scholarly journals An adaptive stepsize algorithm for the numerical solving of initial-value problems

2015 ◽  
Vol 23 (1) ◽  
pp. 185-198
Author(s):  
Romulus Militaru
Keyword(s):  
Exact Solution ◽  
Initial Value Problems ◽  
Truncation Error ◽  
Selection Algorithm ◽  
Computational Accuracy ◽  
Initial Value ◽  
One Step ◽  
Adaptive Stepsize ◽  
Set Of Points ◽  
Step Selection

Abstract The present paper focuses on the efficient numerical solving of initial-value problems (IVPs) using digital computers and one-step numerical methods. We start from considering that the integration stepsize is the crucial factor in determining the number of calculations required and the amount of work involved to obtain the approximate values of the exact solution of a certain problem for a given set of points, within a prescribed computational accuracy, is proportional to the number of accomplished iterations. We perform an analysis of the local truncation error and we derive an adaptive stepsize algorithm which coupled with a certain one-step numerical method makes the use of this structure more computationally effective and insures that the estimated values of the exact solution are in agreement with an imposed accuracy. We conclude with numerical computations proving the efficiency of the proposed step selection algorithm.

scholarly journals New explicit high‐order one‐step methods for singular initial value problems

2020 ◽  
Author(s):  
Bohdan Datsko ◽  
Myroslaw Kutniv ◽  
Andriy Kunynets ◽  
Andrzej Włoch
Keyword(s):  
Initial Value Problems ◽  
High Order ◽  
Initial Value ◽  
One Step

scholarly journals A New Algorithm for Solving Terminal Value Problems of q-Difference Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Yong-Hong Fan ◽  
Lin-Lin Wang
Keyword(s):  
Exact Solution ◽  
Numerical Methods ◽  
Error Estimate ◽  
Numerical Solution ◽  
Numerical Method ◽  
Difference Equations ◽  
Initial Value Problems ◽  
Convergence Speed ◽  
Initial Value ◽  
Delta Derivatives

We propose a new algorithm for solving the terminal value problems on a q-difference equations. Through some transformations, the terminal value problems which contain the first- and second-order delta-derivatives have been changed into the corresponding initial value problems; then with the help of the methods developed by Liu and H. Jafari, the numerical solution has been obtained and the error estimate has also been considered for the terminal value problems. Some examples are given to illustrate the accuracy of the numerical methods we proposed. By comparing the exact solution with the numerical solution, we find that the convergence speed of this numerical method is very fast.

A Nonlinear Explicit One-Step Integration Scheme for Singular Autonomous Initial Value Problems

2007 ◽  
Author(s):  
Higinio Ramos ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras
Keyword(s):  
Initial Value Problems ◽  
Integration Scheme ◽  
Initial Value ◽  
One Step

scholarly journals The Treatment of Second Order Ordinary Differential Equations Using Equidistant One-step Block Hybrid

2019 ◽  
pp. 1-9
Author(s):  
Y. Skwame ◽  
J. Sabo ◽  
M. Mathew
Keyword(s):  
Numerical Analysis ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Second Order ◽  
Direct Solution ◽  
Block Method ◽  
Initial Value ◽  
One Step ◽  
Better Than

A general one-step hybrid block method with equidistant of order 6 has been successfully developed for the direct solution of second order IVPs in this article. Numerical analysis shows that the developed method is consistent and zero-stable which implies its convergence. The analysis of the new method is examined on two highly and mildly stiff second-order initial value problems to illustrate the efficiency of the method. It is obvious that our method performs better than the existing method within which we compare our result with. Hence, the approach is an adequate one for solving special second order IVPs.

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