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Radial Basis Functions
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Radial Basis Functions

A special issue of Mathematical and Computational Applications (ISSN 2297-8747).

Deadline for manuscript submissions: 31 December 2025 | Viewed by 8820

Special Issue Editors

Prof. Dr. Benny Yiu-Chung Hon

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Guest Editor
Department of Mathematics, City University of Hong Kong, Hong Kong, China
Interests: meshfree methods; inverse problems; computational mechanics
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Zhuojia Fu

E-Mail Website
Guest Editor
College of Mechanics and Engineering Science, Hohai University, Nanjing 210098, China
Interests: acoustic and elastic waves; inverse problems; RBF-based meshless methods; Trefftz method; boundary element method
Special Issues, Collections and Topics in MDPI journals
Dr. Junpu Li

E-Mail Website
Guest Editor
School of Mechanics and Safety Engineering, Zhengzhou University, Zhengzhou 450001, China
Interests: radial basis functions; computational mechanics; computational wave dynamic

Special Issue Information

Dear Colleagues,

Radial basis functions (RBFs) are a type of real-value function which invariably involve only one-dimensional distance. With their merit of meshless and independence from dimensionality and geometric complexity, RBFs have gained substantial attention from various scientific computing and engineering applications, such as the recovery of functions from scattered data, the numerical solution of partial differential equations (PDEs), ill-posed and inverse problems, neural networks, machine learning algorithms, and so on. This Special Issue will present recent research results on radial basis functions and their applications in engineering and sciences.

Prof. Dr. Benny Y. C. Hon
Prof. Dr. Zhuojia Fu
Dr. Junpu Li
Guest Editors

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematical and Computational Applications is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

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Published Papers (5 papers)

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Research

22 pages, 1346 KB  
Article
A Hybrid Numerical Framework Based on Radial Basis Functions and Finite Difference Method for Solving Advection–Diffusion–Reaction-Type Interface Models
by Muhammad Asif, Javairia Gul, Mehnaz Shakeel and Ioan-Lucian Popa
Math. Comput. Appl. 2026, 31(1), 1; https://doi.org/10.3390/mca31010001 - 19 Dec 2025
Abstract
Advection–diffusion–reaction-type interface models have wide-ranging applications in environmental science, chemical engineering, and biological systems, particularly in modeling pollutant transport in groundwater, reactive flows, and drug diffusion across biological membranes. This paper presents a novel numerical method for the solution of these models. The [...] Read more.
Advection–diffusion–reaction-type interface models have wide-ranging applications in environmental science, chemical engineering, and biological systems, particularly in modeling pollutant transport in groundwater, reactive flows, and drug diffusion across biological membranes. This paper presents a novel numerical method for the solution of these models. The proposed method integrates the meshless collocation technique with the finite difference method. The temporal derivative is approximated using a finite difference scheme, while spatial derivatives are approximated using radial basis functions. The interface across the fixed boundary is treated with discontinuous diffusion, advection, and reaction coefficients. The proposed numerical scheme is applied to both linear and non-linear models. The Gauss elimination method is used for the linear models, while the quasi-Newton linearization method is employed to address the non-linearity in non-linear cases. The L∞ error is computed for varying numbers of collocation points to assess the method’s accuracy. Furthermore, the performance of the method is compared with the Haar wavelet collocation method and the immersed interface method. Numerical results demonstrate that the proposed approach is more efficient, accurate, and easier to implement than existing methods. The technique is implemented in MATLAB R2024b software. Full article
(This article belongs to the Special Issue Radial Basis Functions)
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20 pages, 3456 KB  
Article
RBF-Based Meshless Collocation Method for Time-Fractional Interface Problems with Highly Discontinuous Coefficients
by Faisal Bilal, Muhammad Asif, Mehnaz Shakeel and Ioan-Lucian Popa
Math. Comput. Appl. 2025, 30(6), 133; https://doi.org/10.3390/mca30060133 - 5 Dec 2025
Viewed by 280
Abstract
Time-fractional interface problems arise in systems where interacting materials exhibit memory effects or anomalous diffusion. These models provide a more realistic description of physical processes than classical formulations and appear in heat conduction, fluid flow, porous media diffusion, and electromagnetic wave propagation. However, [...] Read more.
Time-fractional interface problems arise in systems where interacting materials exhibit memory effects or anomalous diffusion. These models provide a more realistic description of physical processes than classical formulations and appear in heat conduction, fluid flow, porous media diffusion, and electromagnetic wave propagation. However, the presence of complex interfaces and the nonlocal nature of fractional derivatives makes their numerical treatment challenging. This article presents a numerical scheme that combines radial basis functions (RBFs) with the finite difference method (FDM) to solve time-fractional partial differential equations involving interfaces. The proposed approach applies to both linear and nonlinear models with constant or variable coefficients. Spatial derivatives are approximated using RBFs, while the Caputo definition is employed for the time-fractional term. First-order time derivatives are discretized using the FDM. Linear systems are solved via Gaussian elimination, and for nonlinear problems, two linearization strategies, a quasi-Newton method and a splitting technique, are implemented to improve efficiency and accuracy. The method’s performance is assessed using maximum absolute and root mean square errors across various grid resolutions. Numerical experiments demonstrate that the scheme effectively resolves sharp gradients and discontinuities while maintaining stability. Overall, the results confirm the robustness, accuracy, and broad applicability of the proposed technique. Full article
(This article belongs to the Special Issue Radial Basis Functions)
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13 pages, 1838 KB  
Article
Application of Generalized Finite Difference Method and Radial Basis Function Neural Networks in Solving Inverse Problems of Surface Anomalous Diffusion
by Luchuan Shi and Qiang Xi
Math. Comput. Appl. 2025, 30(1), 7; https://doi.org/10.3390/mca30010007 - 9 Jan 2025
Cited by 1 | Viewed by 1248
Abstract
In this study, a new hybrid method based on the generalized finite difference method (GFDM) and radial basis function (RBF) neural network technologies is developed to solve the inverse problems of surface anomalous diffusion. Specifically, the GFDM is utilized to compute the time-fractional [...] Read more.
In this study, a new hybrid method based on the generalized finite difference method (GFDM) and radial basis function (RBF) neural network technologies is developed to solve the inverse problems of surface anomalous diffusion. Specifically, the GFDM is utilized to compute the time-fractional derivative model on the surface, whereas RBF neural networks are employed to invert the diffusion coefficient, source term coefficient, and the fractional order within the anomalous diffusion equation governing the surface. The results of four examples show that for the three parameters of diffusion coefficient, source term coefficient, and fractional order, the errors of inversion results are in the order of 10−2 under different conditions. Therefore, this method can obtain the required parameters quickly and accurately under different conditions. Full article
(This article belongs to the Special Issue Radial Basis Functions)
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16 pages, 4439 KB  
Article
Simulation of Temperature Field in Steel Billets during Reheating in Pusher-Type Furnace by Meshless Method
by Qingguo Liu, Umut Hanoglu, Zlatko Rek and Božidar Šarler
Math. Comput. Appl. 2024, 29(3), 30; https://doi.org/10.3390/mca29030030 - 24 Apr 2024
Cited by 4 | Viewed by 3013
Abstract
Using a meshless method, a simulation of steel billets in a pusher-type reheating furnace is carried out for the first time. The simulation represents an affordable way to replace the measurements. The heat transfer from the billets with convection and radiation is considered. [...] Read more.
Using a meshless method, a simulation of steel billets in a pusher-type reheating furnace is carried out for the first time. The simulation represents an affordable way to replace the measurements. The heat transfer from the billets with convection and radiation is considered. Inside each of the billets, the heat diffusion equation is solved on a two-dimensional central slice of the billet. The diffusion equation is solved in a strong form by the Local Radial Basis Function Collocation Method (LRBFCM) with explicit time-stepping. The ray tracing procedure solves the radiation, where the view factors are computed with the Monte Carlo method. The changing number of billets in the furnace at the start and the end of the loading and unloading of the furnace is considered. A sensitivity study on billets’ temperature evolution is performed as a function of a different number of rays used in the Monte Carlo method, different stopping times of the billets in the furnace, and different spacing between the billets. The temperature field simulation is also essential for automatically optimizing the furnace’s productivity, energy consumption, and the billet’s quality. For the first time, the LRBFCM is successfully demonstrated for solving such a complex industrial problem. Full article
(This article belongs to the Special Issue Radial Basis Functions)
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30 pages, 10410 KB  
Article
Assessment of Local Radial Basis Function Collocation Method for Diffusion Problems Structured with Multiquadrics and Polyharmonic Splines
by Izaz Ali, Umut Hanoglu, Robert Vertnik and Božidar Šarler
Math. Comput. Appl. 2024, 29(2), 23; https://doi.org/10.3390/mca29020023 - 17 Mar 2024
Cited by 9 | Viewed by 2545
Abstract
This paper aims to systematically assess the local radial basis function collocation method, structured with multiquadrics (MQs) and polyharmonic splines (PHSs), for solving steady and transient diffusion problems. The boundary value test involves a rectangle with Dirichlet, Neuman, and Robin boundary conditions, and [...] Read more.
This paper aims to systematically assess the local radial basis function collocation method, structured with multiquadrics (MQs) and polyharmonic splines (PHSs), for solving steady and transient diffusion problems. The boundary value test involves a rectangle with Dirichlet, Neuman, and Robin boundary conditions, and the initial value test is associated with the Dirichlet jump problem on a square. The spectra of the free parameters of the method, i.e., node density, timestep, shape parameter, etc., are analyzed in terms of the average error. It is found that the use of MQs is less stable compared to PHSs for irregular node arrangements. For MQs, the most suitable shape parameter is determined for multiple cases. The relationship of the shape parameter with the total number of nodes, average error, node scattering factor, and the number of nodes in the local subdomain is also provided. For regular node arrangements, MQs produce slightly more accurate results, while for irregular node arrangements, PHSs provide higher accuracy than MQs. PHSs are recommended for use in diffusion problems that require irregular node spacing. Full article
(This article belongs to the Special Issue Radial Basis Functions)
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