Review
Mathematics
Archna Kumari, Vijay K. Kukreja
Summary: This article provides an overview of the widely used Hermite interpolating polynomials and their application in solving various types of differential equations. The use of Hermite interpolation has become an established tool in applied science.
Article
Mathematics
Jimmy Johansson
Summary: We prove that a global holomorphic section of O(d) on a closed complex subspace X subset of P-n has an interpolant if and only if it satisfies a set of moment conditions involving a residue current associated with a locally free resolution of O-X. Specifically, when X is a finite set of points in C-n subset of P-n, this can be interpreted as a set of linear conditions that a function on X must satisfy in order to have a polynomial interpolant of degree at most d.
COMPLEX VARIABLES AND ELLIPTIC EQUATIONS
(2022)
Article
Mathematics, Applied
Zhiying Ma, Xinxiang Li, C. S. Chen
Summary: A new Kansa method with fictitious center approach is proposed in this paper, where the radial basis function (RBF) approximation is augmented by polynomial basis functions. The proposed approach significantly improves the accuracy and stability of the previously proposed ghost point method using radial basis functions, eliminating the difficulty of selecting a good RBF shape parameter. Two numerical examples are presented to demonstrate the effectiveness and improvement of the proposed method over previous methods.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Engineering, Marine
Baiwei Feng, Chengsheng Zhan, Zuyuan Liu, Xide Cheng, Haichao Chang
Summary: The Wendland psi 3,1 (W) function was selected for hull surface modification based on radial basis functions (RBF) interpolation. A case study validated the advantages of this method, resulting in optimized hull form with reduced wave-making resistance and total resistance. The findings support the feasibility and value of RBF interpolation-based surface modification in engineering practice.
JOURNAL OF MARINE SCIENCE AND ENGINEERING
(2021)
Article
Mathematics, Applied
Phung Van Manh, Nguyen Van Trao, Phan Thanh Tung, Le Ngoc Cuong
Summary: New polynomial interpolation schemes based on Taylor and Hermite types are developed in R-n, with interpolation conditions involving real and imaginary parts of specific differential operators. Formulas for interpolation polynomials in Newton form are provided, which can be computed using an algorithm.
NUMERICAL ALGORITHMS
(2022)
Article
Computer Science, Interdisciplinary Applications
Youssef El Seblani, Elyas Shivanian
Summary: This paper introduces an effective technique called MRPHI for solving partial differential equations with Neumann boundary condition by utilizing radial point interpolation and Hermite-type interpolation techniques. The method is tested on various two-dimensional diffusion equations to demonstrate stability across different arbitrary domains over time.
ENGINEERING WITH COMPUTERS
(2021)
Review
Mathematics
Pravin Singh, Nabendra Parumasur, Shivani Singh
Summary: This paper focuses on the application of piecewise polynomial functions in numerical analysis, specifically on the use of spline functions and Hermite functions to solve problems with irregular features. By deriving the quintic Hermite basis and utilizing orthogonal collocation method, error analysis and numerical simulations were conducted to enhance theoretical results.
Article
Mathematics, Applied
Francesco Dell'Accio, Alvise Sommariva, Marco Vianello
Summary: We establish the almost sure unisolvency of interpolation by continuous random sampling with respect to any given density in spaces of multivariate almost everywhere analytic functions. Examples regarding polynomial and RBF approximation are provided.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Engineering, Aerospace
Liang Xie, Zhicong Kang, Haifeng Hong, Zhihua Qiu, Biao Jiang
Summary: A new mesh deformation method, namely the dual-restricted RBF algorithm, is proposed in this study. This algorithm improves the efficiency of the mesh deformation process by introducing a constraint function for stationary wall preservation.
AEROSPACE SCIENCE AND TECHNOLOGY
(2022)
Article
Mathematics
Mohammad Heidari, Maryam Mohammadi, Stefano De Marchi
Summary: This paper discusses the problem of choosing the scale or shape parameter of radial basis functions (RBFs) in kernel-based methods. It introduces a direct relation between the shape parameter and the curvature of RBFs at each point. Based on this relation, RBFs are characterized as scalable or unscalable, and their curvature is used to classify commonly used RBFs. The paper then proposes a curvature-based scaled RBFs method, where the shape parameters depend on the function values and approximate curvature values of the function to be approximated. Numerical experiments show that this method performs better than the standard fixed-scale basis and other shape parameter selection methods.
MATHEMATICAL MODELLING AND ANALYSIS
(2023)
Article
Mathematics
Le Ngoc Cuong, Nguyen Quang Dieu, Phung Van Manh
Summary: This paper constructs a set of explicit differential operators that are evaluated at a singular point on a quadratic hypersurface, providing a divisibility criterion. The result is then utilized to collect interpolation conditions at distinct points, yielding new regular Hermite interpolation schemes in Rn.
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS
(2023)
Article
Engineering, Multidisciplinary
Saeid Abbasbandy, Elyas Shivanian, Khalid Hammood AL-Jizani, Satya N. Atluri
Summary: In this study, an approximate formulation for a generalized form of the biharmonic problem is developed using the PSMRPI method. The method allows for arbitrary distribution of nodal points and implementation of multiple boundary conditions, especially in complex geometric domains. The technique demonstrates the validity and trustworthiness of PSMRPI in solving the generalized biharmonic problem, with comparisons to previously studied methods.
ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
(2021)
Article
Mathematics, Applied
Zywilla Fechner, Eszter Gselmann, Laszlo Szekelyhidi
Summary: This paper investigates characterization theorems for generalized moment functions on commutative groups. By describing moment functions defined on groups, introducing higher rank generalized moment functions and properties on groups, and characterizing exponential polynomials using Bell polynomials, the main result is presented. The description of generalized moment functions of higher rank defined on a commutative group as the product of an exponential and compositions of multivariate Bell polynomial and an additive function is discussed, along with corollaries for rank one generalized moment functions. The paper concludes with discussions on possible directions for further research.
RESULTS IN MATHEMATICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Youssef El Seblani, Elyas Shivanian
Summary: This paper presents a suitable method for treating partial derivative equations, specifically the Laplace equation with Robin boundary conditions. The approach used is a nodal Hermite meshless collocation technique, incorporating radial basis functions to obtain shape functions and applying Hermite interpolation technique to impose boundary conditions directly, known as MRPHI. Trustworthy results were obtained through examples demonstrating the effectiveness of the method.
ENGINEERING WITH COMPUTERS
(2021)
Article
Multidisciplinary Sciences
Deyun Zhong, Ju Zhang, Liguan Wang, Lin Bi
Summary: This paper presents an automatic modeling method for narrow vein type ore bodies based on Boolean combination constraints. By constructing implicit functions and performing Boolean operations, the method can effectively model narrow vein type ore bodies, which is of practical value.
SCIENTIFIC REPORTS
(2022)
Article
Mathematics, Applied
Y. C. Hon, R. Schaback
APPLIED MATHEMATICS AND COMPUTATION
(2015)
Article
Mathematics, Applied
Oleg Davydov, Robert Schaback
NUMERISCHE MATHEMATIK
(2016)
Article
Mathematics, Applied
Robert Schaback
NUMERISCHE MATHEMATIK
(2016)
Article
Mathematics, Applied
G. Santin, R. Schaback
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2016)
Article
Mathematics, Applied
Oleg Davydov, Robert Schaback
IMA JOURNAL OF NUMERICAL ANALYSIS
(2019)
Article
Mathematics, Applied
M. Mohammadi, R. Schaback
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2017)
Article
Mathematics
Robert Schaback
JOURNAL OF APPROXIMATION THEORY
(2018)
Article
Mathematics, Applied
Oleg Davydov, Robert Schaback
NUMERISCHE MATHEMATIK
(2018)
Article
Mathematics, Applied
Ka Chun Cheung, Leevan Ling, Robert Schaback
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2018)
Article
Mathematics, Applied
Robert Schaback
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2015)
Article
Mathematics, Applied
Mira Bozzini, Licia Lenarduzzi, Milvia Rossini, Robert Schaback
IMA JOURNAL OF NUMERICAL ANALYSIS
(2015)
Article
Mathematics, Applied
Klaus Boehmer, Robert Schaback
NUMERICAL ALGORITHMS
(2019)
Article
Mathematics, Applied
Fahimeh Saberi Zafarghandi, Maryam Mohammadi, Robert Schaback
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2019)
Proceedings Paper
Computer Science, Interdisciplinary Applications
Robert Schaback
MESHFREE METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS VIII
(2017)
Article
Mathematics, Applied
Licia Lenarduzzi, Robert Schaback
APPLIED MATHEMATICS AND COMPUTATION
(2017)
Article
Mathematics, Applied
M. S. Bruzon, T. M. Garrido, R. de la Rosa
Summary: We study a family of generalized Zakharov-Kuznetsov modified equal width equations in (2+1)-dimensions involving an arbitrary function and three parameters. By using the Lie group theory, we classify the Lie point symmetries of these equations and obtain exact solutions. We also show that this family of equations admits local low-order multipliers and derive all local low-order conservation laws through the multiplier approach.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Dohee Jung, Changbum Chun
Summary: The paper presents a general approach to enhance the Pade iterations for computing the matrix sign function by selecting an arbitrary three-point family of methods based on weight functions. The approach leads to a multi-parameter family of iterations and allows for the discovery of new methods. Convergence and stability analysis as well as numerical experiments confirm the improved performance of the new methods.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Abhishek Yadav, Amit Setia, M. Thamban Nair
Summary: In this paper, we propose a Galerkin's residual-based numerical scheme for solving a system of Cauchy-type singular integral equations using Chebyshev polynomials. We prove the well-posedness of the system and derive a theoretical error bound and convergence order. The numerical examples validate the theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Fernando Chacon-Gomez, M. Eugenia Cornejo, Jesus Medina, Eloisa Ramirez-Poussa
Summary: The use of decision rules allows for reliable extraction of information and inference of conclusions from relational databases, but the concepts of decision algorithms need to be extended in fuzzy environments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Ilhame Amirali, Gabil M. Amiraliyev
Summary: This paper considers the one-dimensional initial-boundary problem for a pseudoparabolic equation with a time delay. To solve this problem numerically, a higher-order difference method is constructed and the error estimate for its solution is obtained. Based on the method of energy estimates, the fully discrete scheme is shown to be convergent of order four in space and of order two in time. The given numerical results illustrate the convergence and effectiveness of the numerical method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Tong-tong Shang, Guo-ji Tang, Wen-sheng Jia
Summary: The goal of this paper is to investigate a class of linear complementarity problems over tensor-spaces, denoted by TLCP, which is an extension of the classical linear complementarity problem. First, two classes of structured tensors over tensor-spaces (i.e., T-R tensor and T-RO tensor) are introduced and some equivalent characterizations are discussed. Then, the lower bound and upper bound of the solutions in the sense of the infinity norm of the TLCP are obtained when the problem has a solution.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Fabio Difonzo, Pawel Przybylowicz, Yue Wu
Summary: This paper focuses on the existence, uniqueness, and approximation of solutions of delay differential equations (DDEs) with Caratheodory type right-hand side functions. It presents the construction of the randomized Euler scheme for DDEs and investigates its error. Furthermore, the paper reports the results of numerical experiments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Priyanka Roy, Geetanjali Panda, Dong Qiu
Summary: In this article, a gradient based descent line search scheme is proposed for solving interval optimization problems under generalized Hukuhara differentiability. The innovation and importance of these concepts are presented from practical and computational perspectives. The necessary condition for existence of critical point is presented in inclusion form of interval-valued gradient. Suitable efficient descent direction is chosen based on the monotonic property of the interval-valued function and specific interval ordering. Mathematical convergence of the scheme is proved under the assumption of Inexact line search. The theoretical developments are implemented with a set of interval test problems in different dimensions. A possible application in finance is provided and solved by the proposed scheme.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Zhongqian Wang, Changqing Ye, Eric T. Chung
Summary: In this paper, the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with mixed boundary conditions for elasticity equations in high contrast media is developed. The method offers advantages such as independence of target region's contrast from precision and significant impact of oversampling domain sizes on numerical accuracy. Furthermore, this is the first proof of convergence of CEM-GMsFEM with mixed boundary conditions for elasticity equations. Numerical experiments demonstrate the method's performance.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Samaneh Soradi-Zeid, Maryam Alipour
Summary: The Laguerre polynomials are a new set of basic functions used to solve a specific class of optimal control problems specified by integro-differential equations, namely IOCP. The corresponding operational matrices of derivatives are calculated to extend the solution of the problem in terms of Laguerre polynomials. By considering the basis functions and using the collocation method, the IOCP is simplified into solving a system of nonlinear algebraic equations. The proposed method has been proven to have an error bound and convergence analysis for the approximate optimal value of the performance index. Finally, examples are provided to demonstrate the validity and applicability of this technique.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Almudena P. Marquez, Maria L. Gandarias, Stephen C. Anco
Summary: A generalization of the KP equation involving higher-order dispersion is studied. The Lie point symmetries and conservation laws of the equation are obtained using Noether's theorem and the introduction of a potential. Sech-type line wave solutions are found and their features, including dark solitary waves on varying backgrounds, are discussed.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Susanne Saminger-Platz, Anna Kolesarova, Adam Seliga, Radko Mesiar, Erich Peter Klement
Summary: In this article, we study real functions defined on the unit square satisfying basic properties and explore the conditions for generating bivariate copulas using parameterized transformations and other constructions.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Lulu Tian, Nattaporn Chuenjarern, Hui Guo, Yang Yang
Summary: In this paper, a new local discontinuous Galerkin (LDG) algorithm is proposed to solve the incompressible Euler equation in two dimensions on overlapping meshes. The algorithm solves the vorticity, velocity field, and potential function on different meshes. The method employs overlapping meshes to ensure continuity of velocity along the interfaces of the primitive meshes, allowing for the application of upwind fluxes. The article introduces two sufficient conditions to maintain the maximum principle of vorticity.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Cheng Wang, Jilu Wang, Steven M. Wise, Zeyu Xia, Liwei Xu
Summary: In this paper, a temporally second-order accurate numerical scheme for the Cahn-Hilliard-Magnetohydrodynamics system of equations is proposed and analyzed. The scheme utilizes a modified Crank-Nicolson-type approximation for time discretization and a mixed finite element method for spatial discretization. The modified Crank-Nicolson approximation allows for mass conservation and energy stability analysis. Error estimates are derived for the phase field, velocity, and magnetic fields, and numerical examples are presented to validate the proposed scheme's theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Mingyu He, Wenyuan Liao
Summary: This paper presents a numerical method for solving reaction-diffusion equations in spatially heterogeneous domains, which are commonly used to model biological applications. The method utilizes a fourth-order compact alternative directional implicit scheme based on Pade approximation-based operator splitting techniques. Stability analysis shows that the method is unconditionally stable, and numerical examples demonstrate its high efficiency and high order accuracy in both space and time.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)