The accuracy and efficiency of the Multiquadric Radial Basis Function (MQ-RBF) method are highly sensitive to the choice of the shape parameter. This study aims to identify optimal relationships for determining the shape parameter in solving common partial differential equations (PDEs) encountered in water engineering. To this end, the procedure for reconstructing and solving PDEs using the MQ-RBF method is first outlined. Then, optimal values for the shape parameter are derived based on domain length and the number of computational centers. Based on these findings, empirical formulas for the optimal shape parameter are proposed, which significantly reduce computational cost. The performance of the proposed formulas is compared with exact solutions and existing empirical relations. Results show that, unlike previous approaches, the new formulas offer high accuracy, with negligible errors across different examples-sometimes approaching zero. Moreover, compared to optimization-based techniques, the proposed method dramatically improves computational speed by eliminating the need for iterative algorithms. The study also investigates stability conditions, showing that for diffusion, advection-diffusion, and Burgers’ equations, the maximum allowable time step depends on the diffusion coefficient, flow velocity, and Reynolds number.

In this paper, the shape influence of piezoelectric beams including triangle, trapezoid, rectangle, invertedtrapezoid, convex parabola, concave parabola, and comb-shaped (a combination of two triangular beams with aconnector of 4 mm length) are addressed and analyzed by FEM. The analysis is performed for a bimorphpiezoelectric beam. The analyzed parameters include the beam length, thickness and width of the piezoelectriclayer. The study is performed using COMSOL Multiphysics software for all seven shapes. The results show thatdue to the mechanical properties of the beams, the natural frequency of the triangular beam is more for allconsidered parameters. In addition, as the width of the beam end increases, the natural frequency reduces, too. Since natural frequency is inversely related to electric power, the inverted trapezoidal beam has the highestelectric power and the triangular beam has the lowest one.

In most practical cases, to increase parameter estimation accuracy, we need an estimator with the least risk. In this, contraction estimators play a critical role. Our main purpose is to evaluate the efficiency of some shrinkage estimators of the shape parameter of the Pareto-Rayleigh distribution under two classes of shrinkage estimators. In this research, the purpose estimators’,efficiency will be compared with the unbiased estimator obtained under the quadratic loss function. The relationship between these two classes of shrinkage estimators was examined, and then the relative efficiency of the proposed estimators was discussed and concluded via doing a Monte Carlo simulation.

In many water engineering studies, there is a need to fill lost rain data using mapping tools. In this research this has been done by 5 types of RBFs; the data used were extracted by 3 test functions in which the support domain varied from 0.1 by 0.1 meter net to 0.5 by 0.5 meter net with different numbers of stations in a unit area domain. The c parameter was optimized by cross validation method and the Normalized Mean Square Error (NMSE), Percent Average Estimation Error (PAEE) and Coefficient of determination (R2) were the statistical controlling tools for choosing suitable RBF function type. Compared to other works in literature, this work had a better performance in mapping. It is also shown that the c parameter that optimizes the RBF function is highly dependent on the support domain size; the finer the resolutions of support domain, the better the results achieved. The attribute was also found for an arbitrary station point, Z (0.25, 0.35) to show the model capability for an irregular domain. This work may be compared with meshless methods for further research.

Researchers in various engineering fields are increasingly interested in evaluating the influence of the microstructural properties of minerals, such as hematite, on their reactivity. Hematite behaviour and microstructural changes during mechanical activation (MA) were investigated using the shape parameter. The laser diffraction analysis indicates the occurrence of agglomeration in the mechanically activated hematite after 60 min of MA. Based on the microstructural studies, the hematite crystallinity reduced to below 16% after 240 min of MA. Microstrain and crystallite size continuously changed during MA, leading to an increase in hematite leaching efficiency by sulfuric acid. The results revealed that the microstrain variation rate is higher than other microstructural parameters. The obtained shape parameter increased from about 0.31 in the initial hematite to 0.88 and 0.66 in the hematite mechanically activated for 60 and 240 min, respectively. The shape parameter study demonstrated that most mechanical energy is stored in the hematite lattice as microstrain after 240 minutes of MA. Considering the lower impact of microstrain on promoting hematite reactivity, shorter MA times could be preferred over prolonged MA, which would reduce costs and increase capacity.

In this paper, some meshless methods based on the local Newton basis functions are used to solve some time dependent partial differential equations. For stability reasons, used variably scaled radial kernels for constructing Newton basis functions. In continuation, with considering presented basis functions as trial functions, approximated solution functions in the event of spatial variable with collocation method. Then, with aid of method of lines obtained a system of ordinary differential equations according to solution function in the event of time. Methods applied for solving the nonlinear Burgers’ equation and couple Burgers’ equation. The numerical results show that the proposed method is efficient, accurate and stable.

In this paper, a Bayesian approach is proposed for shift point detection in an inverse Gaussian distribution. In this study, the mean parameter of inverse Gaussian distribution is assumed to be constant and shift points in shape parameter is considered. First the posterior distribution of shape parameter is obtained. Then the Bayes estimators are derived under a class of priors and using various loss functions. We assumed uniform, Jeffreys, exponential, gamma and chi square distributions as prior distributions. The squared error loss function (SELF), entropy loss function (ELF), linex loss function (LLF) and precautionary loss function (PLF), are used as loss functions. We attempt to find out the best estimator for shift point under various priors and loss functions. The proposed Bayesian approach can be adapted to any similar problem for shift point detection. Simulation studies were done to investigate the performance of different loss functions. The results of simulation study denote that the Jeffrey prior distribution under PLF has the most accurate estimation of shift point for sample size of 20, and the gamma prior distribution under SELF has the most accurate estimation of shift point for sample size of 50.

The radial basis functions (RBFs) meshless method has high accuracy for the interpolation of scattered data in high dimensions. Most of the RBFs depend on a parameter, called shape parameter which plays a significant role to specify the accuracy of the RBF method. In this paper, we present three algorithms to choose the optimal value of the shape parameter. These are based on Rippa’, s theory to remove data points from the data set and results obtained by Fasshauer and Zhang for the iterative approximate moving least square (AMLS) method. Numerical experiments confirm stable solutions with high accuracy compared to other methods.