Constructing new control points for Bézier interpolating polynomials using new geometrical approach

Interpolation is a mathematical technique employed for estimating the value of missing data between data points. This technique assures that the resulting polynomial passes through all data points. One of the most useful interpolating polynomials is the parametric interpolating polynomial. Bézier interpolating curves and surfaces are parametric interpolating polynomials for two-dimensional (2D) and three-dimensional (3D) datasets, respectively, that produce smooth, flexible, and accurate functions. According to the previous studies, the most crucial component in deriving Bézier interpolating polynomials is the construction of control points. However, most of the existing strategies constructed control points that produce partial smooth functions. As a result, the approximate values of the missing data are not accurate. In this study, nine new strategies of geometrical approach for constructing new 2D and 3D Bézier control points are proposed. The obtained control points from each new strategies are substituted in the relevant Bézier curve and surface equations to derive Bézier piecewise and non-piecewise interpolating polynomials which leads to the development of nine new methods. The proposed methods are proven to preserve the stability and smoothness of the generated Bézier interpolating curves and surfaces. In addition, the numerical results show that most of the resulting polynomials are able to approximate the missing values more accurately compared to those derived by the existing methods. The Bézier interpolating surfaces derived by the proposed method with highest accuracy for 3D datasets are then applied to upscale grey and colour images by the factors of two and three. Not only does the proposed method produces higher quality upscaled images, the numerical results also show that it outperforms the existing methods in terms of accuracy. Therefore, this study has successfully proposed new strategies for constructing new 2D and 3D control points for deriving Bézier interpolating polynomials that are capable of approximating the missing values accurately. In terms of application, the derived Bézier interpolating surfaces have a great potential to be employed in image upscaling

Geometric Piecewise Cubic Bézier Interpolating Polynomial with C2 Continuity

Bézier curve is a parametric polynomial that is applied to produce good piecewise interpolation methods with more advantage over the other piecewise polynomials. It is, therefore, crucial to construct Bézier curves that are smooth and able to increase the accuracy of the solutions. Most of the known strategies for determining internal control points for piecewise Bezier curves achieve only partial smoothness, satisfying the first order of continuity. Some solutions allow you to construct interpolation polynomials with smoothness in width along the approximating curve. However, they are still unable to handle the locations of the inner control points. The partial smoothness and non-controlling locations of inner control points may affect the accuracy of the approximate curve of the dataset. In order to improve the smoothness and accuracy of the previous strategies, а new piecewise cubic Bézier polynomial with second-order of continuity C2 is proposed in this study to estimate missing values. The proposed method employs geometric construction to find the inner control points for each adjacent subinterval of the given dataset. Not only the proposed method preserves stability and smoothness, the error analysis of numerical results also indicates that the resultant interpolating polynomial is more accurate than the ones produced by the existing methods

Dual numerical solutions of Casson SA–hybrid nanofluid toward a stagnation point flow over stretching/shrinking cylinder

A computational study of Casson sodium alginate–hybrid nanofluid of stagnation point flow through a shrinking/stretching cylinder with radius effect was carried out. Since the hybrid nanofluid is considered more contemporary type of nanofluid, it is currently being employed to enhance the efficiency of heat transmission rates. The aim of this study is to scrutinize the effect of particular parameters, such as the shrinking parameter, the Reynold number, the Casson fluid parameter, the solid copper volume fraction, and the Prandtl number, on the temperature and velocity profiles. Furthermore, the research looked into the variation of skin friction coefficient as well as the Nusselt number according to the Casson fluid parameters, and the copper solid volume fraction against shrinking parameter was investigated as part of this study. By including the appropriate similarity variables in the alteration, the nonlinear partial differential equation has been transformed into a set of ordinary differential equations (ODEs). In the end, the MATLAB bvp4c solver program is used to rectify ODEs. The findings revealed the existence of two solutions for shrinking surface with varying copper volume fractions and Casson fluid parameter values. Furthermore, the temperature profile rate was reduced in both solutions as the strength of the Reynold number, Casson fluid parameter, and copper volume fraction increased. Finally, non-unique solutions were obtained in the range of λ≥λci\lambda \ge {\lambda }_{{\rm{ci}}}

Multiple solutions of Hiemenz flow of CNTs hybrid base C2H6O2+H2O nanofluid and heat transfer over stretching/shrinking surface: Stability analysis

The purpose of the current article is to numerically and theoretically examine the flow of two-dimensional (2D) steady Hiemenz with the transfer of heat of carbon nanotubes (CNTs) hybrid base C2H6O2+H2O (Ethylene glycol + water) nanofluid across a linear shrinking/stretching surface. The equations of Navier–Stokes have been converted into equations of self-similar applying suitable transformations of similarity variables, and then numerically resolved using the three-stage Labatto-three-A formula. In addition, an endeavor is made to extend the behavior of asymptotic of the solution to massive stretching. The comparison between the found asymptotic solutions and previously reported numerical results is rather impressive. Observations indicate that equations of self-similar display double solutions within the restricted shrinking parameter range. There exists one solution for every case of stretching. In the first solution, the impacts of nanoparticle solid volume fraction and shrinking parameters on velocity and thermal fields exhibit an increasing trend. Consequently, the linear analysis of temporal stability has been performed to establish the most fundamentally viable option. The smallest eigenvalue sign determines whether a solution is unstable or stable for the purposes of stability analysis. The analysis of stability demonstrates that the first solution describing the primary flow is stable