scholarly journals Efficient k-Step Linear Block Methods to Solve Second Order Initial Value Problems Directly

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1752
Author(s):  
Higinio Ramos ◽  
Samuel N. Jator ◽  
Mark I. Modebei
Keyword(s):  
Computational Efficiency ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Computing Time ◽  
Approximate Solutions ◽  
Second Order ◽  
Initial Value ◽  
Special Equation ◽  
Block Methods ◽  
Grid Points

There are dozens of block methods in literature intended for solving second order initial-value problems. This article aimed at the analysis of the efficiency of k-step block methods for directly solving general second-order initial-value problems. Each of these methods consists of a set of 2k multi-step formulas (although we will see that this number can be reduced to k+1 in case of a special equation) that provides approximate solutions at k grid points at once. The usual way to obtain these formulas is by using collocation and interpolation at different points, which are not all necessarily in the mesh (it may also be considered intra-step or off-step points). An important issue is that for each k, all of them are essentially the same method, although they can adopt different formulations. Nevertheless, the performance of those formulations is not the same. The analysis of the methods presented give some clues as how to select the most appropriate ones in terms of computational efficiency. The numerical experiments show that using the proposed formulations, the computing time can be reduced to less than half.

scholarly journals Direct Solution of Second Order Ordinary Differential Equations Using a Class of Hybrid Block Methods

2020 ◽  
Vol 5 (2) ◽  
Author(s):  
Emmanuel A Areo ◽  
Nosimot O Adeyanju ◽  
Sunday J Kayode
Keyword(s):  
Initial Value Problems ◽  
Hybrid Methods ◽  
Multistep Method ◽  
Second Order ◽  
Block Method ◽  
Initial Value ◽  
Order Error ◽  
Block Methods ◽  
Grid Points ◽  
Step Number

This research proposes the derivation of a class of hybrid methods for solution of second order initial value problems (IVPs) in block mode. Continuous linear multistep method of two cases with step number k = 4 is developed by interpolating the basis function at certain grid points and collocating the differential system at both grid and off-grid points. The basic properties of the method including order, error constant, zero stability, consistency and convergence were investigated. In order to examine the accuracy of the methods, some differential problems of order two were solved and results generated show a better performance when comparison is made with some current methods.Keywords- Block Method, Hybrid Points, Initial Value Problems, Power Series, Second Order 

scholarly journals Solving third order ordinary differential equations directly using hybrid numerical models

2020 ◽  
pp. 69-76
Author(s):  
J. O. Kuboye ◽  
O. F. Quadri ◽  
O. R. Elusakin
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Numerical Models ◽  
Gaussian Elimination ◽  
Initial Value ◽  
Third Order ◽  
Block Methods ◽  
Grid Points

In this work, numerical methods for solving third order initial value problems of ordinary differential equations are developed. Multi-step collocation is used in deriving the methods, where power series approximate solution is employed as a basis function. Gaussian elimination approach is considered in finding the unknown variables $a_j, j=0(1)8$ in interpolation and collocation equations, which are substituted into the power series to give the continuous implicit schemes. The discrete schemes and its derivatives are derived by evaluating the grid and non-grid points. These schemes are arranged in a matrix form to produce block methods. The order of the developed methods are found to be six. The numerical results proved the efficiency of the methods over the existing methods.

scholarly journals Numerical Simulation of One Step Block Method for Treatment of Second Order Forced Motions in Mass-Spring Systems

2021 ◽  
pp. 1-11
Author(s):  
J. Sabo ◽  
T. Y. Kyagya ◽  
W. J. Vashawa
Keyword(s):  
Numerical Simulation ◽  
Initial Value Problems ◽  
Second Order ◽  
Block Method ◽  
Initial Value ◽  
Mathematical Problems ◽  
One Step ◽  
Grid Points ◽  
The One ◽  
Mass Spring

This paper discuss the numerical simulation of one step block method for treatment of second order forced motions in mass-spring systems of initial value problems. The one step block method has been developed with the introduction of off-mesh point at both grid and off- grid points using interpolation and collocation procedure to increase computational burden which may jeopardize the accuracy of the method in terms of error. The basic properties of the one step block method was established and numerical analysis shown that the one step block method was found to be consistent, convergent and zero-stable. The one step block method was simulated on three highly stiff mathematical problems to validate the accuracy of the block method without reduction, and obviously the results shown are more accurate over the existing method in literature.

scholarly journals New Algorithm for First order Stiff Initial Value Problems

2017 ◽  
Vol 58 (1) ◽  
pp. 19-28 ◽  
Author(s):  
A. O. Adesanya ◽  
R. O. Onsachi ◽  
M. R. Odekunle
Keyword(s):  
Initial Value Problems ◽  
Numerical Experiments ◽  
System Of Nonlinear Equations ◽  
Initial Value ◽  
Unknown Parameters ◽  
First Order ◽  
Continuous Scheme ◽  
Discrete Methods ◽  
Grid Points ◽  
The Stability

AbstractIn this paper, we consider the development and implementation of algorithms for the solution of stiff first order initial value problems. Method of interpolation and collocation of basis function to give system of nonlinear equations which is solved for the unknown parameters to give a continuous scheme that is evaluated at selected grid points to give discrete methods. The stability properties of the method is verified and numerical experiments show that the new method is efficient in handling stiff problems.

scholarly journals Modified variational iteration method for variant Boussinesq equation

2015 ◽  
Vol 19 (4) ◽  
pp. 1195-1199 ◽  
Author(s):  
Jun-Feng Lu
Keyword(s):  
Exact Solutions ◽  
Iteration Method ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Boussinesq Equation ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Initial Value ◽  
Variational Iteration ◽  
Modified Variational Iteration Method

In this paper, we solve the variant Boussinesq equation by the modified variational iteration method. The approximate solutions to the initial value problems of the variant Boussinesq equation are provided, and compared with the exact solutions. Numerical experiments show that the modified variational iteration method is efficient for solving the variant Boussinesq equation.

scholarly journals Bernstein polynomial induced two step hybrid numerical scheme for solution of second order initial value problems

2021 ◽  
Vol 2 (1) ◽  
pp. 15-25
Author(s):  
A. O. Adeniran ◽  
Longe Idowu O. ◽  
Edaogbogun Kikelomo
Keyword(s):  
Initial Value Problems ◽  
Numerical Scheme ◽  
Bernstein Polynomial ◽  
Grid Point ◽  
Approximate Solutions ◽  
Second Order ◽  
Variable Step Size ◽  
Initial Value ◽  
Step Size ◽  
Order Four

This paper presents a two-step hybrid numerical scheme with one off-grid point for the numerical solution of general second-order initial value problems without reducing to two systems of the first order. The scheme is developed using the collocation and interpolation technique invoked on Bernstein polynomial. The proposed scheme is consistent, zero stable, and is of order four($4$). The developed scheme can estimate the approximate solutions at both steps and off-step points simultaneously using variable step size. Numerical results obtained in this paper show the efficiency of the proposed scheme over some existing methods of the same and higher orders.

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