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The radial basis function (RBF) approximation is a rapidly developing field of mathematics. In the paper, we are concerned with the measurement of scalar physical quantities at nodes on sphere in the 3D Euclidean space and the spherical RBF interpolation of the data acquired. We employ a multiquadric as the radial basis function and the corresponding trend is a polynomial of degree 2 considered in Cartesian coordinates. Attention is paid to geodesic metrics that define the distance of two points on a sphere. The choice of a particular geodesic metric function is an important part of the construction of interpolation formula. We show the existence of an interpolation formula of the type considered. The approximation formulas of this type can be useful in the interpretation of measurements of various physical quantities. We present an example concerned with the sampling of anisotropy of magnetic susceptibility having extensive applications in geosciences and demonstrate the advantages and drawbacks of the formulas chosen, in particular the strong dependence of interpolation results on condition number of the matrix of the system considered and on round-off errors in general. [ABSTRACT FROM AUTHOR]

This study introduces a deep neural network approach that utilizes radial basis functions (RBFs) to solve forward and inverse problems in the process of diffusion. The input layer incorporates multiquadric (MQ) RBFs, symbolizing the radial distance between the boundary points on the spacetime boundary and the source points positioned outside the spacetime boundary. The output layer is the initial and boundary data given by analytical solutions of the diffusion equation. Utilizing the concept of the spacetime coordinates, the approximations for forward and backward diffusion problems involve assigning initial data on the bottom or top spacetime boundaries, respectively. As the need for discretization of the governing equation is eliminated, our straightforward approach uses only the provided boundary data and MQ RBFs. To validate the proposed method, various diffusion scenarios, including forward, backward, and inverse problems with noise, are examined. Results indicate that the method can achieve high-precision numerical solutions for solving diffusion problems. Notably, only 1/4 of the initial and boundary conditions are known, yet the method still yields precise results. [ABSTRACT FROM AUTHOR]

This is a continuation of our previous study on the shape parameter contained in the shifted surface spline. We insist that the data points be purely scattered without meshes and the domain can be of any shape when conducting function interpolation by shifted surface splines. We also endeavor to make our approach easily accessible for scientists, not only mathematicians. However, the space of interpolated functions is smaller than that used before, leading to sharper function approximation. This function space has particular significance in numerical partial differential equations, especially for equations whose solutions lie in Sobolev space. Although the Fourier transform is deeply involved, scientists without a background in Fourier analysis can easily understand and use our approach. [ABSTRACT FROM AUTHOR]

Elliptic boundary value problems (BVPs) are widely used in various scientific and engineering disciplines that involve finding solutions to elliptic partial differential equations subject to certain boundary conditions. This article introduces a novel approach for solving elliptic BVPs using an artificial neural network (ANN)-based radial basis function (RBF) collocation method. In this study, the backpropagation neural network is employed, enabling learning from training data and enhancing accuracy. The training data consist of given boundary data from exact solutions and the radial distances between exterior fictitious sources and boundary points, which are used to construct RBFs, such as multiquadric and inverse multiquadric RBFs. The distinctive feature of this approach is that it avoids the discretization of the governing equation of elliptic BVPs. Consequently, the proposed ANN-based RBF collocation method offers simplicity in solving elliptic BVPs with only given boundary data and RBFs. To validate the model, it is applied to solve two- and three-dimensional elliptic BVPs. The results of the study highlight the effectiveness and efficiency of the proposed method, demonstrating its capability to deliver accurate solutions with minimal data input for solving elliptic BVPs while relying solely on given boundary data and RBFs. [ABSTRACT FROM AUTHOR]

This study introduces a deep neural network approach that utilizes radial basis functions (RBFs) to solve forward and inverse problems in the process of diffusion. The input layer incorporates multiquadric (MQ) RBFs, symbolizing the radial distance between the boundary points on the spacetime boundary and the source points positioned outside the spacetime boundary. The output layer is the initial and boundary data given by analytical solutions of the diffusion equation. Utilizing the concept of the spacetime coordinates, the approximations for forward and backward diffusion problems involve assigning initial data on the bottom or top spacetime boundaries, respectively. As the need for discretization of the governing equation is eliminated, our straightforward approach uses only the provided boundary data and MQ RBFs. To validate the proposed method, various diffusion scenarios, including forward, backward, and inverse problems with noise, are examined. Results indicate that the method can achieve high-precision numerical solutions for solving diffusion problems. Notably, only 1/4 of the initial and boundary conditions are known, yet the method still yields precise results.

This is a continuation of our previous study on the shape parameter contained in the shifted surface spline. We insist that the data points be purely scattered without meshes and the domain can be of any shape when conducting function interpolation by shifted surface splines. We also endeavor to make our approach easily accessible for scientists, not only mathematicians. However, the space of interpolated functions is smaller than that used before, leading to sharper function approximation. This function space has particular significance in numerical partial differential equations, especially for equations whose solutions lie in Sobolev space. Although the Fourier transform is deeply involved, scientists without a background in Fourier analysis can easily understand and use our approach.

Elliptic boundary value problems (BVPs) are widely used in various scientific and engineering disciplines that involve finding solutions to elliptic partial differential equations subject to certain boundary conditions. This article introduces a novel approach for solving elliptic BVPs using an artificial neural network (ANN)-based radial basis function (RBF) collocation method. In this study, the backpropagation neural network is employed, enabling learning from training data and enhancing accuracy. The training data consist of given boundary data from exact solutions and the radial distances between exterior fictitious sources and boundary points, which are used to construct RBFs, such as multiquadric and inverse multiquadric RBFs. The distinctive feature of this approach is that it avoids the discretization of the governing equation of elliptic BVPs. Consequently, the proposed ANN-based RBF collocation method offers simplicity in solving elliptic BVPs with only given boundary data and RBFs. To validate the model, it is applied to solve two- and three-dimensional elliptic BVPs. The results of the study highlight the effectiveness and efficiency of the proposed method, demonstrating its capability to deliver accurate solutions with minimal data input for solving elliptic BVPs while relying solely on given boundary data and RBFs.

In this study, an enhancement to the Multiquadric Radial Basis Function (MQ-RBF) method is presented for solving Elasticity Boundary Value Problems (E-BVPs). MQ-RBF is utilized to reproduce the system of governing partial differential equations and boundary conditions. Also the inefficacies of some existing methods in determining the Optimal Shape Parameter (OSP) when solving E-BVPs are shown. Afterward, a new algorithm is introduced based on Teaching–Learning-Based Optimization (TLBO). In addition, the lower and upper bounds of the shape parameter and a new objective function were introduced for the TLBO algorithm. Using the algorithm, one OSP is obtained for both MQ-RBF functions in 2D problems, resulting in reduced computational cost. Its efficiency is evidenced by solving several examples. Moreover, its accuracy is demonstrated by comparing its outcomes to those of analytical methods. Unlike the 1D problems, the OSP is found to be a function of loading and material properties as well as the geometry in 2D problems. [ABSTRACT FROM AUTHOR]

In this paper, we totally discard the traditional trial-and-error algorithms of choosing the acceptable shape parameter c in the multiquadrics − c 2 + ∥ x ∥ 2 when dealing with differential equations, for example, the Poisson equation, with the RBF collocation method. Instead, we choose c directly by the MN-curve theory and hence avoid the time-consuming steps of solving a linear system required by each trial of the c value in the traditional methods. The quality of the c value thus obtained is supported by the newly born choice theory of the shape parameter. Experiments demonstrate that the approximation error of the approximate solution to the differential equation is very close to the best approximation error among all possible choices of c. [ABSTRACT FROM AUTHOR]

In the present article, a meshfree numerical scheme based on multiquadric radial basis function to solve the time-fractional Kersten–Krasil'shchik coupled KdV–mKdV system has been proposed. The finite difference scheme and the Kansa method, respectively, are used to discretize the temporal and spatial derivatives. The stability and convergence of the proposed numerical scheme are also theoretically established. Finally, numerical simulations are performed and compared with the exact solutions to establish the accuracy and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we present an easily accessible approach to finding a suitable shape parameter in the shifted surface spline for function interpolation. We aim at helping more readers, including mathematicians and non-mathematicians, to use our method to solve practical problems. Hence, some highly complicated mathematical theorems and definitions are avoided. The major requirement, as in our previous approach, that the data points should be evenly spaced in the domain is also relaxed. This means that the data points are purely scattered without restrictions. The drawback is that the shape parameter thus obtained is not exactly the same as the theoretically predicted optimal value, which can always be achieved by using our previous rigorous approach. However, experiments show that the gap is quite small and the final interpolation errors thus obtained are satisfactory. [ABSTRACT FROM AUTHOR]

In this article, we propose a simplified radial basis function (RBF) method with exterior fictitious sources for solving elliptic boundary value problems (BVPs). Three simplified RBFs, including Gaussian, multiquadric (MQ), and inverse multiquadric (IMQ) without the shape parameter, are adopted in this study. With the consideration of many exterior fictitious sources outside the domain, the radial distance of the RBF is always greater than zero, such that we can remove the shape parameter from RBFs. Additionally, simplified Gaussian, MQ, and IMQ RBFs and their derivatives in the governing equation are always smooth and nonsingular. Comparative analysis is conducted for three different collocation types, including conventional uniform centers, randomly fictitious centers, and exterior fictitious sources. Numerical examples of elliptic BVPs in two and three dimensions are carried out. The results demonstrate that the proposed simplified RBFs with exterior fictitious sources can significantly improve the accuracy, especially for the Laplace equation. Furthermore, the proposed simplified RBFs exhibit the simplicity of solving elliptic BVPs without finding the optimum shape parameter. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical solution of the time-fractional Gilson–Pickering equation using the Kansa method, in which the multiquadrics were utilized as the radial basis function. To achieve this, a meshless numerical scheme based on the finite difference along with the Kansa method has been presented. First, the finite difference approach has been utilized to discretize the temporal derivative, and subsequently, the Kansa method is employed to discretize the spatial derivatives. The stability and convergence analysis of the numerical scheme are also elucidated in this paper. Furthermore, the soliton solutions have been acquired by implementing the Kudryashov method for comparison with the numerical results. Finally, numerical simulations are performed to confirm the applicability and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, a novel meshless numerical scheme to solve the time-fractional Oskolkov–Benjamin–Bona–Mahony–Burgers-type equation has been proposed. The proposed numerical scheme is based on finite difference and Kansa-radial basis function collocation approach. First, the finite difference scheme has been employed to discretize the time-fractional derivative and subsequently, the Kansa method is utilized to discretize the spatial derivatives. The stability and convergence analysis of the time-discretized numerical scheme are also elucidated in this paper. Moreover, the Kudryashov method has been utilized to acquire the soliton solutions for comparison with the numerical results. Finally, numerical simulations are performed to confirm the applicability and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

The localized method is one of the popular approaches in solving large-scale problems in science and engineering. In this paper, we implement the localized method of particular solutions using polynomial basis functions for solving various nonlinear problems. To validate our proposed numerical method, we present four numerical examples in regular and irregular domains which are solved by using localized method of particular solution with polynomial basis functions. We compared our numerical method with localized method of particular solutions using multiquadric radial basis function and numerical results clearly show that our numerical method is highly accurate, efficient, and outperformed the method using multiquadric radial basis function.

In this paper, we totally discard the traditional trial-and-error algorithms of choosing the acceptable shape parameter c in the multiquadrics −c2+∥x∥2 when dealing with differential equations, for example, the Poisson equation, with the RBF collocation method. Instead, we choose c directly by the MN-curve theory and hence avoid the time-consuming steps of solving a linear system required by each trial of the c value in the traditional methods. The quality of the c value thus obtained is supported by the newly born choice theory of the shape parameter. Experiments demonstrate that the approximation error of the approximate solution to the differential equation is very close to the best approximation error among all possible choices of c.

In this paper, we present an easily accessible approach to finding a suitable shape parameter in the shifted surface spline for function interpolation. We aim at helping more readers, including mathematicians and non-mathematicians, to use our method to solve practical problems. Hence, some highly complicated mathematical theorems and definitions are avoided. The major requirement, as in our previous approach, that the data points should be evenly spaced in the domain is also relaxed. This means that the data points are purely scattered without restrictions. The drawback is that the shape parameter thus obtained is not exactly the same as the theoretically predicted optimal value, which can always be achieved by using our previous rigorous approach. However, experiments show that the gap is quite small and the final interpolation errors thus obtained are satisfactory.

In this article, we propose a simplified radial basis function (RBF) method with exterior fictitious sources for solving elliptic boundary value problems (BVPs). Three simplified RBFs, including Gaussian, multiquadric (MQ), and inverse multiquadric (IMQ) without the shape parameter, are adopted in this study. With the consideration of many exterior fictitious sources outside the domain, the radial distance of the RBF is always greater than zero, such that we can remove the shape parameter from RBFs. Additionally, simplified Gaussian, MQ, and IMQ RBFs and their derivatives in the governing equation are always smooth and nonsingular. Comparative analysis is conducted for three different collocation types, including conventional uniform centers, randomly fictitious centers, and exterior fictitious sources. Numerical examples of elliptic BVPs in two and three dimensions are carried out. The results demonstrate that the proposed simplified RBFs with exterior fictitious sources can significantly improve the accuracy, especially for the Laplace equation. Furthermore, the proposed simplified RBFs exhibit the simplicity of solving elliptic BVPs without finding the optimum shape parameter.

A modeling method using multiquadric (MQ) radial basis function (RBF) and k-means clustering is proposed to establish the tropospheric atmospheric refractivity (TAR) profile model. The MQ-RBF interpolation algorithm is trained with the measurement data from 132 meteorological stations in 10 years. Mean Absolute Error (MAE)and Root Mean Squared Error (RMSE) of the prediction results are investigated with different local neighborhood parameters (r, θ, and t)of the MQ-RBF. Considering the tradeoff between RMSE and MAE, the parameters of the search ellipse(r, θ, and t) are optimized. The linear fit coefficients (a1, b1, a2, and b2) of the linear relationships between the coefficients (C1 and G) of TAR distribution formula and atmospheric refractive (N0) at observation station are obtained by using the least square linear regression method with k-means clustering algorithm. This method overcomes the difficulty of measuring the TAR profile under various meteorological conditions at different time and place, and has the characteristics of real-time, convenient and accurate. It provides a promising method for the correction of TAR errors in the radar applications. [ABSTRACT FROM AUTHOR]

The local radial basis functions (RBF) method is becoming increasingly popular as an alternative to the global version that suffers from ill-conditioning. The purpose of this paper is to design and describe the valuation of the real estate index options by a local RBF scheme based multiquadric radial basis function-generated finite difference (RBF-FD) method. As a generalized finite differencing scheme, the RBF-FD method functions without the need for underlying meshes to structure nodes. It offers high-order accuracy approximation and removes the difficulty of the ill-conditioned conventional global collocation methods. This paper employs an optimal variable shape parameter for the multiquadric basis functions at each grid point of the domain. Meanwhile, a local mesh refinement technique is adopted to deal with non-smooth payoffs of option. These techniques are effective and stable in improving the computational accuracy of the RBF-FD method. Several numerical experiments are presented and compared with the FD and compactly supported RBF methods to demonstrate the good performances of the proposed method. Lastly, the RBF-FD method is extended to price the American option of the real estate index. [ABSTRACT FROM AUTHOR]

Abstract In this research, the Kansa or Multiquadric method (MQ) has been developed for solving the seepage problems in 2D and 3D arbitrary domains. This research is the first application of this method for seepage analysis in both confined and unconfined porous media. The domain decomposition approach has been employed for applying MQ method easily in inhomogeneous and irregular complex geometries and decreasing the computational costs. For determining the optimum shape parameter that affects strongly the accuracy of MQ and other RFB methods, a new scheme that decreases drastically the computational time is introduced. The efficiency of the proposed algorithm has been examined under various radial basis functions, variations of number of interpolating points and points distribution, through a numerical example with analytical solution. Eventually, three examples including different boundary conditions are presented. Comparing results of the examples with other numerical methods indicates that the present approach has high capability and accuracy in solving seepage problems. [ABSTRACT FROM AUTHOR]

In this paper, we have studied the effect of conduction band nonparabolicity on wave packet broadening in different semiconducting mediums. Here, through a fourth derivative of the wavefunction, we have included the nonparabolicity effect into the one-dimensional Schrödinger equation. We have solved this equation by means of a meshless radial base function approach. We have compared five different semiconducting mediums GaAs, GaN, AlN, InSb and GaSb and showed that the wave packet broadens slower in systems with larger effective masses and smaller nonparabolicity parameters. Thus, we can manage and control the dispersion of a Gaussian wave packet utilizing the nonparabolicity and effective mass parameters. [ABSTRACT FROM AUTHOR]

In this paper numerical meshless method for solving Fokker–Planck equation is considered. This meshless method is based on multiquadric radial basis function and collocation method to approximate the solution. Here we apply θ-weighted finite difference method. The stability analysis of the method is dealt with, using a linearized stability method. Numerical examples illustrate the validity and applicability of the method.

In this article, we present multiquadric radial basis functions (RBFs), including multiquadric (MQ) and inverse multiquadric (IMQ) functions, without the shape parameter for solving partial differential equations using the fictitious source collocation scheme. Different from the conventional collocation method that assigns the RBF at each center point coinciding with an interior point, we separated the center points from the interior points, in which the center points were regarded as the fictitious sources collocated outside the domain. The interior, boundary, and source points were therefore collocated within, on, and outside the domain, respectively. Since the radial distance between the interior point and the source point was always greater than zero, the MQ and IMQ RBFs and their derivatives in the governing equation were smooth and globally infinitely differentiable. Accordingly, the shape parameter was no longer required in the MQ and IMQ RBFs. Numerical examples with the domain in symmetry and asymmetry are presented to verify the accuracy and robustness of the proposed method. The results demonstrated that the proposed method using MQ RBFs without the shape parameter acquires more accurate results than the conventional RBF collocation method with the optimum shape parameter. Additionally, it was found that the locations of the fictitious sources were not sensitive to the accuracy.

In this article, a collocation method using radial polynomials (RPs) based on the multiquadric (MQ) radial basis function (RBF) for solving partial differential equations (PDEs) is proposed. The new global RPs include only even order radial terms formulated from the binomial series using the Taylor series expansion of the MQ RBF. Similar to the MQ RBF, the RPs is infinitely smooth and differentiable. The proposed RPs may be regarded as the equivalent expression of the MQ RBF in series form in which no any extra shape parameter is required. Accordingly, the challenging task for finding the optimal shape parameter in the Kansa method is avoided. Several numerical implementations, including problems in two and three dimensions, are conducted to demonstrate the accuracy and robustness of the proposed method. The results depict that the method may find solutions with high accuracy, while the radial polynomial terms is greater than 6. Finally, our method may obtain more accurate results than the Kansa method.

The Kansa method with the Multiquadric-radial basis function (MQ-RBF) is inherently meshfree and can achieve an exponential convergence rate if the optimal shape parameter is available. However, it is not an easy task to obtain the optimal shape parameter for complex problems whose analytical solution is often a priori unknown. This has long been a bottleneck for the MQ-Kansa method application to practical problems. In this paper, we present a novel sample solution approach (SSA) for achieving a reasonably good shape parameter of the MQ-RBF in the Kansa method for the solution of problems whose analytical solution is unknown. The basic assumption behind the SSA is that the optimal shape parameter is considered to be largely depended on the shape of computational domain, the type of the boundary conditions, the number and distribution of nodes, and the governing equation. In the procedure of the SSA, we set up a pseudo-problem as the sample solution whose solution is known. It is not difficult to obtain the optimal parameter of the MQ-RBF in the numerical solution of the pseudo-problem. The SSA suggests that the optimal shape parameter of the pseudo-problem can also achieve an approximately optimal accuracy in the solution of the original problem. Numerical examples and comparisons are provided to verify the proposed SSA in terms of accuracy and stability in solving homogeneous problems and non-homogeneous modified Helmholtz problems in several complex domains even using chaotic distribution of collocation points. [ABSTRACT FROM AUTHOR]

The method of approximate particular solutions (MAPS) was first proposed by Chen et al. in Chen, Fan, and Wen, Numer Methods Partial Differential Equations, 28 (2012), 506-522. using multiquadric (MQ) and inverse multiquadric radial basis functions (RBFs). Since then, the closed form particular solutions for many commonly used RBFs and differential operators have been derived. As a result, MAPS was extended to Matérn and Gaussian RBFs. Polyharmonic splines (PS) has rarely been used in MAPS due to its conditional positive definiteness and low accuracy. One advantage of PS is that there is no shape parameter to be taken care of. In this article, MAPS is modified so PS can be used more effectively. In the original MAPS, integrated RBFs, so called particular solutions, are used. An additional integrated polynomial basis is added when PS is used. In the modified MAPS, an additional polynomial basis is directly added to the integrated RBFs without integration. The results from the modified MAPS with PS can be improved by increasing the order of PS to a certain degree or by increasing the number of collocation points. A polynomial of degree 15 or less appeared to be working well in most of our examples. Other RBFs such as MQ can be utilized in the modified MAPS as well. The performance of the proposed method is tested on a number of examples including linear and nonlinear problems in 2D and 3D. We demonstrate that the modified MAPS with PS is, in general, more accurate than other RBFs for solving general elliptic equations. [ABSTRACT FROM AUTHOR]

We consider cardinal interpolation on gridded data by using various radial basis functions associated with one or two parameters, one of which leads asymptotically to so-called flat-limits. Previously it had been shown that the classical Paley–Wiener functions can be recovered by such cardinal interpolations as the parameter tends to infinity. In this article, we extend the results by relaxing the requirements on the approximand functions from several points of view. The radial basis functions that we are concerned with and which are of special interest contain the celebrated multiquadrics, inverse multiquadrics and shifted thin-plate spline radial basis functions for instance. We also generalise the classes of admitted approximands as well as the radial basis functions to generalised multiquadrics in place of the well-known ordinary or for example inverse multiquadrics. An interesting analytical aspect of this work is that – unlike the classical Whittaker–Shannon theorem – functions (approximands) may be reproduced for the parameter c → ∞ in the generalised multiquadrics cardinal approximands, where the usual Shannon series does not converge with theses approximands due to the slow decay of the sinc-function which does not allow e.g. polynomials as approximands. In contrast to the latter, the generalised multiquadrics cardinal functions employed here decay sufficiently fast for each fixed parameter c that even polynomials may be admitted as approximands and are reproduced when then the parameter tends to infinity. [ABSTRACT FROM AUTHOR]

Gaussian processes have become a standard framework for modeling deterministic computer simulations and producing predictions of the response surface. This article investigates a new covariance function that is shown to offer superior prediction compared to the more common covariances for computer simulations of real physical systems. This is demonstrated via a gamut of realistic examples. A simple, closed-form expression for the covariance is derived as a limiting form of a Brownian-like covariance model as it is extended to some hypothetical higher-dimensional input domain, and so we term it a lifted Brownian covariance. This covariance has connections with the multiquadric kernel. Through analysis of the kriging model, this article offers some theoretical comparisons between the proposed covariance model and existing covariance models. The major emphasis of the theory is explaining why the proposed covariance is superior to its traditional counterparts for many computer simulations of real physical systems. Supplementary materials for this article are available online. [ABSTRACT FROM PUBLISHER]

In this paper, an effective simple method is proposed for approximating the solution of Fredholm integral equations systems using radial basis functions (RBFs). We present an algorithm based on interpolation by radial basis functions including multiquadrat- ics (MQs), using Legendre-Gauss-Lobatto nodes and weights. A theorem is presented to prove the convergence of the algorithm as well. Some numerical examples are illustrated and the results compared with those obtained by triangular functions (TFs), delta basis functions (DFs), block-pulse functions (BPFs), Sinc fucntion, Adomian decomposition, Haar wavelet and direct methods demonstrate the validity, accuracy and applicability of the RBF method. [ABSTRACT FROM AUTHOR]

A new shift-adaptive meshfree method for solving a class of time-dependent partial differential equations (PDEs) in a bounded domain (one-dimensional domain) with moving boundaries and nonhomogeneous boundary conditions is introduced. The radial basis function (RBF) collocation method is combined with the finite difference scheme, because, unlike with Kansa's method, nonlinear PDEs can be converted to a system of linear equations. The grid-free property of the RBF method is exploited, and a new adaptive algorithm is used to choose the location of the collocation points in the first time step only. In fact, instead of applying the adaptive algorithm on the entire domain of the problem (like with other existing adaptive algorithms), the new adaptive algorithm can be applied only on time steps. Furthermore, because of the radial property of the RBFs, the new adaptive strategy is applied only on the first time step; in the other time steps, the adaptive nodes (obtained in the first time step) are shifted. Thus, only one small system of linear equations must be solved (by LU decomposition method) rather than a large linear or nonlinear system of equations as in Kansa's method (adaptive strategy applied to entire domain), or a large number of small linear systems of equations in the adaptive strategy on each time step. This saves a lot in time and memory usage. Also, Stability analysis is obtained for our scheme, using Von Neumann stability analysis method. Results show that the new method is capable of reducing the number of nodes in the grid without compromising the accuracy of the solution, and the adaptive grading scheme is effective in localizing oscillations due to sharp gradients or discontinuities in the solution. The efficiency and effectiveness of the proposed procedure is examined by adaptively solving two difficult benchmark problems, including a regularized long-wave equation and a Korteweg-de Vries problem. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1622-1646, 2016 [ABSTRACT FROM AUTHOR]

In order to produce an accurate noise map of a city or a region, it is necessary to make noise measurements at certain locations and these measurements must be modeled with the most suitable mathematical algorithm. A homogeneous and representative distribution of the noise measurement points is the first key factor in the production of sound noise maps. The second key element is the calculation of the noise values of gridding points based on noise measurement points according to the selected mathematical calculation method and the generation of maps according to these gridding points. In this study, a noise map of the Isparta city center and its periphery was produced using inverse distance weighted (IDW), Kriging and multiquadric interpolation methods with different parameters and four grid resolution. Then, the influence of parameter selection for each method was investigated in themself by taking into account grid resolution, namely 10 ∗ 10 m, 50 ∗ 50 m, 100 ∗ 100 m and 200 ∗ 200 m, and the performance of three method with 50 ∗ 50 m grid resolution were compared with each other. In addition, the noise mapping of the city of Isparta were produced by Kriging method with respect to maximum, average and minimum noise data and they were evaluated by considering the national environmental noise thresholds. [ABSTRACT FROM AUTHOR]

This is a continuation of our earlier study of the shape parameter c contained in the multiquadrics ( − 1 ) ⌈ β 2 ⌉ ( c 2 + ‖ x ‖ 2 ) β 2 , β > 0 , and the inverse multiquadrics ( c 2 + ‖ x ‖ 2 ) β , β < 0 . In the previous two papers we presented criteria for the optimal choice of c , based on the exponential-type error bound. In this paper a new set of criteria is developed, based on the improved exponential-type error bound. This results in much sharper error estimates when c is chosen appropriately, with the same size of fill distance as in the previous two papers. What is important is that the optimal value of c can be successfully predicted without any search when fill distance is of reasonable size, making it practically useful. The drawback is that the distribution of the data points is not purely scattered. However, for practical purposes, it does not seem to be a big problem. [ABSTRACT FROM AUTHOR]

In this paper numerical meshless method for solving Fokker–Planck equation is considered. This meshless method is based on multiquadric radial basis function and collocation method to approximate the solution. Here we apply θ -weighted finite difference method. The stability analysis of the method is dealt with, using a linearized stability method. Numerical examples illustrate the validity and applicability of the method. [ABSTRACT FROM AUTHOR]

Many papers discuss the problem of recovering a function when a set of data is known on a domain Ω ⊂ R^2. Most of these papers assume that the size of the data set is large. At our knowledge, little or practically nothing is said about the case in which the sample has a size that is moderate. In this paper we tackle this problem. We indicate a way for its solution and hence we give a concrete example. Some numerical experiments are shown.

A local radial basis function meshless method (LRBFM) is developed to solve radiative heat transfer in participating media, in which multiquadric (MQ) radial basis functions (RBF) augmented with polynomial basis are employed to construct the trial functions, and the radiative transfer equation (RTE) is discretized directly at nodes by collocation method. The LRBFM belongs to a class of truly meshless methods which do not need any mesh, and can be implemented on a set of uniform or irregular nodes without nodes׳ connectivity. To improve numerical stability of LRBFM for the solution to radiative heat transfer in strongly inhomogeneous media, an upwind support domain scheme is introduced. The upwind scheme is implemented by moving the support domain of local radial basis function interpolation approximation to the opposite direction of each streamline, which can fully capture the information from upstream and improve the accuracy and stability of LRBFM. Performances of the LRBFM and upwind LRBFM (LRBFM_U) are compared with analytical solutions and other numerical results reported earlier in the literatures via a variety of problems in 1-D and 2-D geometries with strongly inhomogeneous media. It is demonstrated that the local radial basis function meshless method with upwind support domain scheme (LRBFM_U) provides high accuracy and great stability to solve radiative heat transfer in strongly inhomogeneous media. [ABSTRACT FROM AUTHOR]

Orthometric heights are used in many engineering projects. However, the heights determined by the widely-used Global Navigation Satellite System (GNSS) are ellipsoid heights. Leveling measurements conducted with the purpose of determining the orthometric heights on points are quite arduous and time-consuming processes. To be able to use ellipsoid heights in engineering projects, their transformation to orthometric heights defined in the height datum of the region is necessary. Therefore, in terms of convenience and feasibility GNSS/levelling method is preferred in determining geoid heights. This method is based on the principle of transformation of ellipsoid heights to orthometric heights. In effect, the main purpose of the method can also be regarded as the estimation of geoid undulation values for the study area. During the estimation process, polynomial (surface, curve) models are generally used. Polynomial models produce meaningful results for points which are scattered uniformly on the study area. However, for strip areas where the points scatter along a route (road etc. projects), the accuracy of the geoid heights obtained from these models is low. Therefore, different estimation techniques have to be implemented in strip areas instead of polynomial models. In this study, interpolation methods used in determining the geoid undulation of a strip area were researched and the identification of the best suitable method for the area was examined. For this purpose, geoid undulation values were calculated with the help of least-squares collocation (LSC) and radial basis functions such as Multiquadric (MQ), Thin Plate Spline (TPS) along with polynomial models, and results were presented. According to the results, it was observed that TPS, MQ, LSC methods respectively yield better results compared to polynomial methods. [ABSTRACT FROM AUTHOR]

In this paper, a numerical method based on radial basis functions finite difference (RBF-FD) has been developed for solving the time fractional convection–diffusion equation. We first approximate the time fractional derivative by FD and obtain the semi-discretized scheme. The unconditional stability and convergence of the semi-discretized scheme are given. After that, we use the multiquadric RBF to approximate the spatial derivatives in the numerical experiments. The aim of this paper is to show that the RBF-FD method is better than the classical FD method for solving our mentioned equation. Finally, two numerical examples are proposed respectively to verify the correctness of our theoretical analysis and to demonstrate the superiority of the RBF-FD method. [ABSTRACT FROM AUTHOR]

In this article, we propose two meshless collocation approaches for solving time dependent partial differential algebraic equations (PDAEs) in terms of the multiquadric quasi-interpolation schemes. In presenting the process of the solution, the error is estimated. Furthermore, the comparisons on condition numbers of the collocation matrices using different methods and the sensitivity of the shape parameter c are given. With the use of the appropriate collocation points, the method for PDAEs with index-2 is improved. The results show that the methods have some advantages over some known methods, such as the smaller condition numbers or more accurate solutions for PDAEs which has an modal index-2 or an impulse solution with index-2. Copyright © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 95-119, 2014 [ABSTRACT FROM AUTHOR]