New Efficient Numerical Model for Solving Second, Third and Fourth Order Ordinary Differential Equations Directly (original) (raw)
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In this research, we have proposed the numerical application of second derivative ordinary differential equations using power series for the direct solution of higher order initial value problems. The method was derived using power series, via interpolation and collocation procedure. The analysis of the method was studied, and it was found to be consistent, zero-stable and convergent. The derived method was able to solve highly stiff problems without converting to the equivalents system of first order ODEs. The generated results showed that the derived methods are notable better than those methods in literature. We further sketched the solution graph of our method and it is evident that the new method convergence toward the exact solution.
Journal of the Nigerian Society of Physical Sciences
Continuous hybrid methods are now recognized as efficient numerical methods for problems whose solutions have finite domains or cannot be solved analytically. In this work, the continuous hybrid numerical method for the solution of general second order initial value problems of ordinary differential equations is considered. The method of collocation of the differential system arising from the approximate solution to the problem is adopted using the power series as a basis function. The method is zero stable, consistent, convergent. It is suitable for both non-stiff and mildly-stiff problems and results were found to compete favorably with the existing methods in terms of accuracy.
DergiPark (Istanbul University), 2021
We propose and present a self-starting numerical approximation with a higher order of accuracy for direct solution of a special fourth-order ordinary differential equation (ODE) using a Hybrid Linear Multistep Method (HLMM). The technique utilizes the collocation and interpolation approach with six-step numbers and two off-step points using power series as the basis function. Error constants and basic properties proved the convergence of the method. Numerical experiments involving both linear, nonlinear, and linear systems of fourth-order initial value problems appearing in modeling of physical phenomenon from various areas of applied sciences were used to demonstrate the effectiveness and efficiency of the proposed method. The results revealed that the proposed method is an excellent choice for approximating general fourth-order ODE and shows the impact of choices of step sizes in the numerical solution of the problem considered. In addition, the proposed HLMM outperformed existing methods in the literature in terms of accuracy.
Journal of Mathematics Letters, 2020
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.
Numerical Algorithms for Direct Solution of Fourth Order Ordinary Differential Equations
Journal of the Nigerian Society of Physical Sciences
This paper examines the derivation of hybrid numerical algorithms with step length(k) of five for solving fourth order initial value problems of ordinary differential equations directly. In developing the methods, interpolation and collocation techniques are considered. Approximated power series is used as interpolating polynomial and its fourth derivative as the collocating equation. These equations are solved using Gaussian-elimination approach in finding the unknown variables aj, j=0,...,10 which are substituted into basis function to give continuous implicit scheme. The discrete schemes and its derivatives that form the block are obtainedby evaluating continuous implicit scheme at non-interpolating points. The developed methods are of order seven and the results generated when the methods were applied to fourth order initial value problems compared favourably with existing methods.order initial value problems compared favourably with existing methods.
An Examination of a Second Order Numerical Method for Solving Initial Value Problems
Journal of the Nigerian Society of Physical Sciences, 2020
This paper presents an examination of a Second Order Convergence Numerical Method (SOCNM) for solving Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs). The SOCNM has been derived via the interpolating function comprises of polynomial and exponential forms. The analysis and the properties of SOCNM were discussed. Three numerical examples have been solved successfully to examine the performance of SOCNM in terms of accuracy and stability. The comparative study of SOCNM, Improved Modified Euler Method (IMEM), Fadugba and Olaosebikan Scheme (FOS) and the Exact Solution (ES) is presented. By varying the step length, the absolute relative errors at the final nodal point of the associated integration interval are computed. Furthermore, the analysis of the properties of SOCNM shows that the method is consistent, stable, convergent and has second order accuracy. Moreover, the numerical results show that SOCNM is more accurate than IMEM and FOS and also compared favoura...
2020
Numerical analysis is a subject that is concerned with how to solve real life problems numerically. Numerical methods form an important part of solving differential equations emanated from real life situations, most especially in cases where there is no closed-form solution or difficult to obtain exact solutions. The main aim of this paper is to review some numerical methods for solving initial value problems of ordinary differential equations. The comparative study of the Third Order Convergence Numerical Method (FS), Adomian Decomposition Method (ADM) and Successive Approximation Method (SAM) in the context of the exact solution is presented. The methods will be compared in terms of convergence, accuracy and efficiency. Five illustrative examples/test problems were solved successfully. The results obtained show that the three methods are approximately the same in terms of accuracy and convergence in the case of first order linear ordinary differential equations. It is also observed that FS, ADM and SAM were found to be computationally efficient for the linear ordinary differential equations. In the case of the non-linear ordinary differential equations, SAM is found to be more accurate and converges faster to the exact solution than the FS and ADM. Hence, It is clearly seen that the ADM is found to be better than the FS and SAM in the case of non-linear differential equations in terms of computational efficiency.
Numerical solution of Second order Ordinary Differential Equations
Computational Algorithms and Numerical Dimensions, 2024
Numerical analysis is a modern technology employed by scientists and engineers to tackle complex problems that are challenging to solve through direct methods. In this article, we introduce the Milne-Simpson predictor-corrector technique (MS) and the fourth-order Adam-Bashforth-Moulton predictor-corrector method (ADM) for solving second-order initial value problems (IVPs) of ordinary differential equations. Both methods are highly efficient and particularly suitable for addressing IVPs. We compare the Euler and fourth-order Runge-Kutta methods within the ADM framework to the Milne-Simpson predictor-corrector approach. To validate the accuracy of the numerical solutions, we compare the approximate solutions with the exact solutions and find that they align well. We also observe that initializing the fourth-order ADM method with the fourth-order Runge-Kutta method and using the MS method for approximation yields superior accuracy compared to starting the ADM with the Euler method. Additionally, we evaluate the performance, effectiveness, and computational efficiency of both approaches. Finally, we demonstrate the convergence of these methods and examine the error terms for various step sizes. The numerical experiments are conducted using Matlab 2024.
Zenodo (CERN European Organization for Nuclear Research), 2023
In this research work, a Class of Continuous 4-Step Method (CCSM) for solving second order ordinary differential equations is provided. The step size of the approximate solution affects the coefficients of the developed approach. As a by-product of the continuous technique, a discrete second derivatives method is obtained from the continuous method. At any stage in collocation, the primary predictor needed for the implicit method assessment is of the same order as the method. The stability, consistency, and convergence aspects of the method were discussed and used to the solve linear and non-linear problems in order to demonstrate its applicability and efficiency.