Zurni Omar

Ordinary differential equations (ODEs) that model real life scenarios as initial value problems (IVPs) mostly take the form of higher order IVPs. However, it is Ordinary differential equations (ODEs) that model real life scenarios as initial value problems (IVPs) mostly take the form of higher order IVPs. However, it is

Interpolation is a mathematical technique employed for estimating unknown values between data points. This technique assures that the resulting polynomial passes through all data points. One of the most useful interpolation polynomials is the parametric interpolation polynomial. Bézier interpolating surfaces are parametric interpolation polynomials that produce smooth, flexible, and accurate functions. One of the most crucial parts in deriving Bézier interpolating surfaces is to construct control points. However, the control points constructed by most of the existing strategies produce partial smoothness functions with less accurate solutions. In this paper, a new strategy for constructing control points is proposed. This strategy interpolates diagonal data points to construct control points of three-dimensional (3D) datasets. The obtained control points are then employed to derive Bézier interpolating surfaces that preserve stability and smoothness. The newly derived Bézier interpolating surfaces are then applied in colour image upscaling. Not only do the derived Bézier interpolating surfaces produce sharper upscale images, the numerical results show that they also produce more accurate solutions than other existing methods.

Differential equations are vastly adopted for modelling real life scenarios with certain parameters representing selected variables in the model. In cases where real data is utilized, the parameters of the differential equation need to be estimated based on the data, but majority of the techniques adopted in parameter estimation requires that the exact form of the function is known. This is unfortunately not the case when considering differential equations because in many instances the differential equation cannot be solved analytically, hence numerical methods are introduced to obtain an approximate solution. This article describes the development of a multistep central difference method using Taylor series expansion of a generalised multistep central difference scheme to estimate the parameters in the predator-prey model via derivative approximation in application to real data. The results obtained agree with existing study and the numerical solution of the model based on these parameters is obtained using block method. Hence, this article has presented an approach for solving the predator-prey model when considering real data.

This study introduces new operational matrices to approximate the solution of first and second order delay differential equations (DDEs) using Said-Ball polynomials (SBP). To obtain the required approximate solution, the DDEs are transformed to a simplified set of algebraic equations with the adoption of the residual correction procedure (RCP) for error estimation of the SBP operational matrices. Some examples are given to demonstrate the employability of the new method and a comparison is made with the existing results. The new approach of adopting Said-Ball operational matrix displayed impressive results when compared in terms of absolute error with previous studies.

The processes of diffusion and reaction play essential roles in numerous system dynamics. Consequently, the solutions of reaction–diffusion equations have gained much attention because of not only their occurrence in many fields of science but also the existence of important properties and information in the solutions. However, despite the wide range of numerical methods explored for approximating solutions, the adoption of block methods is yet to be investigated. Hence, this article introduces a new two-step third–fourth-derivative block method as a numerical approach to solve the reaction–diffusion equation. In order to ensure improved accuracy, the method introduces the concept of nonlinearity in the solution of the linear model through the presence of higher derivatives. The method obtained accurate solutions for the model at varying values of the dimensionless diffusion parameter and saturation parameter. Furthermore, the solutions are also in good agreement with previous solutions by existing authors.

In the current study, the characteristics of heat transfer of a steady, two-dimensional, stagnation point, and magnetohydrodynamic (MHD) flow of shear thickening Casson fluid on an exponentially vertical shrinking/stretching surface are examined in attendance of convective boundary conditions. The impact of the suction parameter is also considered. The system of governing partial differential equations (PDEs) and boundary conditions is converted into ordinary differential equations (ODEs) with the suitable exponential similarity variables of transformations and then solved using the shooting method with the fourth order Runge–Kutta method. Similarity transformation is an important class of phenomena in which scale symmetry allows one to reduce the number of independent variables of the problem. It should be noted that solutions of the ODEs show the symmetrical behavior of the PDES for the profiles of velocity and temperature. Similarity solutions are obtained for the case of stretching/shrinking and suction parameters. It is revealed that there exist two ranges of the solutions in the specific ranges of the physical parameters, three solutions depend on the opposing flow case where stagnation point (A) should be equal to 0.1, two solutions exist when λ₁ = 0 where λ₁ is a mixed convection parameter and A > 0.1, and a single solution exists when λ₁ > 0. Moreover, the effects of numerous applied parameters on velocity, temperature distributions, skin friction, and local Nusselt number are examined and given through tables and graphs for both shrinking and stretching surfaces.