In this paper, we introduce a general framework for constructing univariate spline spaces defined on refined non-uniform partitions with variable knot insertion. The proposed approach is developed for spline spaces characterized by a prescribed level of smoothness and a corresponding range of polynomial degrees. Within this setting, we establish suitable continuity conditions at the inserted knots and develop a fully local Hermite interpolation scheme. We then construct a normalized B-spline-type basis consisting of non-negative, compactly supported functions that form a partition of unity. By means of blossoming techniques, we derive adapted B-spline-type representations and design quasi-interpolants exhibiting superconvergence properties. The proposed operators provide high-order accuracy for approximation on non-uniform meshes, as confirmed by theoretical analysis and numerical experiments.

The paper introduces and investigates a cubic collocation method based on a superconvergent quasi-interpolant for approximating solutions of boundary value problems. It is proven to achieve fourth-order convergence. The proposed method accurately approximates the solution along with its first and second derivatives at knots with high precision. Numerical results confirm the predicted convergence order, as established by the analysis. Additionally, we compare this method with those presented in the literature.

This work presents a numerical resolution of the two-dimensional Ripa system using Roe scheme. A discretization of the source term satisfying the conservation property is presented. The mathematical model is based on the ordinary shallow water equations in merging the horizontal temperature gradient. The numerical approach employs unstructured grids and integrates Runge–Kutta method and the minmod limiter to achieve second order accuracy in both time and space domains. A strategy of mesh adaptation based on temperature gradient is implemented to refine the study domain and gain in computational cost. Various numerical tests are conducted to verify the efficiency of the numerical method.

This paper deals with the optimisation of the Multi-Objectif k-Minimum Spanning Tree (MO k-MST) problem. A wide varieties of decision making problems in the real world can be formulated as a MO k-MST, which is known to be NP-complete. In order to solve a such problem, we propose two approximate approaches based on simulated annealing method: the first one will integrates the static weighted sum method while the second one uses the dynamic weighted sum method. Computational experiments were carried out in order to compare the performance of each method.

This paper presents an efficient approximate hybrid algorithm designed to tackle the multi-objective k-Minimum Spanning Tree (MO k-MST) problem. Instead of aiming to identify the entire Pareto optimal solution set, we opt to convert the MO nature of the problem to a single objective one using the weighted sum method. Subsequently, we integrate both simulated annealing (SA) and ant colony optimization (ACO) algorithms in order discover practical solutions to the problem. The MO k-MST dilemma arises in various real-world decision-making scenarios. Numerical experiments demonstrate that our proposed hybrid approach outperforms the standalone simulated annealing method, thus offering enhanced performance.

In the current generation of cellular networks, energy efficiency is considered as an important issue due to their high-energy consumption. To meet the rising traffic resulting from the growing mobile stations requests and covering the entire transmission area, base stations must be increasingly deployed. However, increasing the number of these stations increases the cost and energy consumption, which leads to conflicting goals. In this paper, we introduce a multi-objective mathematical model on cellular networks that aimed to minimize the expected total cost of base stations and maximize total coverage. This optimization must take into account the traffic demand profile. Given that the studied model corresponds to an NP-hard multi-objective problem, we use a meta-heuristic algorithm to solve it. The simulation results show the effectiveness of our approach to cover the grid while reducing costs and energy consumption.

This paper presents a study of the cubic spline collocation method for solving boundary value problems. The proposed method is based on super-convergent quasi-interpolant.The convergence analysis of this method is discussed, and we have given numerical results to illustrate the error estimates and the order of convergence.

In this study, we employ a blossoming technique and smoothness criteria to devise a two-step method for creating a $$C^2$$ septic spline quasi-interpolant on any given triangulation. This approach ensures an optimal approximation order without the need for coefficient masks associated with smoothness or B-spline basis. To demonstrate the validity of our theoretical findings, we provide numerical experiments.

This work deals with the numerical solution of dam-break flow over an erodible bed. The mathematical model is a combination of the shallow water, the transport diffusion and the bed morphology change equations. The system is solved by a well-Balanced central upwind scheme with conservative property. Several tests are illustrated in order to validate the accuracy and the performance of the model. A comparison of central upwind scheme and Roe scheme is presented.

This work deals with the numerical modeling of dam-break problem over mobile bed. The governing equations are a combination of the coupled model and the non-capacity model. The mathematical model is solved by the finite-volume Roe scheme, associated with a new discretization of the source term. Attention is given to the behavior of the water flow and the bed rate change. The numerical scheme is applied on bottoms composed of very fine sand.

. The Richards' equation attracts the attention of several scientific researchers due to its importance in the hydrogeology field especially porous soil. This work presents a numerical method to solve the two dimensional Richards' equation. The pressure form and the mixed form of Richards' equation are solved numerically using a bivariate diamond finite volumes scheme. Euler explicit scheme is used for the time discretization. Different test cases are done to validate the accuracy and the efficiency of our numerical model and to compare the possible numerical strategies. We started with a first simple test case of Richards' pressure form where the hydraulic capacity and the hydraulic conductivity are taken constant and then a second test case where the hydrodynamics parameters are linear variables. Finally, a third test case where the soil parameters are taken according the Van Gunchten empirical model is presented.

Islamic banking and finance are increasingly attracting attention among investors and researchers worldwide. However, there is a paucity of the studies which have focused on Islamic derivatives. These Shariah compliant derivatives were created in order to enable Muslims to invest in international markets and hedge risks according to their beliefs. In this paper, we bridge the gap in the financial literature by proposing a new mathematical model for pricing the "urbun" in Islamic finance (used as a call option), taking into consideration its similarities and differences with conventional options. Also we provide many numerical examples in order to illustrate the model's solutions.

In the light of present-day research, as we all know, the financial market owns the characteristics of self-similarity and long-range dependence and Fractional Brownian motion has these properties. From that point, The model with fractional stochastic volatility could be more businesslike than the model with standard Brownian motion. Furthermore, for more hardheaded models we can use hybrid models by incorporating stochastic interest rate into stochastic volatility. In this paper, we bridge the gap in the financial literature by proposing a new mathematical model, more general and realistic model based on concocting the stochastic interest rate following the CIR process together with fractional stochastic volatility. By using the replication technique, Itos lemma, and Malliavin calculus, we derive a partial differential equation for valuing European options, also we provide many numerical examples in order to illustrate the model solutions.

In this paper we use the developed B-spline representation to construct a cubic super-superconvergent quasi-interpolant with an optimal approximation order which improves the efficiency and accuracy over traditional methods.

The flow through saturated porous soil is described by Darcy equation that generally has no analytical solution. Therefore, we have to use numerical methods to solve it. There exist several numerical methods and the well-known most used one is the finite elements method. In this work, we use a Diamond finite volumes scheme characterized with its conservation property to simulate Darcy equation in the linear and the non linear case in two dimensional case, based on permeability tensor. Different test cases are studied to validate the accuracy and the efficiency of the numerical model. The numerical scheme proves, through the obtained results, high level of accuracy comparing with the exact solution. The diamond finite volumes scheme for its ability to ensure the mass conservation of our model, knowing that the finite elements or the finite differences methods are less efficient on this side, our finite volumes method is the most suited to the Darcy flow and more faithful to physical problems comparing to other methods. We focused in our study on different aspects of the finite volumes scheme, the mass conservation, the non-linearity of the hydraulic conductivity and their interaction with the finite volumes method.

In this paper, we use the finite element method to construct a new normalized basis of a univariate quadratic C-1 spline space. We give a new representation of Hermite interpolant of any piecewise polynomial of class at least C-1 in terms of its polar form. We use this representation for constructing several superconvergent and super-superconvergent discrete quasi-interpolants which have an optimal approximation order. This approach is simple and provides an interesting approximation. Numerical results are given to illustrate the theoretical ones.

In the current generation of cellular networks, practitioners and researchers have shown a keen interest in green wireless communication due to its capability to create eco-efficient networks. To adapt to the increase in traffic and services for all mobile subscribers, Base stations (BSs) and relay stations (RSs) must be deployed more and more in order to meet the growth in this demand. However, increasing the number of BSs or RSs can increase energy consumption and reduce efficiency as it is responsible for large carbon dioxide (CO2) emissions. In this paper, we introduce a new approach to the management of the base stations and the relay stations using a fuzzy dynamic distribution of the BSs and RSs in the center according to the distribution of users. We use, secondly, a sleep mode activation of these stations which is seen as the key to reducing grid power consumption. The performance and the effectiveness of the proposed approach are clarified by a simulation example that reveals the capacity of our strategy in reducing energy consumption.

In graph theory, the k-minimum spanning tree problem is considered to be one of the well-known NP hard problems to solve. This paper address this problem by proposing several hybrid approximate approaches based on the combination of simulated annealing, tabu search and ant colony optimization algorithms. The performances of the proposed methods are compared to other approaches from the literature using the same well-known library of benchmark instances.

The objective of this paper is to develop a numerical method for solving a bidimensional unilateral obstacle problem. This is based on the bicubic splines collocation method and the generalized Newton method. In this paper, we obtain an approximate expression for solving a bidimensional unilateral obstacle problem. We show that the approximate formula obtained by the bicubic splines collocation method is effective. Next, we prove the convergence of the proposed method. The method is applied to some test examples and the numerical results have been compared with the exact solutions. The obtained results show the computational efficiency of the method. It can be concluded that computational efficiency of the method is effective for the two-dimensional obstacle problem. 2010 Mathematics Subject Classification. 65L10, 34B15, 65M22.