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 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.

Continuous linear multistep method for the general solution of first order initial value problems for Volterra integro-differential equations

2018 ◽  
Vol 14 (5) ◽  
pp. 960-969
Author(s):  
Nathaniel Mahwash Kamoh ◽  
Terhemen Aboiyar
Keyword(s):  
General Solution ◽  
Approximation Method ◽  
Legendre Polynomial ◽  
Basis Function ◽  
Initial Value Problems ◽  
Block Method ◽  
Initial Value ◽  
Content Type ◽  
First Order ◽  
Predictor Corrector

Purpose The purpose of this paper is to develop a block method of order five for the general solution of the first-order initial value problems for Volterra integro-differential equations (VIDEs). Design/methodology/approach A collocation approximation method is adopted using the shifted Legendre polynomial as the basis function, and the developed method is applied as simultaneous integrators on the first-order VIDEs. Findings The new block method possessed the desirable feature of the Runge–Kutta method of being self-starting, hence eliminating the use of predictors. Originality/value In this paper, some information about solving VIDEs is provided. The authors have presented and illustrated the collocation approximation method using the shifted Legendre polynomial as the basis function to investigate solving an initial value problem in the class of VIDEs, which are very difficult, if not impossible, to solve analytically. With the block approach, the non-self-starting nature associated with the predictor corrector method has been eliminated. Unlike the approach in the predictor corrector method where additional equations are supplied from a different formulation, all the additional equations are from the same continuous formulation which shows the beauty of the method. However, the absolute stability region showed that the method is A-stable, and the application of this method to practical problems revealed that the method is more accurate than earlier methods.

scholarly journals A Real-Time Implementation of Novel and Stable Variable Step Size MPPT

Energies ◽  
2020 ◽  
Vol 13 (18) ◽  
pp. 4668 ◽  
Author(s):  
Maissa Farhat ◽  
Oscar Barambones ◽  
Lassaâd Sbita
Keyword(s):  
Real Time ◽  
Maximum Power ◽  
Variable Step Size ◽  
System Control ◽  
Resistive Load ◽  
Step Size ◽  
Pv System ◽  
Variable Step ◽  
Improved Performance ◽  
Power Point

This paper presents a complete study of a standalone photovoltaic (PV) system including a maximum power tracker (MPPT) driving a DC boost converter to feed a resistive load. Here, a new MPPT approach using a modification on the original perturb and observe (P&O) algorithm is proposed; the improved algorithm is founded on a variable step size (VSZ). This novel algorithm is realized and efficiently implemented in the PV system. The proposed VSZ algorithm is compared both in simulation and in real time to the P&O algorithm. The stability analysis for the VSZ algorithm is performed using Lyapunov’s stability theory. In this paper, a detailed study and explanation of the modified P&O MPPT controller is presented to ensure high PV system performance. The proposed algorithm is practically implemented using a DSP1104 for real-time testing. Significant results are achieved, proving the validity of the proposed PV system control scheme. The obtained results show that the proposed VSZ succeeds at harvesting the maximum power point (MPP), as the amount of harvested power using VSZ is three times greater than the power extracted without the tracking algorithm. The VSZ reveals improved performance compared to the conventional P&O algorithm in term of dynamic response, signal quality and stability.

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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