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Scattered data interpolation is important in sciences, engineering, and medical-based problems. The degree smoothness attained is either C1 or C2. However, based on our simulations, we found that, Hussain and Hussain (2011) scheme is not producing C1 everywhere. Motivated by this, in this study we will propose an improve sufficient condition for the C1 on each adjacent triangle. Our scheme is guaranteed to produce C1 everywhere. To support our claim, we have implemented both schemes and perform some numerical comparison including error measurement. From the results, our proposed scheme is producing C1 interpolating surface while the interpolating surface produced by Hussain and Hussain (2011) is not C1 everywhere. Furthermore, our proposed scheme give smaller error compared with Hussain and Hussain (2011) scheme. All numerical and graphical results are presented by using MATLAB programming.

Scattered data interpolation is an essential technique used in environmental, geospatial and meteorological analysis. It enables the estimation of unknown values within a dataset at the domain by utilising known data points. The utilisation of this method is crucial to generate smooth and continuous surfaces from discrete data, facilitating the creation of more precise models and visualisations of environmental events. The Malaysian rainfall and terrain data introduced in this paper, are essential datasets for these analyses. This paper provides detailed rainfall data from various meteorological stations across Malaysia, sourced from the Department of Irrigation and Drainage Malaysia. Additionally, terrain data for Kalumpang was obtained, which was based on the Agricultural Station in Selangor. These primary datasets are essential for understanding local climatic and topographic variations. However, challenges such as the size of the dataset, time constraints, weather conditions, and difficulties in obtaining data due to sensitivity and confidentiality issues limit the scope of the data collection. Although there are certain limits, the datasets have been carefully processed and prepared to be easily accessible in Microsoft Excel, MATLAB, and Python. This allows for additional research and applications in machine learning. This paper presents raw and pre-processed datasets, enabling comprehensive analyses and advancing scattered data interpolation techniques.

Computer graphics is a dynamic field that heavily relies on mathematical techniques. For instance, subdivision scheme is used to create smooth and visually appealing curves and surfaces of arbitrary topology. The primary focus of this study is to transform two 5-point binary subdivision schemes into a single 7-point binary subdivision scheme with shape control. We have merged two binary approximating schemes that were constructed using the uniform B-spline basis function and the Lagrange basis function into a new subdivision scheme. It has been demonstrated that, for fixed values of the global shape control parameters, the curves generated by the proposed 7-point binary subdivision scheme maintain C6 continuity everywhere. Furthermore, a brief discussion on the analysis of the Gibbs phenomenon in the new subdivision scheme has been presented. This is also a reminder of the challenges and intricacies involved in computer graphics and geometric modeling.

This article develops two new algorithms of three-term conjugate gradient (TTCG) coefficients to handle nonlinear least-squares (NLS) problems using the structured secant equation. Motivated by improved conjugacy and sufficient descent conditions, we developed two new formulations of CG coefficients. Furthermore, with these parameters, two search directions are proposed that satisfy the sufficient descent condition, which further enhances the efficiency of the proposed strategies. One key advantage of the proposed techniques is their low memory requirements, rendering them suitable for large-scale nonlinear least squares problems. Moreover, some mild suppositions and a non-monotone line search are used to establish the global convergence properties of the methods. More so, we investigate the robustness and effectiveness of the proposed methods numerically by performing experiments on benchmark test problems, and their performance is compared against existing methods. The outcomes of these experiments indicate that the proposed methods outperform the other techniques regarding the metrics of comparison adopted. Finally, the algorithms are applied to an extension of the model of robotic motion control of four degrees of freedom (4DOF), resulting in positive outcomes for the robot’s motion traits.

This paper presents an approach to scattered data interpolation using q-Bézier triangular patches via an efficient algorithm. While existing studies have formed q-Bézier triangular patches through convex combination, their application to scattered data interpolation has not been previously explored. Therefore, this study aims to extend the use of q-Bézier triangular patches to scattered data interpolation by achieving C1 continuity throughout the data points. We test the proposed scheme using both established data points and real-life engineering problems. We compared the performance of the proposed interpolation scheme with a well-known existing scheme by varying the q parameter. The comparison was based on visualization and error analysis. Numerical and graphical results were generated using MATLAB. The findings indicate that the proposed scheme outperforms the existing scheme, demonstrating a higher coefficient of determination (R2), smaller root mean square error (RMSE), and faster central processing unit (CPU) time. These results highlight the potential of the proposed q-Bézier triangular patches scheme for more accurate and reliable scattered data interpolation via the proposed algorithm.

In quantile regression models, numerous penalization methods have been developed to deal with ordinary least-squares method problems. Such methods are ridge penalized quantile regression, lasso penalized quantile regression, and elastic net penalized quantile regression which are used for variable selection and regularization and deals with the multicollinearity problem when it exists between the predictor variables. However, the variables of interest are often represented through time series processes, in which such time series data are often non-stationary and non-linear, which leads to poor accuracy of the resultant regression models and hence results with less reliability. The EMD-EnetQR method is proposed to address this issue, which consists of applying the empirical mode decomposition (EMD) algorithm to time series data and then using the resulting components in penalized quantile regression models. This study aims to apply the proposed EMD-QREnet method to determine the influence of the decomposition components of the original time series predictor variables on the response variable to build a model fit and improve prediction accuracy. Furthermore, this study addressed the multicollinearity between the decomposition components. Simulation studies and real dataset applications were conducted. The results show that the proposed EMD-QREnet method, in most cases, outperforms the other methods by improving prediction accuracy.

Internet of Things (IoT) is the paramount virtual network that enables remote users to access connected multimedia devices. It has dragged the attention of the community because it encompasses real-world scenarios with implicit environs. Despite several beneficial aspects, IoT is surrounded by provocations for successful implementation, as data travels in different layers. One of the critical challenges is the security of the data in these layers. Researchers conducted numerous studies focusing on the level of security at a single technique, creating loopholes to address the entire scenario of securing an IoT network. This study aims to comprehensively review current security issues, wireless communication techniques, and technologies for securing IoT. This work’s utmost significance is addressing all the security perspectives at a glance. For this purpose, research contributions from the previous years are investigated for better understanding. Some countermeasures and snags from security perspectives have also been analyzed in detail concerning the current industry trends. Blockchain, machine learning, fog, and edge computing are possible solutions to secure IoT. After studying these techniques and their immunity to attacks, machine learning can become a hope if incorporated with end-to-end security. This comprehensive review will provide adequate understanding and knowledge in defining security lines of action for the successful implementation of IoT.

This paper presents the advanced classes of linear symmetric subdivision schemes for the fitting of data and the creation of geometric shapes. These schemes are derived from the B-spline and Lagrange’s blending functions. The important characteristics of the derived schemes, including continuity, support, and the impact of parameters on the magnitude of the artifact and Gibbs oscillations are discussed. Schemes additionally generalize various subdivision schemes. Linear symmetric subdivision schemes can produce Gibbs oscillations when the initial data is taken from discontinuous functions. Additionally, these schemes may generate unwanted artifacts in the limit curve that do not exist in the original polygon. One solution is to use non-linear schemes, but this approach increases the computational complexity of the scheme. An alternative approach is proposed that involves modifying the linear symmetric schemes by introducing parameters into the linear rules. The suitable values of these parameters reduce or eliminate Gibbs oscillations and artifacts while still using linear symmetric schemes. Our approach provides a balance between reducing or eliminating Gibbs oscillations and artifacts while maintaining computational efficiency.

The system of ordinary differential equations has many uses in contemporary mathematics and engineering. Finding the numerical solution to a system of ordinary differential equations for any arbitrary interval is very appealing to researchers. The numerical solution of a system of fourth-order ordinary differential equations on any finite interval [a,b] is found in this work using a symmetric Bernstein approximation. This technique is based on the operational matrices of Bernstein polynomials for solving the system of fourth-order ODEs. First, using Chebyshev collocation nodes, a generalised approximation of the system of ordinary differential equations is discretized into a system of linear algebraic equations that can be solved using any standard rule, such as Gaussian elimination. We obtain the numerical solution in the form of a polynomial after obtaining the unknowns. The Hyers–Ulam and Hyers–Ulam–Rassias stability analyses are provided to demonstrate that the proposed technique is stable under certain conditions. The results of numerical experiments using the proposed technique are plotted in figures to demonstrate the accuracy of the specified approach. The results show that the suggested Bernstein approximation method for any interval is quick and effective.

This paper proposes a modification to the Hestenes-Stiefel (HS) method by devising a spectral parameter using a modified secant relation to solve nonlinear least-squares problems. Notably, in the implementation, the proposed method differs from existing approaches, in that it does not require a safeguarding strategy and its Hessian matrix is positive and definite throughout the iteration process. Numerical experiments are conducted on a range of test problems, with 120 instances to demonstrate the efficacy of the proposed algorithm by comparing it with existing techniques in the literature. However, the results obtained validate the effectiveness of the proposed method in terms of the standard metrics of comparison. Additionally, the proposed approach is applied to address motion-control problems in a robotic model, resulting in favorable outcomes in terms of the robot’s motion characteristics.

This study deals with the numerical solution of a class of linear systems of second-order boundary value problems (BVPs) using a new symmetric cubic B-spline method (NCBM). This is a typical cubic B-spline collocation method powered by new approximations for second-order derivatives. The flexibility and high order precision of B-spline functions allow them to approximate the answers. These functions have a symmetrical property. The new second-order approximation plays an important role in producing more accurate results up to a fifth-order accuracy. To verify the proposed method’s accuracy, it is tested on three linear systems of ordinary differential equations with multiple step sizes. The numerical findings by the present method are quite similar to the exact solutions available in the literature. We discovered that when the step size decreased, the computational errors decreased, resulting in better precision. In addition, details of maximum errors are investigated. Moreover, simple implementation and straightforward computations are the main advantages of the offered method. This method yields improved results, even if it does not require using free parameters. Thus, it can be concluded that the offered scheme is reliable and efficient.

Subdivision schemes are equipped with some rules that take a polygon as an input and produce smooth curves or surfaces as an output. This presents the issue of how accurately the polygon approximates the limit curve and surface. What number of iterations/levels would be necessary to achieve the required shape at a user-specified error tolerance? In fact, several methods have been introduced in the case of stationary schemes to address the issue in terms of the error bounds (distance between polygon/polyhedron and limiting shape) and subdivision depth (the number of iterations required to obtain the result at a user-specified error tolerance). However, in the case of non-stationary schemes, this topic needs to be further studied to meet the requirements of new practical applications. This paper highlights a new approach based on a convolution technique to estimate error bounds and subdivision depth for non-stationary schemes. The given technique is independent of any condition on the coefficient of the non-stationary subdivision schemes, and it also produces the best results with the least amount of computational effort. In this paper, we first associated constants with the vectors generated by the given non-stationary schemes, then formulated an expression for the convolution product. This expression gives real values, which monotonically decrease with the increase in the order of the convolution in both the curve and surface cases. This convolution feature plays an important role in obtaining the user-defined error tolerance with fewer iterations. It achieves a trade-off between the number of iterations and user-specified errors. In practice, more iterations are needed to achieve a lower error rate, but we achieved this goal by using fewer iterations.

The fractional Fourier transform is a natural generalization of the Fourier transform. In this work, we recall the definition of the fractional Fourier transform and its relation to the conventional Fourier transform. We exhibit that this relation permits one to obtain easily the main properties of the fractional Fourier transform. We investigate the sharp Hausdorff-Young inequality for the fractional Fourier transform and utilize it to build Matolcsi-Szücs inequality related to this transform. The other versions of the inequalities concerning the fractional Fourier transform is also discussed in detail. The results obtained in this paper are very significant, especially in the field of fractional differential equations.

Motivated by the fact that the quaternion Fourier transform is a powerful tool in quaternion signal analysis, here, we study the quaternion quadratic-phase Fourier transform, which is a generalized version of the quaternion Fourier transform. We first give a definition of the quaternion quadratic-phase Fourier transform. We derive in detail some essential properties related to this generalized transformation. We explore how the quaternion quadratic-phase Fourier transform is related to the quaternion Fourier transform. It is shown that this relation allows us to obtain several versions of uncertainty principles concerning the quaternion quadratic-phase Fourier transform.

Abstract This paper discusses the functional scattered data interpolation to interpolate the general scattered data. Compared with the previous works, we construct a new cubic Bézier-like triangular basis function controlled by three shape parameters. This is an advantage compared with the existing schemes since it gives more flexibility for the shape design in geometric modeling. By choosing some suitable value of the parameters, this new triangular basis is reduced to the cubic Ball and cubic Bézier triangular patches, respectively. In order to apply the proposed bases to general scattered data, firstly the data is triangulated using Delaunay triangulation. Then the sufficient condition for C 1 $C^{1}$ continuity using cubic precision method on each adjacent triangle is implemented. Finally, the interpolation scheme is constructed based on a convex combination between three local schemes of the cubic Bézier-like triangular patches. The detail comparison in terms of maximum error and coefficient of determination r 2 $r^{2}$ with some existing meshfree methods i.e. radial basis function (RBF) such as linear, thin plate spline (TPS), Gaussian, and multiquadric are presented. From graphical results, the proposed scheme gives more visually pleasing interpolating surfaces compared with all RBF methods. Based on error analysis, for all four functions, the proposed scheme is better than RBFs except for data from the third function. Overall, the proposed scheme gives r 2 $r^{2}$ value between 0.99920443 and 0.99999994. This is very good for surface fitting for a large scattered data set.

In this note, we present some properties of a certain space of delta m-subharmonic functions. We prove that the convergence in this space implies the convergence in m-capacity.

A new rational bi-quartic spline interpolation scheme is constructed in this study. This rational bi-quartic spline has six free parameters that can be used to refine the final interpolating surface. The degree of smoothness attained is C1 continuity. Image interpolation is one of the methods in image processing that is used to rescale the images, either downscaling or upscaling. The proposed rational bi-quartic spline will be used to interpolate the missing pixel in image interpolation. We employ an algorithm that will speed up the process as well as be able to produce the quality upscaling images. We measure the quality of the proposed method by calculating peak-signal-to-noise-ratio (PSNR). Based on numerical comparison, the proposed scheme gives higher PSNR (consequently smaller RMSE) unlike some existing schemes. Thus, the new rational bi-quartic spline with six parameters is better than the existing scheme via an efficient algorithm. All numerical simulations and graphical results are presented using MATLAB.

This paper discusses the construction of scattered data interpolation scheme based on rational quartic triangular patches with C1 continuity. The C1 sufficient condition is derived on each adjacent triangle. We employ rational corrected scheme comprising three local schemes defined on each triangle. The final scheme is then being applied for scattered data interpolation. In this study, we tested the proposed scheme on regular and irregular scattered data. For regular data, we used 36, 65 and 100 data sets, while for irregular data, we used some real data sets. Besides that, we compared the performance between the proposed scheme with cubic Ball and cubic Bézier schemes. We measure the error analysis by calculating the Root Mean Square Error (RMSE), maximum error (Max error), coefficient of determination $(R^{2}) $ and CPU time (in seconds). Based on the comparisons, our proposed scheme performs better than the other two existing schemes. Furthermore, for large scattered data sets, the proposed scheme requires less computation time than existing schemes. The free parameters in the description of the proposed scheme provide greater flexibility in controlling the quality of the final interpolating surface. This is very useful in the designing processes.

The heat transfer ratio has some important applications in industries and the engineering sector. In this model, the authors used the hybrid nanofluid because the heat transfer ratio of hybrid nanofluid is more as compared to the base fluid; the key objective of this research work is to boost up the heat transfer ratio, for example, to regulate the energy is possible only by adding the heat transmission mechanism in the flow model. The current research paper investigates the steady 3D MHD mixed convection couple stress flow of hybrid nanofluid with hall and ion slip effect. The objective of the current research work is to increase the heat consignment ratio, which is the demand of the manufacturing and engineering sector, this type of flow has some important applications in the industries sector and engineering sector for the purpose of cooling and hotness effect, also hotness and cooling play some important role in daily life. To transform the nonlinear partial differential equation to a nonlinear ordinary differential equation we used the defined similarity transformation. The transform nonlinear ordinary differential equations are solved by an approximate analytical method. The important obtained results are presented in the graphs. The influence of different parameters such as couple stress parameter, mixed convection parameter, nanoparticle volume friction, Hall parameter, magnetic field parameter, thermophores parameter, Eckert number, and prandtl number are taken over graphs. The Cf(skin friction coefficient), Nu(nusselt number), convergence control parameter, and comparison of the present work with the published work are described in the form of tables.

The linear canonical wavelet transform is a nontrivial generalization of the classical wavelet transform in the context of the linear canonical transform. In this article, we first present a direct interaction between the linear canonical transform and Fourier transform to obtain the generalization of the uncertainty principles related to the linear canonical transform. We develop these principles for constructing some uncertainty principles concerning the linear canonical wavelet transform.

From the observed datasets, we should be able to produce curve surfaces that have the same characteristics as the original datasets. For instance, if the given data are positive, then the resulting curve or surface must be positive on entire given intervals, i.e., everywhere. In this study, a new partial blended rational bi-quartic spline with C1 continuity is constructed through the partially blended scheme. This rational spline is defined on four corners of the rectangular meshes. The sufficient condition for the positivity of rational bi-quartic spline is derived on four boundary curve networks. There are eight free parameters that can be used for shape modification. The first-order partial derivatives are estimated by using numerical techniques. We also show that the proposed scheme is local quadratic reproducing such that it can exactly reproduce the quadratic surface. We test the proposed scheme to interpolate various types of positive surface data. Based on statistical indicators such as the root mean square error (RMSE) and coefficient of determination (R2), we found that the proposed scheme is on par with some established schemes. In fact, it requires less CPU times (in seconds) to generate the interpolating surface on rectangular meshes. Furthermore, by combining the statistical indicators’ result and graphically visualizing the test functions, the proposed method has the capability to reconstruct very comparable smoothing interpolating positive surfaces compared to some existing schemes. This finding is significant in producing a better interpolating surface for computer graphics applications since the proposed scheme has a smaller error compared with existing schemes.

This paper discusses the constraint data interpolation or range restricted interpolation for surface data arranges on rectangular meshes that lie above or below an arbitrary plane and between two arbitrary planes by using partially blended rational bi-cubic spline function with 12 parameters. Common research in range restricted surface interpolation is to construct the constrained surface lie above linear plane. However, in this paper, we consider the constraint surfaces up to degree three (cubic). To construct the constrained surface with shape preserving properties, i.e., the resulting surface will lie below or above single planes or between two respective planes, the data dependent sufficient conditions are derived on four parameters; meanwhile, the remaining eight parameters are free parameters to change the shape of the interpolating surface locally. The proposed scheme is tested with various types of data test, including some well-known functions. From the numerical results, we found that the proposed scheme is easy to use, locally control via free parameters, and require less computation compared with some existing schemes as well as visually pleasant for visualization. Furthermore, based on root mean square error (RMSE) and coefficient of determination (R2), the proposed scheme is better than existing scheme with the value of R2 achieved is in between 0.9701 (97.01%) and 0.9954 (99.54%). This is quite good for range restricted surface data interpolation since we can explain at least 97.01% of the variance in the interpolation by using the proposed scheme. Furthermore, the proposed scheme requires less CPU time (in seconds) compared with the existing scheme.

The quaternion linear canonical transform is an important tool in applied mathematics and it is closely related to the quaternion Fourier transform. In this work, using a symmetric form of the two-sided quaternion Fourier transform (QFT), we first derive a variation on the Heisenberg-type uncertainty principle related to this transformation. We then consider the general two-sided quaternion linear canonical transform. It may be considered as an extension of the two-sided quaternion linear canonical transform. Based on an orthogonal plane split, we develop the convolution theorem that associated with the general two-sided quaternion linear canonical transform and then derive its correlation theorem. We finally discuss how to apply general two-sided quaternion linear canonical transform to study the generalized swept-frequency filters.

In this paper, a new numerical technique is introduced to find the solution of the system of Volterra integral equations based on symmetric Bernstein polynomials. The use of Bernstein polynomials to find the numerical solutions of differential and integral equations increased due to its fast convergence. Here, the numerical solution of the system of Volterra integral equations on any finite interval [m,n] is obtained by replacing the unknown functions with the generalized Bernstein basis functions. The proposed technique converts the given system of equations into the system of algebraic equations which can be solved by using any standard rule. Further, Hyers–Ulam stability criteria are used to check the stability of the given technique. The comparison between exact and numerical solution for the distinct nodes is demonstrated to show its fast convergence.

The present work discusses the basic features of bifurcation properties with chaotic motion of oblique plane wave in the discrete nonlinear electrical transmission lines having conformable derivative evolution. The planar dynamical systems are obtained to understand such physical issues for this model. In addition, the analytical plane wave solutions of nonlinear mathematical leading equation are determined by using two different mathematical methods. Both methods are supported only by the shock plane wave solutions. The conditions for other types of oscillatory plane wave solutions are also determined. The effect of obliqueness on the bifurcation properties, chaotic motion and shock plane waves structures are focused graphically with discussions by considering both of locality and non-locality in the system. It is found that the parameter of non-local operator is only affected on the nonlinear shock wave phenomena, whereas all basic features of discrete nonlinear electrical transmission line are changed with the changes of obliqueness.

In this study, we introduce a new modification of fractional reduced differential transform method (m-FRDTM) to find exact and approximate solutions for nonhomogeneous linear multiterm time-fractional diffusion equations (MT-TFDEs) of constant coefficients in a bounded domain with suitable initial conditions. Different applications in two and three fractional order terms are given to illustrate our new modification. The approximate solutions are given in the form of series solutions. The results show that the m-FRDTM for MT-TFDEs is a powerful method and can be generalized to other types of multiterm time-fractional equations.

The extraversion personality trait has a positive correlation with social interaction. In neuroimaging studies, investigations on extraversion in face-to-face verbal interactions are still scarce. This study presents an electroencephalography (EEG)-based investigation of the extraversion personality trait in relation to eye contact during face-to-face interactions, as this is a vital signal in social interactions. A sample of healthy male participants were selected (consisting of sixteen more extraverted and sixteen less extraverted individuals) and evaluated with the Eysenck's Personality Inventory (EPI) and Big Five Inventory (BFI) tools. EEG alpha oscillations in the occipital region were measured to investigate extraversion personality trait correlates of eye contact during a face-to-face interaction task and an eyes-open condition. The results revealed that the extraversion personality trait has a significant positive correlation with EEG alpha coherence in the occipital region, presumably due to its relationship with eye contact during the interaction task. Furthermore, the decrease in EEG alpha power during the interaction task compared to the eyes-open condition was found to be greater in the less extraverted participants; however, no significant difference was observed between the less and more extraverted participants. Overall, these findings encourage further research towards the understanding of neural mechanism correlates of the extraversion personality trait-particularly in social interaction.

In this study, we introduce a new cubic B-spline (CBS) approximation method to solve linear two-point boundary value problems (BVPs). This method is based on cubic B-spline basis functions with a new approximation for the second-order derivative. The theoretical new approximation for a second-order derivative and the error analysis have been successfully derived. We found that the second-order new approximation was O(h3) accurate. By using this new second-order approximation, the proposed method was O(h5) accurate. Four numerical problems consisting of linear ordinary differential equations and trigonometric equations with different step sizes were performed to validate the accuracy of the proposed methods. The numerical results were compared with the least squares method, finite difference method, finite element method, finite volume method, B-spline interpolation method, extended cubic B-spline interpolation method and the exact solutions. By finding the maximum errors, the results consistently showed that the proposed method gave the best approximations among the existing methods. We also found that our proposed method involved simple implementation and straightforward computations. Hence, based on the results and the efficiency of our method, we can say that our method is reliable and a promising method for solving linear two-point BVPs.

In this work, the effects of plasma parameters on overtaking collisions of ion acoustic multi-shocks are studied in an unmagnetized collisionless plasma with positive and negative ions, and (α,q)-distributed electrons. To investigate such phenomena, the reductive perturbation technique is implemented to derive the Burgers equation. The N-shock wave solution is determined for this equation by directly implementing the exponential function. The result reveals that both the amplitudes and thicknesses of overtaking collisions of N-shock wave compressive and rarefactive electrostatic potential are significantly modified with the influences of viscosity coefficients of positive and negative ions. In addition, the density ratios also play an essential role to the formation of overtaking collisions of N-shocks. It is observed from all theoretical and parametric investigations that the outcomes may be very useful in understanding the dynamical behavior of overtaking collisions of multi-shocks in various environments, especially the D- and F-regions of the Earth’s ionosphere and the future experimental investigations in Q-machine laboratory plasmas.

Scattered data interpolation is important in sciences, engineering, and medical-based problems. Quartic Bézier triangular patches with 15 control points (ordinates) can also be used for scattered data interpolation. However, this method has a weakness; that is, in order to achieve C 1 continuity, the three inner points can only be determined using an optimization method. Thus, we cannot obtain the exact Bézier ordinates, and the quartic scheme is global and not local. Therefore, the quartic Bézier triangular has received less attention. In this work, we use Zhu and Han’s quartic spline with ten control points (ordinates). Since there are only ten control points (as for cubic Bézier triangular cases), all control points can be determined exactly, and the optimization problem can be avoided. This will improve the presentation of the surface, and the process to construct the scattered surface is local. We also apply the proposed scheme for the purpose of positivity-preserving scattered data interpolation. The sufficient conditions for the positivity of the quartic triangular patches are derived on seven ordinates. We obtain nonlinear equations that can be solved using the regula-falsi method. To produce the interpolated surface for scattered data, we employ four stages of an algorithm: (a) triangulate the scattered data using Delaunay triangulation; (b) assign the first derivative at the respective data; (c) form a triangular surface via convex combination from three local schemes with C 1 continuity along all adjacent triangles; and (d) construct the scattered data surface using the proposed quartic spline. Numerical results, including some comparisons with some existing mesh-free schemes, are presented in detail. Overall, the proposed quartic triangular spline scheme gives good results in terms of a higher coefficient of determination (R2) and smaller maximum error (Max Error), requires about 12.5% of the CPU time of the quartic Bézier triangular, and is on par with Shepard triangular-based schemes. Therefore, the proposed scheme is significant for use in visualizing large and irregular scattered data sets. Finally, we tested the proposed positivity-preserving interpolation scheme to visualize coronavirus disease 2019 (COVID-19) cases in Malaysia.

In this study, a new scheme for positivity preserving interpolation is proposed by using C1 rational quartic spline of (quartic/quadratic) with three parameters. The sufficient condition for the positivity rational quartic interpolant is derived on one parameter meanwhile the other two are free parameters for shape modification. These conditions will guarantee to provide positive interpolating curve everywhere. We tested the proposed positive preserving scheme with four positive data and compared the results with other established schemes. Based on the graphical and numerical results, we found that the proposed scheme is better than existing schemes, since it has extra free parameter to control the positive interpolating curve.

In this study, a relatively new method to solve partial differential equations (PDEs) called the fractional reduced differential transform method (FRDTM) is used. The implementation of the method is based on an iterative scheme in series form. We test the proposed method to solve nonlinear fractional Burgers equations in one, two coupled, and three dimensions. To show the efficiency and accuracy of this method, we compare the results with the exact solutions, as well as some established methods. Approximate solutions for different values of fractional derivatives together with exact solutions and absolute errors are represented graphically in two and three dimensions. From all numerical results, we can conclude the efficiency of the proposed method for solving different types of nonlinear fractional partial differential equations over existing methods.

The Subdivision Schemes (SSs) have been the heart of Computer Aided Geometric Design (CAGD) almost from its origin, and various analyses of SSs have been conducted. SSs are commonly used in CAGD and several methods have been invented to design curves/surfaces produced by SSs to applied geometry. In this article, we consider an algorithm that generates the 5-point approximating subdivision scheme with varying arity. By applying the algorithm, we further discuss several properties: continuity, Hölder regularity, limit stencils, error bound, and shape of limit curves. The efficiency of the scheme is also depicted with assuming different values of shape parameter along with its application.

This paper discusses scattered data interpolation by using cubic Timmer triangular patches. In order to achieve C1 continuity everywhere, we impose a rational corrected scheme that results from convex combination between three local schemes. The final interpolant has the form quintic numerator and quadratic denominator. We test the scheme by considering the established dataset as well as visualizing the rainfall data and digital elevation in Malaysia. We compare the performance between the proposed scheme and some well-known schemes. Numerical and graphical results are presented by using Mathematica and MATLAB. From all numerical results, the proposed scheme is better in terms of smaller root mean square error (RMSE) and higher coefficient of determination (R2). The higher R2 value indicates that the proposed scheme can reconstruct the surface with excellent fit that is in line with the standard set by Renka and Brown’s validation.

This study considers the construction a new C 1 rational cubic spline (cubic numerator and quadratic denominator) with three parameters. Some mathematical properties of this rational function will be investigated in details. In order to use the proposed rational cubic spline for positivity preserving interpolation and constrained data that lie above any arbitrary straight line, the automatic choice of one parameter will be developed through some mathematical derivation. On each sub-interval this idea gives extra two free parameters that very useful for the user to manipulate the shape of the curve while at the same time maintaining the shape preservations of the data. Numerical comparison with some existing schemes shows that, the proposed scheme is on par. Furthermore, for some data sets, the proposed scheme is better as well as produce more visually pleasing interpolating curve for scientific visualization.

This study deals with the numerical solution of the non-linear differential equations (DEs) arising in the study of hydrodynamics and hydro-magnetic stability problems using a new cubic B-spline scheme (CBS). The main idea is that we have modified the boundary value problems (BVPs) to produce a new system of linear equations. The algorithm developed here is not only for the approximation solutions of the 10th order BVPs but also estimate from 1st derivative to 10th derivative of the exact solution as well. Some examples are illustrated to show the feasibility and competence of the proposed scheme.

This study deals with the application of new rational bi-cubic Ball function with six parameters in image interpolation, especially for the grayscale image. These six free parameters can be modified to get better and quality image resolution, and refine the shape of the interpolating surface. This bivariate rational Ball function has been extended from univariate cases by using a tensor product approach. The proposed scheme is tested for image upscaling with factors of two and four through an efficient algorithm. The effectiveness of the proposed scheme is measured by using an image quality assessment (IQA), such as peak-signal-to-noise-ratio (PSNR), root mean square error (RMSE) or feature similarity (FSIM) index. Numerical and graphical results with comparisons against some existing scheme are presented by using MATLAB. The proposed scheme resulted in higher PSNR and FSIM, and smaller RMSE. Thus, the new rational bi-cubic Ball with six parameters is better than the existing scheme via an efficient algorithm.

This paper discusses the construction of new C2 rational cubic spline interpolant with cubic numerator and quadratic denominator. The idea has been extended to shape preserving interpolation for positive data using the constructed rational cubic spline interpolation. The rational cubic spline has three parameters αi, βi, and γi. The sufficient conditions for the positivity are derived on one parameter γi while the other two parameters αi and βi are free parameters that can be used to change the final shape of the resulting interpolating curves. This will enable the user to produce many varieties of the positive interpolating curves. Cubic spline interpolation with C2 continuity is not able to preserve the shape of the positive data. Notably our scheme is easy to use and does not require knots insertion and C2 continuity can be achieved by solving tridiagonal systems of linear equations for the unknown first derivatives di, i=1,…,n-1. Comparisons with existing schemes also have been done in detail. From all presented numerical results the new C2 rational cubic spline gives very smooth interpolating curves compared to some established rational cubic schemes. An error analysis when the function to be interpolated is ft∈C3t0,tn is also investigated in detail.

Using Autoregressive Distributed Lag (ARDL) and Vector Autoregressive (VAR) frameworks, this study examines the integration between the emerging stock market of Indonesia and its major trading partners (i.e., Japan, the U.S., Singapore, and China). During the period of July 1998 to December 2007, the Indonesian stock market is found to be integrated with its major trading partners. Thus, this implies that there is a limited room available for investors to gain risk-reduction benefits through diversifying their portfolio in those markets. Meanwhile, in the short run, the Indonesian market responds more to shocks in the U.S. and Singapore than in Japan and China. In designing policies pertaining to its stock market, the Indonesian government should take into account any development in the stock markets of its major trading partners, particularly the U.S. and Singaporean markets.

This paper discusses the positivity preserving interpolation for positive surfaces data by extending the C1 rational cubic spline interpolant of Karim and Kong to the bivariate cases. The partially blended rational bicubic spline has 12 parameters in the descriptions where 8 of them are free parameters. The sufficient conditions for the positivity are derived on every four boundary curves network on the rectangular patch. Numerical comparison with existing schemes also has been done in detail. Based on Root Mean Square Error (RMSE), our partially blended rational bicubic spline is on a par with the established methods.

This paper discussed the local control of interpolating function by using rational cubic spline (cubic/quadratic) with three parameters originally proposed by the authors. The rational spline has C1 continuity. The bounded properties of the rational cubic interpolants and shape controls of the rational interpolants are discussed in detail. The value control, inflection point control, and convexity control at a point by using the proposed rational cubic spline are constructed. Several numerical results are presented to show the capability of the method. Numerical comparisons with the existing scheme are also further elaborated. From the results, it was indicated that the scheme works well and it is comparable with the established existing scheme.