Research Article
An Efficient A-Stable Linear Multistep Hybrid Block Method for Solving Fifth-Order Initial Value Problems in Ordinary Differential Equations
Duromola Monday Kolawole*
Issue:
Volume 13, Issue 3, June 2025
Pages:
174-193
Received:
9 April 2025
Accepted:
23 April 2025
Published:
29 May 2025
Abstract: Furthering research on linear multistep hybrid block methods is essential to enhance accuracy, stability, and efficiency in solving ordinary differential equations, enabling advanced modelling of complex systems across science and engineering for better predictive analysis and real-world applications. In this study, an efficient linear multistep hybrid block method with a single-step and seven off-step points for the direct numerical integration of fifth-order initial value problems (IVPs) in ordinary differential equations (ODEs), eliminating the need for reduction to a system of first-order ODEs is proposed. The method is constructed using a collocation approach at both grid and off-grid points, alongside interpolation at five off-grid points, to approximate the solution via a power series polynomial. The resulting system of equations is solved to obtain the necessary discrete and additional formulae that constitute the block approach. A comprehensive theoretical analysis confirms that the method possesses desirable numerical properties, including a well-defined order, zero stability, consistency, convergence, and absolute stability. Comparative numerical experiments against existing methods demonstrate that the proposed approach achieves superior accuracy and efficiency, making it a promising tool for solving both linear and nonlinear fifth-order ODEs.
Abstract: Furthering research on linear multistep hybrid block methods is essential to enhance accuracy, stability, and efficiency in solving ordinary differential equations, enabling advanced modelling of complex systems across science and engineering for better predictive analysis and real-world applications. In this study, an efficient linear multistep hy...
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Research Article
Mathematical Modelling and Analytical Expressions for Steady-state Concentrations of Non-linear Glucose-responsive Composite Membranes for Closed-loop Insulin Delivery: Akbari-Ganji and Differential Transform Methods
Issue:
Volume 13, Issue 3, June 2025
Pages:
194-204
Received:
15 May 2025
Accepted:
29 May 2025
Published:
11 June 2025
DOI:
10.11648/j.ajam.20251303.12
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Abstract: The dynamic mechanism comprising an enzymatic reaction and the diffusion of reactants and products inside a glucose-sensitive composite membrane is described using a mathematical model created by Abdekhodaie and Wu. A set of non-linear steady-state reaction-diffusion equations is presented in this theoretical model. These equations have been meticulously and accurately solved analytically, considering the concentrations of glucose, oxygen, and gluconic acid, using a novel approach of Akbari Ganji and differential transform methods. The high level of agreement between these analytical results and the numerical results for steady-state conditions is a testament to the model's precision. A numerical simulation was produced via the precise and widely used MATLAB software. A comprehensive graphic representation of the model's various kinetic parameters' effects has also been provided. Additionally, a theoretical analysis of the kinetic parameters, such as the maximal reaction velocity (Vmax) and the Michaelis-Menten constants (Kg and Kox) for oxygen and glucose, pH profiles with membranes is presented. This expressed model is incredibly helpful when creating glucose-responsive composite membranes for closed-loop insulin delivery.
Abstract: The dynamic mechanism comprising an enzymatic reaction and the diffusion of reactants and products inside a glucose-sensitive composite membrane is described using a mathematical model created by Abdekhodaie and Wu. A set of non-linear steady-state reaction-diffusion equations is presented in this theoretical model. These equations have been meticu...
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