Quasi-Interpolation in a Space of C 2 Sextic Splines over Powell–Sabin Triangulations

In this work, we study quasi-interpolation in a space of sextic splines defined over Powell– Sabin triangulations. These spline functions are of class C 2 on the whole domain but fourth-order regularity is required at vertices and C 3 regularity is imposed across the edges of the refined triangulation and also at the interior point chosen to define the refinement. An algorithm is proposed to define the Powell–Sabin triangles with a small area and diameter needed to construct a normalized basis. Quasi-interpolation operators which reproduce sextic polynomials are constructed after deriving Marsden’s identity from a more explicit version of the control polynomials introduced some years ago in the literature. Finally, some tests show the good performance of these operators.Erasmus+ International Dimension programme, European CommissionPAIDI programme, Junta de Andalucía, Spai

An Algebraic Hyperbolic Spline Quasi-Interpolation Scheme for Solving Burgers-Fisher Equations

In this work, the results on hyperbolic spline quasi-interpolation are recalled to establish the numerical scheme to obtain approximate solutions of the generalized Burgers-Fisher equation. After introducing the generalized Burgers-Fisher equation and the algebraic hyperbolic spline quasi-interpolation, the numerical scheme is presented. The stability of our scheme is well established and discussed. To verify the accuracy and reliability of the method presented in this work, we select two examples to conduct numerical experiments and compare them with the calculated results in the literature

A Reverse Non-Stationary Generalized B-Splines Subdivision Scheme

In this paper, two new families of non-stationary subdivision schemes are introduced. The schemes are constructed from uniform generalized B-splines with multiple knots of orders 3 and 4, respectively. Then, we construct a third-order reverse subdivision framework. For that, we derive a generalized multi-resolution mask based on their third-order subdivision filters. For the reverse of the fourth-order scheme, two methods are used; the first one is based on least-squares formulation and the second one is based on solving a linear optimization problem. Numerical examples are given to show the performance of the new schemes in reproducing different shapes of initial control polygons

<i>C</i>² Cubic Algebraic Hyperbolic Spline Interpolating Scheme by Means of Integral Values

In this paper, a cubic Hermite spline interpolating scheme reproducing both linear polynomials and hyperbolic functions is considered. The interpolating scheme is mainly defined by means of integral values over the subintervals of a partition of the function to be approximated, rather than the function and its first derivative values. The scheme provided is C2 everywhere and yields optimal order. We provide some numerical tests to illustrate the good performance of the novel approximation scheme

Study of the Physico-Chemical and Microbiological Quality of Water from River Innaouene, Taza Province, Morocco

In recent decades, the province of Taza has undergone extensive agricultural modernization, marked by an increase in industrial units such as modern oil mills and olive canneries. At the same time, rapid population growth has exacerbated an important problem, the deterioration of the quality of water resources in limited quantities. This study explores the heart of matter by meticulously sampling physico-chemical and bacteriological parameters at eight strategically chosen sites. Conducted in late July 2021, the survey focuses on the surface waters of the Innaouene River, providing a timely and insightful examination of the complex dynamics of water quality in the context of modernization and demographic change. This work focused on the determination of physicochemical and microbiological parameters of water. The results of the analyses show that this river is subject to multiple constraints and reaches its tolerance level on most of its sections. A multivariate statistical approach: principal component analysis (PCA) has shown that water pollution in the Innaouene River is mainly related to untreated wastewater discharges into streams. Water quality is considered bad to very bad. Therefore, ecological and health impacts can occur