Implicit two-derivative Runge–Kutta collocation methods for systems of initial value problems

Yakubu, D. G.; Kwami, A.M.

doi:10.1016/j.jnnms.2015.01.001

Cited by 7 publications

(4 citation statements)

References 23 publications

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“…Runge-Kutta method is a high-precision single-step algorithm for numerical solutions of ordinary differential equations [21][22][23][24]. For integral differential equation 12, from to +1 , we obtain…”

Section: Runge-kutta Methods For Numerical Conformal Mapping Of Doublymentioning

confidence: 99%

Improving the Accuracy of the Charge Simulation Method for Numerical Conformal Mapping

Fuming

Wang

et al. 2017

Mathematical Problems in Engineering

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In this paper, we present a method to improve the accuracy of the charge simulation method for numerical conformal mapping. The method constructs the constraint equation by using the charge simulation method. The charges and the conformal radius are computed by using the Runge-Kutta method based on the dynamic system of the constraint equation. By using this method, we can obtain new approximated conformal mapping function and improve the accuracy of numerical conformal mapping compared to the charge simulation method for numerical conformal mapping proposed by Amano. Furthermore, the corresponding numerical results are shown to illustrate the performance of the proposed method.

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Section: Runge-kutta Methods For Numerical Conformal Mapping Of Doublymentioning

confidence: 99%

Improving the Accuracy of the Charge Simulation Method for Numerical Conformal Mapping

Fuming

Wang

et al. 2017

Mathematical Problems in Engineering

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“…Lastly, the method outlined in Lambert [ 61 ] and Yakubu et al [ 66 ] is used to analyze and discuss the stability of the ISBS. …”

Section: The Analysis Of the Isbsmentioning

confidence: 99%

Unveiling the Power of Implicit Six-Point Block Scheme: Advancing numerical approximation of two-dimensional PDEs in physical systems

Olaoluwa Omole,

Olusheye Adeyefa,

Iyabo Apanpa

et al. 2024

PLoS ONE

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In the era of computational advancements, harnessing computer algorithms for approximating solutions to differential equations has become indispensable for its unparalleled productivity. The numerical approximation of partial differential equation (PDE) models holds crucial significance in modelling physical systems, driving the necessity for robust methodologies. In this article, we introduce the Implicit Six-Point Block Scheme (ISBS), employing a collocation approach for second-order numerical approximations of ordinary differential equations (ODEs) derived from one or two-dimensional physical systems. The methodology involves transforming the governing PDEs into a fully-fledged system of algebraic ordinary differential equations by employing ISBS to replace spatial derivatives while utilizing a central difference scheme for temporal or y-derivatives. In this report, the convergence properties of ISBS, aligning with the principles of multi-step methods, are rigorously analyzed. The numerical results obtained through ISBS demonstrate excellent agreement with theoretical solutions. Additionally, we compute absolute errors across various problem instances, showcasing the robustness and efficacy of ISBS in practical applications. Furthermore, we present a comprehensive comparative analysis with existing methodologies from recent literature, highlighting the superior performance of ISBS. Our findings are substantiated through illustrative tables and figures, underscoring the transformative potential of ISBS in advancing the numerical approximation of two-dimensional PDEs in physical systems.

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“…In this research paper, we have proposed new efficient techniques as alternative methods to some well-known methods 5,8,11 to solve pharmacokinetic models. The new techniques are called continuous block implicit hybrid one-step collocation methods, developed based on Legendre polynomial nodes that have better stability regions over a wide range of parameters (see Yakubu et al 20 and Yakubu and Kwami 21 ). The motivation for these methods, particularly the use of Gauss family of methods, is that collocation at Gauss points lead to methods which are symmetric and algebraically stable (see e.g.…”

Section: Numerical Computational Techniquesmentioning

confidence: 99%

Accurate numerical integration of highly stiff pharmacokinetics models using continuous block implicit hybrid one-step collocation methods

Yakubu

Abdulhameed

Adamu

et al. 2023

Journal of Algorithms & Computational Technology

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The differential equations of pharmacokinetic models, obtained from the formulation based on the Fick's perfusion principle and law of conservation of mass action, deals with absorption, distribution, and elimination of drugs by the body systems. In the formulation, the ordinary differential equations obtained may result into highly stiff systems whereby suitable implicit methods have to be employed to solve accurately drug concentration in the body systems, owing to the complex nature of the exact solutions, if they exist. Further, because of the large and increasing interest in the problems of drug kinetics, it is necessary to apply considerably accurate implicit numerical methods in evaluating and in applying them to specific cases. These solutions are to help determine the distribution of drug concentration in different compartments (parts) of the human body network systems as a fundamental step to help in understanding and improving treatments. This is necessary since drug concentration in each compartment is different from one another because of the differences in drug affinity to tissues. From the study, the solution curves obtained show that drug level in the gastrointestinal tract decreases with the passage of time, while drug concentration in the blood increases from zero and reaches its maximum level and then decreases steadily again. We further, compared the model curves with some experimental data plots published in scientific papers. The results obtained from the study can be used extensively for various drug diffusion problems arising in pharmaceutical studies. The presented results widen the applicability of the continuous block implicit hybrid one-step collocation methods to diffusion process which have good number of applications in the drug control, drug dosage and other related problems in pharmaceutical and biomedical industries.

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Implicit two-derivative Runge–Kutta collocation methods for systems of initial value problems

Cited by 7 publications

References 23 publications

Improving the Accuracy of the Charge Simulation Method for Numerical Conformal Mapping

Improving the Accuracy of the Charge Simulation Method for Numerical Conformal Mapping

Unveiling the Power of Implicit Six-Point Block Scheme: Advancing numerical approximation of two-dimensional PDEs in physical systems

Accurate numerical integration of highly stiff pharmacokinetics models using continuous block implicit hybrid one-step collocation methods

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