Research

13 pages, 497 KiB

Open AccessArticle

The Optimal Homotopy Asymptotic Method for Solving Two Strongly Fractional-Order Nonlinear Benchmark Oscillatory Problems

by Mohd Taib Shatnawi, Adel Ouannas, Ghenaiet Bahia, Iqbal M. Batiha and Giuseppe Grassi

Mathematics 2021, 9(18), 2218; https://0-doi-org.brum.beds.ac.uk/10.3390/math9182218 - 10 Sep 2021

Cited by 8 | Viewed by 1801

Abstract

This paper proceeds from the perspective that most strongly nonlinear oscillators of fractional-order do not enjoy exact analytical solutions. Undoubtedly, this is a good enough reason to employ one of the major recent approximate methods, namely an Optimal Homotopy Asymptotic Method (OHAM), to [...] Read more.

This paper proceeds from the perspective that most strongly nonlinear oscillators of fractional-order do not enjoy exact analytical solutions. Undoubtedly, this is a good enough reason to employ one of the major recent approximate methods, namely an Optimal Homotopy Asymptotic Method (OHAM), to offer approximate analytic solutions for two strongly fractional-order nonlinear benchmark oscillatory problems, namely: the fractional-order Duffing-relativistic oscillator and the fractional-order stretched elastic wire oscillator (with a mass attached to its midpoint). In this work, a further modification has been proposed for such method and then carried out through establishing an optimal auxiliary linear operator, an auxiliary function, and an auxiliary control parameter. In view of the two aforesaid applications, it has been demonstrated that the OHAM is a reliable approach for controlling the convergence of approximate solutions and, hence, it is an effective tool for dealing with such problems. This assertion is completely confirmed by performing several graphical comparisons between the OHAM and the Homotopy Analysis Method (HAM). Full article

(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)

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15 pages, 1369 KiB

Open AccessArticle

A Fourth Order Symplectic and Conjugate-Symplectic Extension of the Midpoint and Trapezoidal Methods

by Felice Iavernaro and Francesca Mazzia

Mathematics 2021, 9(10), 1103; https://0-doi-org.brum.beds.ac.uk/10.3390/math9101103 - 13 May 2021

Cited by 3 | Viewed by 1416

Abstract

The paper presents fourth order Runge–Kutta methods derived from symmetric Hermite–Obreshkov schemes by suitably approximating the involved higher derivatives. In particular, starting from the multi-derivative extension of the midpoint method we have obtained a new symmetric implicit Runge–Kutta method of order four, for [...] Read more.

The paper presents fourth order Runge–Kutta methods derived from symmetric Hermite–Obreshkov schemes by suitably approximating the involved higher derivatives. In particular, starting from the multi-derivative extension of the midpoint method we have obtained a new symmetric implicit Runge–Kutta method of order four, for the numerical solution of first-order differential equations. The new method is symplectic and is suitable for the solution of both initial and boundary value Hamiltonian problems. Moreover, starting from the conjugate class of multi-derivative trapezoidal schemes, we have derived a new method that is conjugate to the new symplectic method. Full article

(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)

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22 pages, 746 KiB

Open AccessArticle

A Singularly P-Stable Multi-Derivative Predictor Method for the Numerical Solution of Second-Order Ordinary Differential Equations

by Ali Shokri, Beny Neta, Mohammad Mehdizadeh Khalsaraei, Mohammad Mehdi Rashidi and Hamid Mohammad-Sedighi

Mathematics 2021, 9(8), 806; https://0-doi-org.brum.beds.ac.uk/10.3390/math9080806 - 08 Apr 2021

Cited by 4 | Viewed by 1993

Abstract

In this paper, a symmetric eight-step predictor method (explicit) of 10th order is presented for the numerical integration of IVPs of second-order ordinary differential equations. This scheme has variable coefficients and can be used as a predictor stage for other implicit schemes. First, [...] Read more.

In this paper, a symmetric eight-step predictor method (explicit) of 10th order is presented for the numerical integration of IVPs of second-order ordinary differential equations. This scheme has variable coefficients and can be used as a predictor stage for other implicit schemes. First, we showed the singular P-stability property of the new method, both algebraically and by plotting the stability region. Then, having applied it to well-known problems like Mathieu equation, we showed the advantage of the proposed method in terms of efficiency and consistency over other methods with the same order. Full article

(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)

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22 pages, 612 KiB

Open AccessArticle

A Family of Functionally-Fitted Third Derivative Block Falkner Methods for Solving Second-Order Initial-Value Problems with Oscillating Solutions

by Higinio Ramos, Ridwanulahi Abdulganiy, Ruth Olowe and Samuel Jator

Mathematics 2021, 9(7), 713; https://0-doi-org.brum.beds.ac.uk/10.3390/math9070713 - 25 Mar 2021

Cited by 9 | Viewed by 1785

Abstract

One of the well-known schemes for the direct numerical integration of second-order initial-value problems is due to Falkner. This paper focuses on the construction of a family of adapted block Falkner methods which are frequency dependent for the direct numerical solution of second-order [...] Read more.

One of the well-known schemes for the direct numerical integration of second-order initial-value problems is due to Falkner. This paper focuses on the construction of a family of adapted block Falkner methods which are frequency dependent for the direct numerical solution of second-order initial value problems with oscillatory solutions. The techniques of collocation and interpolation are adopted here to derive the new methods. The study of the properties of the proposed adapted block Falkner methods reveals that they are consistent and zero-stable, and thus, convergent. Furthermore, the stability analysis and the algebraic order conditions of the proposed methods are established. As may be seen from the numerical results, the resulting family is efficient and competitive compared to some recent methods in the literature. Full article

(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)

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11 pages, 276 KiB

Open AccessArticle

High Order Two-Derivative Runge-Kutta Methods with Optimized Dispersion and Dissipation Error

by Theodoros Monovasilis and Zacharoula Kalogiratou

Mathematics 2021, 9(3), 232; https://0-doi-org.brum.beds.ac.uk/10.3390/math9030232 - 25 Jan 2021

Cited by 4 | Viewed by 1597

Abstract

In this work we consider explicit Two-derivative Runge-Kutta methods of a specific type where the function f is evaluated only once at each step. New 7th order methods are presented with minimized dispersion and dissipation error. These are two methods with constant coefficients [...] Read more.

In this work we consider explicit Two-derivative Runge-Kutta methods of a specific type where the function f is evaluated only once at each step. New 7th order methods are presented with minimized dispersion and dissipation error. These are two methods with constant coefficients with 5 and 6 stages. Also, a modified phase-fitted, amplification-fitted method with frequency dependent coefficients and 5 stages is constructed based on the 7th order method of Chan and Tsai. The new methods are applied to 4 well known oscillatory problems and their performance is compared with the methods in that of Chan and Tsai.The numerical experiments show the efficiency of the derived methods. Full article

(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)

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21 pages, 2458 KiB

Open AccessArticle

Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

by Janez Urevc and Miroslav Halilovič

Mathematics 2021, 9(2), 174; https://0-doi-org.brum.beds.ac.uk/10.3390/math9020174 - 16 Jan 2021

Viewed by 2496

Abstract

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation [...] Read more.

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature. Full article

(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)

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6 pages, 208 KiB

Open AccessArticle

Asymptotic Behavior of Solution to Nonlinear Eigenvalue Problem

by Tetsutaro Shibata

Mathematics 2020, 8(11), 2064; https://0-doi-org.brum.beds.ac.uk/10.3390/math8112064 - 19 Nov 2020

Cited by 2 | Viewed by 1102

Abstract

We study the following nonlinear eigenvalue problem: [...] Read more.

We study the following nonlinear eigenvalue problem:

- u^{″} (t) = λ f (u (t)), u (t) > 0, t \in I : = (- 1, 1), u (\pm 1) = 0,

where

f (u) = log (1 + u)

and

λ > 0

is a parameter. Then

λ

is a continuous function of

α > 0

, where

α

is the maximum norm

α = ∥ u_{λ} ∥_{\infty}

of the solution

u_{λ}

associated with

λ

. We establish the precise asymptotic formula for

L^{1}

-norm of the solution

∥ u_{α} ∥_{1}

as

α \to \infty

up to the second term and propose a numerical approach to obtain the asymptotic expansion formula for

∥ u_{α} ∥_{1}

. Full article

(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)

16 pages, 460 KiB

Open AccessArticle

Implicit Three-Point Block Numerical Algorithm for Solving Third Order Initial Value Problem Directly with Applications

by Reem Allogmany and Fudziah Ismail

Mathematics 2020, 8(10), 1771; https://0-doi-org.brum.beds.ac.uk/10.3390/math8101771 - 14 Oct 2020

Cited by 8 | Viewed by 2067

Abstract

Recently, direct methods that involve higher derivatives to numerically approximate higher order initial value problems (IVPs) have been explored, which aim to construct numerical methods with higher order and very high precision of the solutions. This article aims to construct a fourth and [...] Read more.

Recently, direct methods that involve higher derivatives to numerically approximate higher order initial value problems (IVPs) have been explored, which aim to construct numerical methods with higher order and very high precision of the solutions. This article aims to construct a fourth and fifth derivative, three-point implicit block method to tackle general third-order ordinary differential equations directly. As a consequence of the increase in order acquired via the implicit block method of higher derivatives, a significant improvement in efficiency has been observed. The new method is derived in a block mode to simultaneously evaluate the approximations at three points. The derivation of the new method can be easily implemented. We established the proposed method’s characteristics, including order, zero-stability, and convergence. Numerical experiments are used to confirm the superiority of the method. Applications to problems in physics and engineering are given to assess the significance of the method. Full article

(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)

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17 pages, 584 KiB

Open AccessFeature PaperArticle

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

by Higinio Ramos, Samuel N. Jator and Mark I. Modebei

Mathematics 2020, 8(10), 1752; https://0-doi-org.brum.beds.ac.uk/10.3390/math8101752 - 12 Oct 2020

Cited by 6 | Viewed by 1705

Abstract

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 [...] Read more.

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

2 k

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. Full article

(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)

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12 pages, 301 KiB

Open AccessArticle

Analysis of Perturbed Volterra Integral Equations on Time Scales

by Eleonora Messina, Youssef N. Raffoul and Antonia Vecchio

Mathematics 2020, 8(7), 1133; https://0-doi-org.brum.beds.ac.uk/10.3390/math8071133 - 10 Jul 2020

Cited by 4 | Viewed by 1638

Abstract

This paper describes the effect of perturbation of the kernel on the solutions of linear Volterra integral equations on time scales and proposes a new perspective for the stability analysis of numerical methods. Full article

(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)

19 pages, 451 KiB

Open AccessArticle

Numerical Approach for Solving Delay Differential Equations with Boundary Conditions

by Nur Tasnem Jaaffar, Zanariah Abdul Majid and Norazak Senu

Mathematics 2020, 8(7), 1073; https://0-doi-org.brum.beds.ac.uk/10.3390/math8071073 - 02 Jul 2020

Cited by 7 | Viewed by 3917

Abstract

In the present paper, a fifth-order direct multistep block method is proposed for solving the second-order Delay Differential Equations (DDEs) directly with boundary conditions using constant step size. In many life sciences applications, a delay plays an essential role in modelling natural phenomena [...] Read more.

In the present paper, a fifth-order direct multistep block method is proposed for solving the second-order Delay Differential Equations (DDEs) directly with boundary conditions using constant step size. In many life sciences applications, a delay plays an essential role in modelling natural phenomena with data simulation. Thus, an efficient numerical method is needed for the numerical treatment of time delay in the applications. The proposed direct block method computes the numerical solutions at two points concurrently at each computed step along the interval. The types of delays involved in this research are constant delay, pantograph delay, and time-dependent delay. The shooting technique is utilized to deal with the boundary conditions by applying a Newton-like method to guess the next initial values. The analysis of the proposed method based on the order, consistency, convergence, and stability of the method are discussed in detail. Four tested problems are presented to measure the efficiency of the developed direct multistep block method. The numerical simulation indicates that the proposed direct multistep block method performs better than existing methods in terms of accuracy, total function calls, and execution times. Full article

(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)

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18 pages, 3931 KiB

Open AccessArticle

Semi-Implicit Multistep Extrapolation ODE Solvers

by Denis Butusov, Aleksandra Tutueva, Petr Fedoseev, Artem Terentev and Artur Karimov

Mathematics 2020, 8(6), 943; https://0-doi-org.brum.beds.ac.uk/10.3390/math8060943 - 08 Jun 2020

Cited by 9 | Viewed by 2030

Abstract

Multistep methods for the numerical solution of ordinary differential equations are an important class of applied mathematical techniques. This paper is motivated by recently reported advances in semi-implicit numerical integration methods, multistep and extrapolation solvers. Here we propose a novel type of multistep [...] Read more.

Multistep methods for the numerical solution of ordinary differential equations are an important class of applied mathematical techniques. This paper is motivated by recently reported advances in semi-implicit numerical integration methods, multistep and extrapolation solvers. Here we propose a novel type of multistep extrapolation method for solving ODEs based on the semi-implicit basic method of order 2. Considering several chaotic systems and van der Pol nonlinear oscillator as examples, we implemented a performance analysis of the proposed technique in comparison with well-known multistep methods: Adams–Bashforth, Adams–Moulton and the backward differentiation formula. We explicitly show that the multistep semi-implicit methods can outperform the classical linear multistep methods, providing more precision in the solutions for nonlinear differential equations. The analysis of stability regions reveals that the proposed methods are more stable than explicit linear multistep methods. The possible applications of the developed ODE solver are the long-term simulations of chaotic systems and processes, solving moderately stiff differential equations and advanced modeling systems. Full article

(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)

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