The Mixed LEAST SQUARES Meshfree (MDLSM) method has shown its appropriate efficiency for solving Partial Differential Equations (PDEs) governing the engineering problems. The method is based on the minimizing the residual functional. The residual functional is defined as a summation of the weighted residuals on the governing PDEs and the boundaries. The MOVING LEAST SQUARES ((MLS)) is usually applied in the MDLSM method for constructing the shape functions. Although the required consistency and compatibility for the APPROXIMATION function is satisfied by the (MLS), the method loss its appropriate efficiency when the nodal points cluster too much. In the current study, the mentioned drawback is overcome using the novel APPROXIMATION function called Mapped MOVING LEAST SQUARES (M(MLS)). In this approach, the cluster of closed nodal points maps to standard nodal distribution. Then the APPROXIMATION function and its derivatives compute noting the some consideration. The efficiency of suggested M(MLS) for overcoming the drawback of (MLS) is evaluated by approximating the mathematical function. The obtained results show the ability of suggested M(MLS) method to solve the drawback. The suggested APPROXIMATION function is applied in MDLSM method, and used for solving the Burgers equations. Obtained results approve the efficiency of suggested method.

In terms of observational data, there are some problems in the standard Big Bang cosmological model. Inflation era, early accelerated phase of the evolution of the universe, can successfully solve these problems. The inflation epoch can be explained by scalar inflaton field. The evolution of this field is presented by a non-linear differential equation. This equation is considered in FLRW model. In FLRW model, we consider the universe as the warped product of real line with a three dimensional homogeneous and isotropic manifold  which could have positive, negative or zero curvature. The main aim of this paper is the numerical solution of the inflation evolution differential equations using of a meshless discrete Galerkin method. The method reduces the solution of these types of differential equations to the solution of Volterra integral equations of the second kind. Therefore, we solve these integral equations using MOVING LEAST SQUARES method. Finally, a numerical example is included to show the validity and efficiency of the new technique.

The concise review systematically summarises the state-of-the-art variants of MOVING LEAST SQUARES ((MLS)) method. (MLS) method is a mathematical tool which could render cogent support in data interpolation, shape construction and formulation of meshfree schemes, particularly due to its flexibility to form complex arithmetic equation. However, the conventional (MLS) method is suffering to deal with discontinuity of field variables. Varied strategies of overcoming such shortfall are discussed in current work. Although numerous (MLS) variants were proposed since the introduction of (MLS) method in numerical/statistical analysis, there is no technical review made on how the methods evolve. The current review is structured according to major strategies on how to improvise (MLS) method: the modification of weight function, the manipulation of discrete norms, the inclusion of iterative feature for residuals minimising and integration of these strategies for more robust computation. A wide range of advanced (MLS) variants have been compiled, summarised, and reappraised according to its underlying principle of improvement. In addition, inherent limitation of (MLS) method and its possible strategy of improvement is discussed too in this article. The current work could render valuable reference to implement and develop advanced (MLS) schemes, whenever complexity of the specific scientific problems arose.

Management and exploitation in mines require a continuous and relatively smooth surface of the mineral grades. While assessing the various mineral elements, the scattered exploratory cavities are irregularly excavated. Producing a continuous surface from measured data requires interpolation methods. Several factors, including the characteristics of the data, affect the efficiency of the interpolation methods. For this reason, the efficiency of different methods in various cases is inconsistence, and choosing the appropriate interpolation method is also challenging. Interpolation methods can be categorized into two groups of mesh-based and meshless methods. Despite the efficiency and capabilities of meshless methods, they have a fundamental shortcoming due to the fixed size of the support domain. On the one hand, the distribution of exploratory cavities in mines is usually irregular, and in some areas, it is very dense, and in others, it is very sparse. On the other hand, the grade values of minerals at the surface of the region can be very variable with high changes. Conventional interpolation methods do not have sufficient efficiency and flexibility in confronting these two aforementioned issues. In this study, a precise, reliable, and flexible method is developed for interpolation of minerals through integrating the MOVING LEAST SQUARES and recursive LEAST SQUARES methods. In the proposed method for crack detection, the residuals statistical test of LEAST SQUARES computations is used. In this method, for the central point, a continuity threshold (non-continuity) is determined based on the standard deviation of field values, so that points with crack are revealed and removed from the calculation of the value of the central point. Moreover, the size of the support domain is determined dynamically based on the recursive property of the method. In this method, an individual radius for the support domain is assigned to each central point according to the values and distributions of the surrounding field points. The dynamic size of the support domain allows a precise and reliable estimation of polynomial coefficients and the values of the central points. The efficiency of the proposed method is evaluated by applying it to simulated data as well as comparing it with the results of conventional interpolation methods on real mineral data. The results of the simulation data indicate the ability of the proposed method to reveal the non-continuity and fractures of surfaces with determining the dynamics size of the support domain based on the data structure. To compare the results of the proposed method with conventional interpolation methods including LPI, IDW, Kriging, and RBF, the root mean square error (RMSE), mean and median of errors are used. In this way, in addition to the overall accuracy of each method, the distribution of errors is also determined. The RMSE, mean and median errors of the proposed method, using the 10-fold cross-validation method for chromium (Cr), are 28. 020, 0. 2. 201 and 2. 874, respectively, and for iron (Fe) are 1. 074, 0. 017 and 0. 094, respectively. Comparison of these results with conventional interpolation methods indicates the efficiency of the proposed method for both groups of high concentration and significant changes in the values and low concentration and almost uniform level of values. The results indicate the ability of the proposed method in detecting the jumps and non-continuity in the support domain and removal of some field points within the dynamic process, lead to a significant increase in the efficiency of the method compared to conventional methods.

In this paper, the interpolating MOVING LEAST-SQUARES (I(MLS)) method is discussed. The interpolating MOVING LEAST square methodology is an e, ective technique for the APPROXIMATION of an unknown function by using a set of disordered data. Then we apply the I(MLS) method for numerical solution of Volterra{Fredholm integral equations, and , nally some examples are given to show the accuracy and applicability of the method.

This paper presents element-free Galerkin (EFG) method as a computational technique that can effectively avoid the disadvantage of mesh entanglement. The present method is used to analyze the static defelection of beams. The MOVING LEAST SQUARES ((MLS)) APPROXIMATION has been used for constructing the shape function based on a set of nodes arbitrarily distributed in the analysis domain.

Many ranking methods have been proposed so far. However, there is not any method which can always give a satisfactory solution for every situation. In this paper, we propose a method for ranking fuzzy numbers based on the distance method and compare the results with other ranking methods in some examples.

It has been more than a decade since mesh-free methods gained considerable attention in engineering and science. The main distinction of the methods with respect to the FEM is their shape functions. In this regard, Diffuse Element Method (DEM) employs LEAST-SQUARES (LS) APPROXIMATION, while Element-Free Galerkin Method (EFGM) benefits from MOVING-LEAST-SQUARES ((MLS)) APPROXIMATION for shape function construction. In order to compare the results of the methods, Boussinesq problem, which has closed-form solution, is used as a benchmark. The results are compared in displacement and stress fields. Finally, it is concluded that EFGM error is less than DEM in the both fields. This is more pronounced in stress field due to weight function being constant in DEM.

One of the main difficulties in the development of meshless methods using the MOVING LEAST SQUARES APPROXIMATION, such as Mixed Discrete LEAST SQUARES Meshless (MDLSM) method, is the imposition of the essential boundary conditions. In this paper the RPIM shape function, which satisfies the properties of the Kronecker delta condition, is employed in the Mixed Discrete LEAST SQUARES Meshless (MDLSM)method for solving the elasticity problems. Accordingly, two new MDLSM formulation is proposed in this article namely RPIM-based MDLSM and coupled (MLS)-RPIM MDLSM formulation. The essential boundary conditions can be imposed directly in both presented methods. The proposed methods are used for the solution of three benchmark elasticity problems and the results are presented and compared with the available analytical solutions and those of (MLS)-based MDLSM formulation. In addition, in each example different types of nodal distributions, regular and irregular configurations, are considered to test the performance of the presented methods. The numerical tests indicates higher accuracy of the suggested approaches in comparison with the (MLS)-based MDLSM method.