021 ◽  
pp. 38-56
Author(s):  
Mahmoud Hawamdeh ◽  
Idris Adamu
Keyword(s):  
Problem Solving ◽  
Teaching And Learning ◽  
Real Life ◽  
Problem Based Learning ◽  
Higher Order ◽  
Educational Institutions ◽  
Learning Skills ◽  
Knowledge And Skills ◽  
Life Problems ◽  
Approach To Teaching

This chapter discuss how Problem-Based learning (PBL) helps to achieve this century's approach to teaching and learning for students in higher educational institutions. If adopted, this method of teaching will enable student to attain learning skills (skills, abilities, problem solving, and learning dispositions that have been identified) to acquire a lifelong habit of approaching problems with initiative and diligence and a drive to acquire the knowledge and skills needed for an effective resolution. And they will develop a systematic approach to solving real-life problems using higher-order skills.

scholarly journals A Review on a Class of Second Order Nonlinear Singular BVPs

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1045
Author(s):  
Amit K. Verma ◽  
Biswajit Pandit ◽  
Lajja Verma ◽  
Ravi P. Agarwal
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Real Life ◽  
Monotone Iterative Technique ◽  
Singular Boundary ◽  
Singular Boundary Value Problems ◽  
Numerical Techniques ◽  
Singular Differential Equations ◽  
Monotone Iterative ◽  
Life Problems

Several real-life problems are modeled by nonlinear singular differential equations. In this article, we study a class of nonlinear singular differential equations, explore its various aspects, and provide a detailed literature survey. Nonlinear singular differential equations are not easy to solve and their exact solution does not exist in most cases. Since the exact solution does not exist, it is natural to look for the existence of the analytical solution and numerical solution. In this survey, we focus on both aspects of nonlinear singular boundary value problems (SBVPs) and cover different analytical and numerical techniques which are developed to deal with a class of nonlinear singular differential equations − ( p ( x ) y ′ ( x ) ) ′ = q ( x ) f ( x , y , p y ′ ) for x ∈ ( 0 , b ) , subject to suitable initial and boundary conditions. The monotone iterative technique has also been briefed as it gained a lot of attention during the last two decades and it has been merged with most of the other existing techniques. A list of SBVPs is also provided which will be of great help to researchers working in this area.

scholarly journals A New L-Stable Third Derivative Hybrid Method for Solving First Order Ordinary Differential Equations

2021 ◽  
pp. 58-69
Author(s):  
Lawrence Osa Adoghe
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hybrid Method ◽  
Multistep Method ◽  
Stability And Convergence ◽  
Stiff Systems ◽  
Convergence Properties ◽  
Main Method ◽  
The Stability ◽  
Third Derivative

In this paper, an L-stable third derivative multistep method has been proposed for the solution of stiff systems of ordinary differential equations. The continuous hybrid method is derived using interpolation and collocation techniques of power series as the basis function for the approximate solution. The method consists of the main method and an additional method which are combined to form a block matrix and implemented simultaneously. The stability and convergence properties of the block were investigated and discussed. Numerical examples to show the efficiency and accuracy of the new method were presented.

scholarly journals Two-Point Block variable order step size Multistep Method for Solving Higher Order Ordinary Differential Equations Directly

2021 ◽  
pp. 101376
Author(s):  
Ahmad Fadly Nurullah Rasedee ◽  
Mohammad Hasan Abdul Sathar ◽  
Siti Raihana Hamzah ◽  
Norizarina Ishak ◽  
Wong Tze Jin ◽  
...  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
Multistep Method ◽  
Variable Order ◽  
Step Size

scholarly journals Optimal representation to High Order Random Boolean kSatisability via Election Algorithm as Heuristic Search Approach in Hopeld Neural Networks

2021 ◽  
pp. 201-208
Author(s):  
Hamza Abubakar ◽  
Abdu Sagir Masanawa ◽  
Surajo Yusuf ◽  
G. I. Boaku
Keyword(s):  
Neural Network ◽  
Optimization Problems ◽  
Real Life ◽  
Model Performance ◽  
Statistical Error ◽  
Hopfield Neural Network ◽  
Higher Order ◽  
Learning Phase ◽  
Life Problems ◽  
Election Algorithm

This study proposed a hybridization of higher-order Random Boolean kSatisfiability (RANkSAT) with the Hopfield neural network (HNN) as a neuro-dynamical model designed to reflect knowledge efficiently. The learning process of the Hopfield neural network (HNN) has undergone significant changes and improvements according to various types of optimization problems. However, the HNN model is associated with some limitations which include storage capacity and being easily trapped to the local minimum solution. The Election algorithm (EA) is proposed to improve the learning phase of HNN for optimal Random Boolean kSatisfiability (RANkSAT) representation in higher order. The main source of inspiration for the Election Algorithm (EA) is its ability to extend the power and rule of political parties beyond their borders when seeking endorsement. The main purpose is to utilize the optimization capacity of EA to accelerate the learning phase of HNN for optimal random k Satisfiability representation. The global minima ratio (mR) and statistical error accumulations (SEA) during the training process were used to evaluate the proposed model performance. The result of this study revealed that our proposed EA-HNN-RANkSAT outperformed ABC-HNN-RANkSAT and ES-HNN-RANkSAT models in terms of mR and SEA.This study will further be extended to accommodate a novel field of Reverse analysis (RA) which involves data mining techniques to analyse real-life problems. 

Sign in / Sign up

Export Citation Format

Share Document