Mathematics and Computing for Scientific Innovation Vol. 1

Fixed Point Theorems for Contractive Type Mappings in B2 Metric Space

2026-08-27T09:06:07+00:00

Fixed-point theory plays a fundamental role in nonlinear analysis and has been extensively extended from classical metric spaces to various generalised distance spaces. Among these, b₂-metric spaces provide a unified framework that combines the characteristics of b-metric and 2-metric spaces. However, the behaviour of contractive-type mappings in complete b₂-metric spaces, particularly without assuming continuity of the b₂-metric function, has received limited attention. This study examines fixed-point results for contractive-type self-mappings in complete b₂-metric spaces, a framework that extends b-metric and 2-metric spaces through a rectangular inequality involving three variables and a coefficient. After presenting the relevant definitions of 2-metric spaces, b₂-metric spaces, convergence, Cauchy sequences, and completeness, two existence and uniqueness results are developed. The first result considers a pair of self-mappings satisfying a generalized contractive condition. Beginning with an arbitrary point, an iterative sequence is constructed, and successive estimates are used to establish that the sequence is Cauchy. Completeness then guarantees convergence to a limit in the space. The contractive condition is subsequently applied to show that this limit is a common fixed point of the two mappings, while a contradiction argument establishes uniqueness. The second result concerns a single self-mapping governed by a contractive condition involving a monotone, upper semicontinuous comparison function. A corresponding iterative procedure proves convergence to a fixed point, and the same contractive framework yields uniqueness. Corollaries are stated for continuous mappings under the respective assumptions. The analysis demonstrates that standard iterative fixed-point techniques can be formulated in the b₂-metric setting without requiring continuity of the b₂-metric function itself. The validity of the conclusions depends on completeness and on the specified restrictions imposed on the contraction parameters and comparison function.

Stability and Consistency of a Finite Difference Scheme for CO₂ Reaction–Diffusion–Advection Transport with Source and Sink Terms

2026-08-27T09:15:47+00:00

The transport of carbon dioxide (CO₂) through reactive porous media involves the simultaneous interplay of molecular diffusion, bulk advection, and concentration-dependent chemical reactions. Modelling this system accurately requires numerical schemes that are not only computationally efficient but are also rigorously verified to be stable and consistent. In this chapter, the numerical properties of an explicit DuFort–Frankel finite difference scheme applied to a dimensionless reaction–diffusion–advection (RDA) equation that incorporates a linear reactive source term, a linear sink term, and a uniform external source are analysed. Starting from the standard three-point central-difference stencil in space and the DuFort–Frankel time-averaged update in time, we derive the matrix representation of the discrete system. Stability is examined by two complementary routes. Analytically, the amplification factor satisfies a quadratic characteristic equation whose roots are shown to obey a modulus bound of unity for every admissible Fourier mode and every mesh ratio, so that the diffusive march is unconditionally stable; the centred advection term preserves this bound provided that the cell Péclet number does not exceed two. Numerically, the eigenvalues of the amplification matrix were computed in MATLAB for a physically representative parameter set drawn from the In Salah CO₂ storage project, and all eight eigenvalues were found to lie strictly inside the unit circle. Consistency is verified through a formal Taylor series expansion of the discrete stencil about the exact solution. The local truncation error is second-order in both space and time, but the DuFort–Frankel time averaging introduces an additional term involving the ratio of the time step to the grid spacing. The scheme is therefore conditionally consistent, recovering the parabolic limit only when the time step is refined faster than the grid spacing. Subject to that refinement path, the Lax equivalence theorem yields second-order convergence. These findings establish the discretisation as a mathematically sound tool for simulating CO₂ transport dynamics and explicitly state the conditions under which it may be relied upon.

Continuous Double-Diffusive Convection in a Binary Alloy Using the Lapwood-Brinkman Model with Cross-Diffusion Effects

2026-08-27T09:19:55+00:00

This study investigates the onset of continuous double-diffusive convection in a binary liquid contained in a two-dimensional rectangular channel filled with an anisotropic porous material. The channel is heated and salted from below, and the coupled effects of thermal and solutal transport are represented through the Soret and Dufour cross-diffusion terms. The momentum equation is formulated using the Lapwood-Brinkman approach with the Darcy model and without a time-derivative term, while the Boussinesq approximation is applied to describe density variations. Linear stability analysis is used to determine the critical conditions for the onset of convection. The governing equations are non-dimensionalised, expanded using Fourier series, and reduced to a sixth-order differential equation. The resulting characteristic relation is solved numerically using the Newton-Raphson method to obtain the critical Rayleigh number for selected values of the solute Rayleigh number, Darcy number, mechanical anisotropy parameter, diffusivity ratio, Soret parameter, and Dufour parameter. The graphical results indicate that increasing the Dufour parameter raises the critical Rayleigh number and therefore stabilises the system. The Soret parameter generally lowers the critical Rayleigh number, although small values may produce a stabilising influence. The diffusivity ratio stabilises the system at small solute Rayleigh numbers but becomes destabilising at larger values. These findings clarify how anisotropy and cross-diffusion jointly influence the linear stability threshold in the considered porous channel.

Medium Domination Number Algorithms for Graphs

2026-08-27T09:26:11+00:00

A network can be modelled as a graph, where nodes represent stations (or network antennas) and links represent the communication links between them. In graph theory, several structural and stability parameters, including vertex cover, connectivity, tenacity, colouring, integrity, and domination, have been extensively studied. These parameters primarily examine the relationships among vertices and edges through their neighbourhood structures. In the classical domination concept, every vertex in a graph is either a dominating vertex or is adjacent to at least one dominating vertex, thereby ensuring that every vertex is protected. In this article, we consider simple, connected, and undirected graphs and investigate the concept of the medium domination number, which extends the traditional notion of domination by considering the neighbourhoods of every pair of nodes rather than individual nodes. The fundamental idea of medium domination is that every pair of nodes (representing stations or network antennas) should be simultaneously protected. As a result, the number of nodes that can dominate both nodes in each pair must be ascertained. The graph’s medium domination number is defined as the total number of nodes that dominate each pair of nodes plus its average over all vertex pairs. In this work, we establish several new theoretical results concerning the medium domination number and verify existing results using two efficient computational approaches, namely the Adjacency Algorithm and the Neighbourhood Algorithm, proposed in this article. These algorithms provide a systematic framework for computing the medium domination number and facilitate its application to network analysis and related graph-theoretic problems.

A 2-Step Implicit Method for Numerical Solution of First-Order IVPs with Third Derivative

2026-08-29T10:23:40+00:00

for the numerical solution of first-order initial value problems (IVPs). The proposed scheme is derived via interpolation and collocation techniques, using Euler polynomials as basis functions and uniquely incorporating derivatives up to the third order to enhance accuracy. Theoretical analysis confirms that the method is zero-stable, consistent, and convergent, achieving a high algebraic order of seven (p = 7). The method is implemented as a selfstarting simultaneous integrator, thereby eliminating the need for predictors or separate procedures to generate starting values. Numerical experiments conducted on a variety of problems, including linear, nonlinear, and stiff differential equations, such as the Prothero–Robinson and Blasius equations, demonstrate that the proposed method yields superior accuracy and smaller absolute errors when compared with existing methods in the literature.

Structural Analysis of Sequential Total Network Interactions: Connectivity and Proximity Perspectives

2026-09-12T12:12:20+00:00

This manuscript examines a network structure obtained through sequential total interactions among an independent sequence of finite networks. The study is framed within simple and undirected network theory, where nodes, links, adjacency, connectedness, independent sets, induced subnetworks, connectivity, and proximity are defined in terms of the underlying node-set and link-set. The construction joins each network in the sequence to its immediate successor by adding links between all nodes of two consecutive networks while retaining the internal links of each component network. From this definition, the manuscript describes the order and size of the resulting network in terms of the orders and sizes of the component networks. The main results determine the connectivity of each node according to its position within the sequence. Nodes in the first and last networks gain additional connections only from the adjacent network, whereas nodes in intermediate networks gain additional connections from both neighbouring networks. The manuscript also establishes proximity results for pairs of nodes. Nodes belonging to different component networks have proximity determined by the distance between their network indices, while non-adjacent nodes within the same component network have proximity two in the resulting structure. These results provide a concise structural description of connectivity and pairwise proximity in sequential total network interactions.

On the Edge Domatic Decomposition of Ladder Graphs

2026-09-23T09:47:53+00:00

Edge domination extends classical domination concepts to the edge set of a graph. In this study, an edge domatic decomposition is considered as a decomposition G1,G2, . . . ,Gn in which each connected subgraph Gi has edge domatic number i, and the maximum number of such subgraphs is termed the edge domatic decomposition number. The work examines three ladder-type graph families: the ladder graph Lp, the circular ladder graph CLp, and the triangular ladder graph T Lp. The analysis proceeds by constructing pairwise edge-disjoint edge dominating sets to determine the edge domatic numbers required for subsequent decompositions. For Lp, the edge domatic number is 3 for p ≥ 3, and the edge domatic decomposition number is also 3. For CLp, the edge domatic number is 3 when p = 3k + 2 and 4 otherwise, while the edge domatic decomposition number is 3 for p ≥ 3. The auxiliary graph Cp ⊙ K1 has edge domatic number 3 for p ≥ 3. For T Lp, the edge domatic number equals 3 when p = 2 and 4 when p > 2. Its edge domatic decomposition number is 2 for p = 2, 3 for p = 3, and 4 for p > 3. These results provide explicit edge domatic decompositions for the ladder graph families considered.

Optimizing Distributed Databases for Enhanced Service Delivery in Mobile and Cloud Applications

2026-09-28T11:24:29+00:00

A distributed database is a collection of multiple, logically interrelated databases distributed over a computer network. A Distributed Database Management System (DDBMS) plays a significant role in big data analysis using a cloud software stack in SaaS cloud computing models. Many issues associated with managing a centralised database include data isolation, redundancy, inconsistency, and atomicity of updates, among others; however, distributed database implementation over high-performance compute nodes maximises information value across networks. In addition, the analysis of big data generated or consumed over the mobile Internet, the Internet of Things (IoT), and cloud computing necessitates low-latency reads and faster updates across clusters. Conventionally, services in distributed systems demand optimised transactions. This paper examines transaction generation over a distributed storage pool using the suggested reference architectures of fragmentation with hybrid semi-join operations. To offer mobility transparency as an additional element of the integrity transparency offered by a DDBMS, the distributed storage pool is simulated using a configured WLAN with multiple concurrent file transfers involving mobile nodes and large file sizes. The major functionality of the storage pool is enhanced by storage virtualisation, whereby a global schema query optimiser manages transactions to characterise latency-driven throughput. Through the joint optimisation of network and storage virtualisation, an integrated virtualised storage system leverages embedded replication technology to deliver robustness and fault-tolerance characteristics. The Heterogeneous Local Schema (LS) Multi-Database design framework enabled measurements of low-latency reads and updates. An appreciable improvement in service delivery, attributed to the distributed storage pool (DSP), was facilitated by hybridised RAID construction and copy mechanisms. Low-latency read and update transactions, improved response time, and accelerated transmissions indicated improved services. Evaluation of the simulated DDBMS model and DSP architecture showed that all complexity (overheads) associated with conventional shared systems was eliminated. Finally, the provisioned mobile Transmission Control Protocol (M-TCP) gave the best overall performance for concurrent transactions across heterogeneous multi-database sites in the integrated network.

Optimal Control Modelling of Malaria Transmission among Young Children in Bungoma County, Kenya

2026-10-09T11:02:05+00:00

Malaria remains a major source of morbidity among children under five years of age in western Kenya, and the effectiveness of control measures depends on biological efficacy, adherence, vector behaviour, insecticide susceptibility, and local transmission conditions. This study develops a deterministic human–mosquito SIR–SI model for young children in a Bungoma County context and formulates five control measures representing effective vaccine protection, long-lasting insecticidal-net protection, treatment effort, larval-source management, and adult-vector removal. The model incorporates maturation out of the under-five population and assigns the vector-directed controls to biologically appropriate demographic processes. Positivity, boundedness, the effective reproduction number, disease-free stability, and local control sensitivity are analysed. A weighted Lyapunov function establishes global asymptotic stability of the disease-free equilibrium in the feasible region when the effective reproduction number is below one. A 100-day forward–backward sweep is used for structural scenario analysis rather than empirical calibration. The no-control scenario gives a basic reproduction number of 1.551, whereas the reference control profile yields an effective reproduction number of 0.292. Under the time-varying control solution, cumulative modelled incident infections decrease by 73.2% relative to no control, with lower integrated control effort than sustained high-intensity fixed-control scenarios. The results are model-based and should not be interpreted as current estimates of incidence, programme coverage, or economic cost-effectiveness in Bungoma County.