scholarly journals Approximate solutions of general second–order initial value problems using differential evolution

2017 ◽  
Keyword(s):  
Differential Evolution ◽  
Initial Value Problems ◽  
Approximate Solutions ◽  
Second Order ◽  
Initial Value

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 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.

scholarly journals Delta-Nabla Type Maximum Principles for Second-Order Dynamic Equations on Time Scales and Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-28
Author(s):  
Jiang Zhu ◽  
Dongmei Liu
Keyword(s):  
Time Scales ◽  
Initial Value Problems ◽  
Second Order ◽  
Dynamic Equations ◽  
Maximum Principles ◽  
Uniqueness Theorems ◽  
Initial Value ◽  
Approximation Theorems ◽  
Construction Techniques ◽  
Linear And Nonlinear

Some delta-nabla type maximum principles for second-order dynamic equations on time scales are proved. By using these maximum principles, the uniqueness theorems of the solutions, the approximation theorems of the solutions, the existence theorem, and construction techniques of the lower and upper solutions for second-order linear and nonlinear initial value problems and boundary value problems on time scales are proved, the oscillation of second-order mixed delat-nabla differential equations is discussed and, some maximum principles for second order mixed forward and backward difference dynamic system are proved.

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