Research Article
Three Stage MERK Methods for Delay Differential Equations
Issue:
Volume 12, Issue 2, June 2026
Pages:
65-69
Received:
14 March 2026
Accepted:
26 March 2026
Published:
26 May 2026
Abstract: The application of Runge- Kutta (R-K) methods in obtaining numerical solutions to a wide class of ordinary and partial differential equations arising in various fields of applied sciences has been widely documented. Their efficiency, accuracy, and ease of implementation have made them one of the most popular techniques for solving initial value problems. This success has motivated their extension to more complex systems, particularly Delay Differential Equations (DDEs), which naturally arise in modeling real-life phenomena where time delays are inherent, such as in population dynamics, control systems, epidemiology, and engineering processes. In this paper, we present a numerical approach for solving DDEs by adapting a three-stage Multiderivative Explicit Runge-Kutta (MERK) method. The presence of delayed arguments in DDEs introduces additional computational challenges, especially in the evaluation of past states. To address this, Lagrange interpolation is employed to approximate the delayed terms. Furthermore, the stability properties of the proposed method are investigated through the derivation of the associated stability polynomials. These polynomials provide insight into the convergence behavior and robustness of the methods when applied to stiff and non-stiff delay systems. The performance of these methods are analyzed by solving DDEs, and comparisons are made with existing methods in the literature. The results demonstrate reliability of the three-stage MERK methods for solving DDEs.
Abstract: The application of Runge- Kutta (R-K) methods in obtaining numerical solutions to a wide class of ordinary and partial differential equations arising in various fields of applied sciences has been widely documented. Their efficiency, accuracy, and ease of implementation have made them one of the most popular techniques for solving initial value pro...
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Research Article
A Convective Heat and Mass Transfer Model: A Differential Geometry Approach
Kande Dickson Kinyua*
,
Karimi Kennedy John Mwangi
Issue:
Volume 12, Issue 2, June 2026
Pages:
70-78
Received:
18 March 2026
Accepted:
14 April 2026
Published:
7 July 2026
DOI:
10.11648/j.ijamtp.20261202.12
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Views:
Abstract: This paper uses the mathematical formalism of differential geometry to present a geometry-consistent framework for modeling convective heat and mass transfer. Conventional methods are usually limited to Euclidean spaces and frequently ignore how curvature and torsion affect scalar transport. On the other hand, we develop the advection-diffusion equations for a general Riemannian manifold (M, g), utilizing curvature-aware operators to capture the impact of geometric complexity on thermal and concentration fields, specifically the Laplace--Beltrami operator and invariant advection terms. For canonical curved geometries, such as annular, helical, and torsional domains, analytical solutions are obtained. These disclose modified Nusselt and Sherwood number scalings and other curvature-induced corrections to classical transport laws. Specifically, we demonstrate that the Nusselt number scales as Nu ∝ Ra1/4 for natural convection in curved annuli, consistent with prior asymptotic analyses. We recover Dean-number-dependent transport behavior in torsional duct flows, which is consistent with empirical findings from nanofluidic heat exchangers. We use mesh-based and meshless discretizations designed for curved surfaces for numerical validation, such as Lattice Boltzmann methods for porous or irregular domains, weighted finite volume schemes, and DEC. Even in highly non-Euclidean environments, these methods maintain geometric fidelity and exhibit second-order convergence. We're bringing performance analysis under one roof by introducing curvature-adjusted, dimensionless groups-- mainly, geometric Peclet, Nusselt, and Sherwood numbers. These capture how the shape of a system affects the way things like heat and mass move through it. What really stands out is how field synergy -- basically, the angle between the way stuff moves and the direction gradients point -- changes with curvature. This opens up fresh ways to boost transport in systems that aren't flat. With this approach, we lay out a unified platform, both in theory and computation, to look at heat and mass transfer in all kinds of complex shapes. Think microfluidic devices, porous media reactors, cutting-edge heat exchangers, or even thermal systems inspired by nature. Next up, we're tackling multi-phase and reactive transport, plus real-world tests in truly three-dimensional curved spaces.
Abstract: This paper uses the mathematical formalism of differential geometry to present a geometry-consistent framework for modeling convective heat and mass transfer. Conventional methods are usually limited to Euclidean spaces and frequently ignore how curvature and torsion affect scalar transport. On the other hand, we develop the advection-diffusion equ...
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