Research Article
Numerical Solution of the Mildly Non-linear Boundary Value Problems (MNBVP) Using Newton-Lieberstein Algorithm
Issue:
Volume 11, Issue 3, September 2026
Pages:
53-61
Received:
7 August 2026
Accepted:
17 August 2026
Published:
27 August 2026
Abstract: Several numerical and classical approaches have been employed to determine the Boundary Value Problems solutions, but not many of these methods have been adopted to work out Mildly Non-Linear Boundary Value Problems (MNBVP); hence, there is a necessity to think of ways of achieving this feat. The MNBVP are problems that are neither linear nor explicitly nonlinear. This paper discusses the process of intertwining the Newton-Lieberstein method into the upwind differencing technique in solving mildly non-linear boundary value problems. The method requires two independent stages that involve different techniques. First, transform the BVP to a tri-diagonal system. Secondly, solve the system of equations by applying the Newton-Lieberstein Algorithm. The resulting equations from the nonlinear boundary problems considered in this paper will generate a nonlinear but will relate structurally to tridiagonal equations. Here, the authors developed a method described as a hybridized iterative algorithm, for which the number of steps cannot be determined a priori. It is a variant of the Newton’s technique. Unlike other classical methods, it applies to nonlinear systems and to systems with more than a thousand equations, after which round off error tends to accumulate. In real life, mathematical problems can now be expressed in the BVPs that sprang largely in the fields of physics, science, and engineering, such as fluid dynamics, electric circuits, the motion of rockets or satellites, and other areas of application. The algorithm proposed will estimate the approximate solutions irrespective of the number of equations involved. Numerical results showed the robustness, efficiency, and flexibility of the intertwined technique.
Abstract: Several numerical and classical approaches have been employed to determine the Boundary Value Problems solutions, but not many of these methods have been adopted to work out Mildly Non-Linear Boundary Value Problems (MNBVP); hence, there is a necessity to think of ways of achieving this feat. The MNBVP are problems that are neither linear nor expli...
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Research Article
Climate-Driven Differential Dynamics of Vector-Borne Diseases in the Niger-Delta: Mathematical Modelling and Environmental Health Policy Implications
Suit Patrick Oghenerhoro
,
Henry Etaroghene Egbogho*
Issue:
Volume 11, Issue 3, September 2026
Pages:
62-77
Received:
13 June 2026
Accepted:
13 June 2026
Published:
23 September 2026
DOI:
10.11648/j.ijssam.20261103.12
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Views:
Abstract: Climate change has become one of the biggest environmental problems shaping public health worldwide, mainly when viewing its impact on vector-borne diseases. In the Niger-Delta area of Nigeria, there are ecological conditions that promote the increase of disease vectors, like mosquitoes, sandflies, and ticks, and this is a significant concern. Higher temperatures, shifting rainfall trends, repeated flooding episodes, and humidity swings have altered vector ecology and increased the danger of disease spread. In this work, a climate-sensitive host vector mathematical model is developed to explore how vector-borne diseases change over time under evolving environmental conditions in the Niger-Delta region. The model is developed based on a system of nonlinear differential equations that tracks the interactions between people in susceptible, exposed, infectious, and recovered groups, along with climate- dependent vector populations. In this setting, several core mathematical properties are confirmed, like positivity and boundedness, plus the existence and uniqueness of solutions. Then the disease-free equilibrium, as well as the endemic equilibrium, are derived and analyzed. With the next-generation matrix method, the basic reproduction number is computed, and it is shown to act as the key quantity that determines whether the disease continues inside the population. Finally, the local stability study indicates that the disease-free equilibrium stays locally asymptotically stable as long as the reproduction threshold is held strictly below one. The numerical investigations across different climate settings, show that rising temperature, higher rainfall intensity, and greater humidity tend to raise vector abundance and disease prevalence in a noticeable way. The sensitivity analysis, on the other hand, points out vector recruitment rate and climate- modified transmission coefficients as the leading drivers behind how quickly disease spreads. The results also show that using integrated vector management along with environmental sanitation, plus climate adaptive interventions, can substantially lower infection prevalence and the risk that outbreaks will happen. Overall, these findings give measurable evidence for reinforcing environmental health policies, and also for building climate adaptation strategies in the Niger-Delta region. The basic structure that was developed in this work serves as a practical decision support instrument, prepared for policymakers, public health agencies, and environmental regulators who are professionally engaged in prevention efforts and climate resilience planning.
Abstract: Climate change has become one of the biggest environmental problems shaping public health worldwide, mainly when viewing its impact on vector-borne diseases. In the Niger-Delta area of Nigeria, there are ecological conditions that promote the increase of disease vectors, like mosquitoes, sandflies, and ticks, and this is a significant concern. High...
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