Solving first-order initial-value problems by using an explicit non-standard A -stable one-step method in variable step-size formulation

2015 ◽  
Vol 268 ◽  
pp. 796-805 ◽  
Author(s):  
Higinio Ramos ◽  
Gurjinder Singh ◽  
V. Kanwar ◽  
Saurabh Bhatia
Keyword(s):  
Initial Value Problems ◽  
Variable Step Size ◽  
Initial Value ◽  
Step Size ◽  
Step Method ◽  
First Order ◽  
Variable Step ◽  
One Step

scholarly journals A 4-Point Block Method for Solving Higher Order Ordinary Differential Equations Directly

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Nazreen Waeleh ◽  
Zanariah Abdul Majid
Keyword(s):  
Initial Value Problems ◽  
Adaptive Strategy ◽  
Variable Step Size ◽  
Block Method ◽  
Initial Value ◽  
Step Size ◽  
First Order ◽  
Variable Step ◽  
Improved Performance ◽  
Fifth Order

An alternative block method for solving fifth-order initial value problems (IVPs) is proposed with an adaptive strategy of implementing variable step size. The derived method is designed to compute four solutions simultaneously without reducing the problem to a system of first-order IVPs. To validate the proposed method, the consistency and zero stability are also discussed. The improved performance of the developed method is demonstrated by comparing it with the existing methods and the results showed that the 4-point block method is suitable for solving fifth-order IVPs.

L-stable Explicit Nonlinear Method with Constant and Variable Step-size Formulation for Solving Initial Value Problems

2018 ◽  
Vol 19 (7-8) ◽  
pp. 741-751 ◽  
Author(s):  
Sania Qureshi ◽  
Higinio Ramos
Keyword(s):  
Initial Value Problems ◽  
Blow Up ◽  
Singular Solutions ◽  
Variable Step Size ◽  
Explicit Method ◽  
Nonlinear Method ◽  
Initial Value ◽  
Step Size ◽  
Third Order ◽  
Variable Step

AbstractIn this work, we develop a nonlinear explicit method suitable for both autonomous and non-autonomous type of initial value problems in Ordinary Differential Equations (ODEs). The method is found to be third order accurate having L-stability. It is shown that if a variable step-size strategy is employed then the performance of the proposed method is further improved in comparison with other methods of same nature and order. The method is shown to be working well for initial value problems having singular solutions, singularly perturbed and stiff problems, and blow-up ODE problems, which is illustrated using a few numerical experiments.

scholarly journals Half Step Numerical Method for Solution of Second Order Initial Value Problems

2021 ◽  
pp. 77-81
Author(s):  
Adeniran Adebayo O. ◽  
Edaogbogun Kikelomo
Keyword(s):  
Numerical Method ◽  
Initial Value Problems ◽  
Second Order ◽  
Variable Step Size ◽  
Initial Value ◽  
Step Size ◽  
Stable Order ◽  
Variable Step ◽  
Interpolation Technique ◽  
Order Four

This paper presents a half step numerical method for solving directly general second order initial value problems. The scheme is developed via collocation and interpolation technique invoked on power series polynomial. The proposed method is consistent, zero stable, order four and three. This method can estimate the approximate solution at both step and off step points simultaneously by using variable step size. Numerical results are given to show the efficiency of the proposed scheme over some existing schemes of same and higher order.

scholarly journals An efficient hybrid numerical scheme for solvinggeneral second order initial value problems (IVPs)

2015 ◽  
Vol 4 (2) ◽  
pp. 411 ◽  
Author(s):  
Oluwadare Adeniran ◽  
Babatunde Ogundare
Keyword(s):  
Initial Value Problems ◽  
Numerical Scheme ◽  
Second Order ◽  
Variable Step Size ◽  
Initial Value ◽  
Step Size ◽  
Interpolation Technique ◽  
One Step ◽  
Grid Points ◽  
Order Four

<p>The paper presents a one step hybrid numerical scheme with two off grid points for solving directly the general second order initial value problems of ordinary differential equations. The scheme is developed using collocation and interpolation technique. The proposed scheme is consistent, zero stable and of order four. This scheme can estimate the approximate solution at both step and off step points simultaneously by using variable step size. Numerical results are given to show the efficiency of the proposed scheme over the existing schemes.</p>

scholarly journals An Accurate Block Solver for Stiff Initial Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
H. Musa ◽  
M. B. Suleiman ◽  
F. Ismail ◽  
N. Senu ◽  
Z. B. Ibrahim
Keyword(s):  
Initial Value Problems ◽  
Variable Step Size ◽  
Initial Value ◽  
Step Size ◽  
Differentiation Formula ◽  
Stiff Initial Value Problems ◽  
Variable Step ◽  
Compute Solution ◽  
The Stability ◽  
Size Mode

New implicit block formulae that compute solution of stiff initial value problems at two points simultaneously are derived and implemented in a variable step size mode. The strategy for changing the step size for optimum performance involves halving, increasing by a multiple of 1.7, or maintaining the current step size. The stability analysis of the methods indicates their suitability for solving stiff problems. Numerical results are given and compared with some existing backward differentiation formula algorithms. The results indicate an improvement in terms of accuracy.

THE FORWARD MAGNETOTELLURIC PROBLEM FOR AN INHOMOGENEOUS AND ANISOTROPIC STRUCTURE

Geophysics ◽  
1974 ◽  
Vol 39 (1) ◽  
pp. 56-68 ◽  
Author(s):  
Flavian Abramovici
Keyword(s):  
Three Dimensional ◽  
Middle Layer ◽  
Conductivity Tensor ◽  
Dimensional Structure ◽  
Impedance Tensor ◽  
Variable Step Size ◽  
Step Size ◽  
Variable Step ◽  
One Step ◽  
Numerical Integration Procedure

The impedance tensor corresponding to the magnetotelluric field for a nonisotropic one‐dimensional structure is given in terms of the solutions of a sixth‐order differential system. The conductivity tensor is three‐dimensional. Its components depend upon depth only in an arbitrary manner such that the corresponding matrix is positive definite. The impedance tensor components are found by a numerical integration procedure based on a set of one‐step methods and a variable step‐size to insure a given accuracy in the final result. Calculations were made for three models having sharp boundaries and also transitional layers. The first of these models has a middle layer of high conductivity, sandwiched between two layers of linearly varying conductivity, while in the second model the middle layer has a very low conductivity. In the third model the conductivity tensor is three‐dimensional and is linearly varying in one of the layers.

scholarly journals A Quantitative Comparison of Numerical Method for Solving Stiff Ordinary Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
S. A. M. Yatim ◽  
Z. B. Ibrahim ◽  
K. I. Othman ◽  
M. B. Suleiman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Computation Time ◽  
Quantitative Comparison ◽  
Initial Value ◽  
Step Size ◽  
Stiff Ordinary Differential Equations ◽  
Block Methods ◽  
Variable Step

We derive a variable step of the implicit block methods based on the backward differentiation formulae (BDF) for solving stiff initial value problems (IVPs). A simplified strategy in controlling the step size is proposed with the aim of optimizing the performance in terms of precision and computation time. The numerical results obtained support the enhancement of the method proposed as compared to MATLAB's suite of ordinary differential equations (ODEs) solvers, namely, ode15s and ode23s.

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