The nonlinear stability analysis of a ferrofluid layer system is formulated mathematically. This system considered the upper and lower free isothermal boundary with the system heated from below. A mathematical formulation is produced to study the behaviour of the chaotic convection in a ferrofluid layer system using Galerkin truncated expansion. The Boussinesq approximation is opted with the existence of internal heating and the magnetic number. It is found that the transition to chaos in this present study is identical to the Lorenz attractor and thus validate the method and analysis of this study. The impact of elevating the internal heat generation is found to hasten the instability of the system and as for the magnetic number, at M1 = 2.5 the homoclinic bifurcation occurs and thus accelerates the convection process.

Real life phenomena found in various fields such as engineering, physics, biology and communication theory can be modeled as nonlinear higher order ordinary differential equations, particularly the Duffing oscillator. Analytical solutions for these differential equations can be time consuming whereas, conventional numerical solutions may lack accuracy. This research propose a block multistep method integrated with a variable order step size (VOS) algorithm for solving these Duffing oscillators directly. The proposed VOS Block method provides an alternative numerical solution by reducing computational cost (time) but without loss of accuracy. Numerical simulations are compared with known exact solutions for proof of accuracy and against current numerical methods for proof of efficiency (steps taken).

Convergence problems has been the focus of interest for researchers that are working in the fields of spectral theory. In the current research we investigate issues relating to the summability of the Fourier-Laplace series on the unit sphere. The necessary conditions which are required to obtain good estimation for summability of the Fourier-Laplace series investigated. This research will also provide new and sufficient conditions in the form of theorems and lemmas which will validate the uniform summability of the Fourier-Laplace series on the sphere.

A linear stability evaluation is conducted to explore the effect on the onset of Marangoni-Bénard convection in a ferrofluid layer system. The system is heated from below with treatment of both the lower and upper boundaries to completely insulate the temperature disturbance. The eigenvalue problem is solved by using regular perturbation technique to obtain the critical number of Marangoni and also the critical number of Rayleigh. It is observed that the increase in the value Crispation, the magnetic number of Rayleigh and also the magnetic number will destabilize the system while the increasing number of Bonds will delay the convection.