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 (MMLS). 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 MMLS for overcoming the drawback of MLS is evaluated by approximating the mathematical function. The obtained results show the ability of suggested MMLS 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.

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.

Determination of the diffusion coefficient on the base of solution of a linear inverse problem of the parameter estimation using the LEAST-square METHOD is presented in this research. For this propose a set of temperature measurements at a single sensor location inside the heat conducting body was considered. The corresponding direct problem was then solved by the application of the heat fundamental solution.

The main purpose of this article is to increase the efficiency of the LEAST SQUARES METHOD in numerical solution of ill-posed functional and physical equations. Determining the LEAST SQUARES of a given function in an arbitrary set is often an ill-posed problem. In this article, by defining artificial constraint and using Lagrange multipliers METHOD, the attempt is to turn n-dimensional LEAST SQUARES problems into (n-1) ones, in a way that the condition number of the corresponding system with(n-1) -dimensional problem will be low. At first, the new METHOD is introduced for2 and 3-term basis, then the presented METHOD is generalized for n-term basis. Finally, the numerical solution of some ill-posed problems like Fredholm integral equations of the first kind and singularly perturbed linear Fredholm integral equations of the second kind are approximated by chain LEAST SQUARES METHOD. Numerical comparisons indicate that the chain LEAST SQUARES METHOD yields accurate and stable approximations in many cases.

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.

Numerical crack modeling is an important and basic problem for researchers. Also, Finite element has good availability in crack modeling, but there are a few problems in using standard shape functions. Meshless METHODs shape functions, so Discrete LEAST square, which is used in this research, makes them more efficient with Finite element, especially in high-gradient problems. In this study, the cohesive crack theory leaving the usual numerical METHODs for crack behavior is investigated. In this way, doing the model conditions that do not occur crack with force out on the edges crack, and gradually become more open crack, and decrease the cohesive tension, that done zero, it means the whole crack is opening. Therefore, meshless METHODs use some techniques, such as visibility criterion and diffraction METHOD, to encounter this problem. This technique is the simplest METHOD for discontinuity modeling in meshless METHODs and has more compatibility with general domain discretization in meshless METHODs. Through engineering problems, the domain of the problem may contain nonconvex boundaries, particularly the fractured ones having discontinuous displacement fields. In such conditions, the shape functions associated with particles, whose supports intersect with the discontinuity, should be modified. One of these criteria is the visibility. In this approach, if the assumed light beam meets the discontinuity line, the shape function after the barrier will be cut. Therefore, discontinuity is applied to the geometry. In other words, the shape function of the particles, which prevents the crack or discontinuity from reaching the light beam, will be modified to amount to a zero. In this study, Cohesive Crack Theory is investigated for modeling and simulating crack behavior in DLS meshless METHOD. This METHOD gradually decreases the resistance of the cracked area to simulate splitting of the material. Finally, the high efficiency and accuracy of DLSM is given by comparing the DLSM results with experimental or FEM ones.

This paper introduces a novel METHOD for solving the generalized total LEAST SQUARES problem, an extension of the total LEAST SQUARES problem. The generalized total LEAST SQUARES problem emerges when solving overdetermined linear systems with the multiple right-hand sides $\mathbf{AX} \thickapprox \mathbf{B}$, where both the observation matrix $\mathbf{B}$ and the data matrix $\mathbf{A}$ contain errors. Our approach involves extending the Taylor series expansion to reformulate the generalized total LEAST SQUARES problem into a linear problem, allowing us to employ the tensor form of the generalized LEAST SQUARES algorithm for efficient computation. This technique streamlines the computational process and enhances solution accuracy. For a more detailed survey, we compare the proposed METHOD for solving the generalized total LEAST SQUARES problem with one of the matrix format METHODs for the associated total LEAST SQUARES problem. Empirical results show that our METHOD significantly improves computational efficiency and solution precision. Additionally, we demonstrate its practical application in the context of image blurring.

Summary Global and regional geomagnetic field models give the components of the geomagnetic field as functions of place and time. Most of these models utilize polynomials or Fourier series to map the input variables to the geomagnetic field values. The only temporal variation in these models is the long term secular variation. However, there is an increasing need amongst certain users for the models that can provide shorter term temporal variations, such as the geomagnetic daily variation. In this research, we have constructed an empirical model of the quiet daily geomagnetic field variation based on functional fitting...