A trigonometrically fitted intra-step block Falkner method for the direct integration of second-order delay differential equations with oscillatory solutions

Abdulganiy, R. I.; Ramos, Higinio; Okunuga, S. A.; Majid, Zanariah Abdul

doi:10.1007/s13370-023-01075-3

Afr. Mat.

2023

DOI: 10.1007/s13370-023-01075-3

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A trigonometrically fitted intra-step block Falkner method for the direct integration of second-order delay differential equations with oscillatory solutions

R. I. Abdulganiy

Higinio Ramos

S. A. Okunuga

et al.

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2024

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Cited by 5 publications

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A New Methodology for the Development of Efficient Multistep Methods for First-Order Initial Value Problems with Oscillating Solutions: III the Role of the Derivative of the Phase Lag and the Derivative of the Amplification Factor

Simos

2024

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Recently, the author developed a theory for the computation of the phase lag and amplification factor for explicit and implicit multistep methods for first-order differential equations. In this paper, we will investigate the role of the derivatives of the phase lag and the derivatives of the amplification factor on the efficiency of the newly developed methods. We will also present the stability regions of the newly developed methods. We will also present numerical experiments and conclusions on the newly developed methodologies.

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A New Methodology for the Development of Efficient Multistep Methods for First-Order Initial Value Problems with Oscillating Solutions: III the Role of the Derivative of the Phase Lag and the Derivative of the Amplification Factor

Simos

2024

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A New Methodology for the Development of Efficient Multistep Methods for First–Order IVPs with Oscillating Solutions IV: The Case of the Backward Differentiation Formulae

Simos

2024

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A theory for the calculation of the phase–lag and amplification–factor for explicit and implicit multistep techniques for first–order differential equations was recently established by the author. His presentation also covered how the approaches’ efficacy is affected by the elimination of the phase–lag and amplification–factor derivatives. This paper will apply the theory for computing the phase–lag and amplification–factor, originally developed for implicit multistep methods, to a subset of implicit methods, called backward differentiation formulae (BDF), and will examine the impact of the phase–lag and amplification–factor derivatives on the efficiency of these strategies. Next, we will show you the stability zones of these brand-new approaches. Lastly, we will discuss the results of numerical experiments and draw some conclusions about the established approaches.

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A New Methodology for the Development of Efficient Multistep Methods for First-Order IVPs with Oscillating Solutions

Simos

2024

Mathematics

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In this research, we provide a novel approach to the development of effective numerical algorithms for the solution of first-order IVPs. In particular, we detail the fundamental theory behind the development of the aforementioned approaches and show how it can be applied to the Adams–Bashforth approach in three steps. The stability of the new scheme is also analyzed. We compared the performance of our novel algorithm to that of established approaches and found it to be superior. Numerical experiments confirmed that, in comparison to standard approaches to the numerical solution of Initial Value Problems (IVPs), including oscillating solutions, our approach is significantly more effective.

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A trigonometrically fitted intra-step block Falkner method for the direct integration of second-order delay differential equations with oscillatory solutions

Cited by 5 publications

References 24 publications

A New Methodology for the Development of Efficient Multistep Methods for First-Order Initial Value Problems with Oscillating Solutions: III the Role of the Derivative of the Phase Lag and the Derivative of the Amplification Factor

A New Methodology for the Development of Efficient Multistep Methods for First-Order Initial Value Problems with Oscillating Solutions: III the Role of the Derivative of the Phase Lag and the Derivative of the Amplification Factor

A New Methodology for the Development of Efficient Multistep Methods for First–Order IVPs with Oscillating Solutions IV: The Case of the Backward Differentiation Formulae

A New Methodology for the Development of Efficient Multistep Methods for First-Order IVPs with Oscillating Solutions

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