Weiwu Jiang¹, Xiaowei Gao^1,2,*

CMES-Computer Modeling in Engineering & Sciences, Vol.140, No.1, pp. 41-76, 2024, DOI:10.32604/cmes.2024.048313

Abstract The collocation method is a widely used numerical method for science and engineering problems governed by partial differential equations. This paper provides a comprehensive review of collocation methods and their applications, focused on elasticity, heat conduction, electromagnetic field analysis, and fluid dynamics. The merits of the collocation method can be attributed to the need for element mesh, simple implementation, high computational efficiency, and ease in handling irregular domain problems since the collocation method is a type of node-based numerical method. Beginning with the fundamental principles of the collocation method, the discretization process in the continuous domain is elucidated, and how… More >

Sergiy Reutskiy¹, Yuhui Zhang^2,*, Jun Lu^3,*, Ciren Pubu⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.139, No.2, pp. 1583-1612, 2024, DOI:10.32604/cmes.2023.044878

Abstract This paper presents an efficient numerical technique for solving multi-term linear systems of fractional ordinary differential equations (FODEs) which have been widely used in modeling various phenomena in engineering and science. An approximate solution of the system is sought in the form of the finite series over the Müntz polynomials. By using the collocation procedure in the time interval, one gets the linear algebraic system for the coefficient of the expansion which can be easily solved numerically by a standard procedure. This technique also serves as the basis for solving the time-fractional partial differential equations (PDEs). The modified radial basis… More >

Tao Hu¹, Cheng Huang², Sergiy Reutskiy^3,*, Jun Lu⁴, Ji Lin^5,*

CMES-Computer Modeling in Engineering & Sciences, Vol.138, No.2, pp. 1521-1548, 2024, DOI:10.32604/cmes.2023.030449

Abstract A novel accurate method is proposed to solve a broad variety of linear and nonlinear (1+1)-dimensional and (2+1)- dimensional multi-term time-fractional partial differential equations with spatial operators of anisotropic diffusivity. For (1+1)-dimensional problems, analytical solutions that satisfy the boundary requirements are derived. Such solutions are numerically calculated using the trigonometric basis approximation for (2+1)-dimensional problems. With the aid of these analytical or numerical approximations, the original problems can be converted into the fractional ordinary differential equations, and solutions to the fractional ordinary differential equations are approximated by modified radial basis functions with time-dependent coefficients. An efficient backward substitution strategy that… More >

Xiaowei Gao^1,*, Huayu Liu¹, Weilong Fan¹

The International Conference on Computational & Experimental Engineering and Sciences, Vol.27, No.1, pp. 1-1, 2023, DOI:10.32604/icces.2023.09510

Abstract In this paper, a novel numerical method, Zonal Free Element Method (ZFLM), is proposed and used to solve thermal-mechanical problems composed of multiple and functionally graded materials. ZFLM is a collocation method, in which two or three lines in 2D or 3D problems, called as line-set, are used at each node to establish the solution scheme solving engineering problems governed by partial differential equations. In ZFLM, the Lagrange polynomial is adopted to approximate physical variables varying over each line of the line-set. The first-order partial derivative is derived by using a directional derivative technique along arclength of a line, and… More >

Zhuojia Fu^1,*, Qiang Xi², Wenzhi Xu¹

The International Conference on Computational & Experimental Engineering and Sciences, Vol.27, No.1, pp. 1-1, 2023, DOI:10.32604/icces.2023.09393

Abstract In the past few decades, although traditional computational methods such as finite element have been successfully used in many scientific and engineering fields, they still face several challenging problems such as expensive computational cost, low computational efficiency, and difficulty in mesh generation in the numerical simulation of wave propagation under infinite domain, large-scale-ratio structures, engineering inverse problems and moving boundary problems. This paper introduces a class of collocation discretization techniques based on physical-informed kernel function (PIKF) to efficiently solve the above-mentioned problems. The key issue in the physical-informed kernel function collocation methods (PIKFCMs) is to construct the related basis functions,… More >

T. M. Agbaje^a,b, S. S. Motsa^a,* , S. Mondal^c,† , P. Sibanda^a

Frontiers in Heat and Mass Transfer, Vol.9, pp. 1-13, 2017, DOI:10.5098/hmt.9.36

Abstract In this work, we present a compliment of the spectral perturbation method (SPM) for solving nonlinear partial differential equations (PDEs) with applications in fluid flow problems. The (SPM) is a series expansion based approach that uses the Chebyshev spectral collocation method to solve the governing sequence of differential equation generated by the perturbation series approximation. Previously the SPM had the limitation of being used to solve problems with small parameters only. This current investigation seeks to improve the performance of the SPM by doing the series expansion about a large parameter. The new method namely the large parameter spectral perturbation… More >

A.Wakif^a,* , Z. Boulahia^a, A. Amine^b , I.L. Animasaun^c , M. I. Afridi^d, M. Qasim^d , R. Sehaqui^a

Frontiers in Heat and Mass Transfer, Vol.12, pp. 1-15, 2019, DOI:10.5098/hmt.12.3

Abstract The significance of an externally applied magnetic field and an imposed negative temperature gradient on the onset of natural convection in a thin horizontal layer of alumina-water nanofluid for various sizes of spherical alumina nanoparticles (e.g., 30 More >

Jingwen Ren¹, Hongwei Lin^1,2,*

CMES-Computer Modeling in Engineering & Sciences, Vol.136, No.3, pp. 2957-2984, 2023, DOI:10.32604/cmes.2023.025983

Abstract Isogeometric analysis (IGA) is introduced to establish the direct link between computer-aided design and analysis. It is commonly implemented by Galerkin formulations (isogeometric Galerkin, IGA-G) through the use of nonuniform rational B-splines (NURBS) basis functions for geometric design and analysis. Another promising approach, isogeometric collocation (IGA-C), working directly with the strong form of the partial differential equation (PDE) over the physical domain defined by NURBS geometry, calculates the derivatives of the numerical solution at the chosen collocation points. In a typical IGA, the knot vector of the NURBS numerical solution is only determined by the physical domain. A new perspective… More > Graphic Abstract

Mohammad Aslefallah¹, Şuayip Yüzbaşi², Saeid Abbasbandy^1,*

CMES-Computer Modeling in Engineering & Sciences, Vol.136, No.2, pp. 1687-1706, 2023, DOI:10.32604/cmes.2023.025647

Abstract In this work, the exponential approximation is used for the numerical simulation of a nonlinear SITR model as a system of differential equations that shows the dynamics of the new coronavirus (COVID-19). The SITR mathematical model is divided into four classes using fractal parameters for COVID-19 dynamics, namely, susceptible (S), infected (I), treatment (T), and recovered (R). The main idea of the presented method is based on the matrix representations of the exponential functions and their derivatives using collocation points. To indicate the usefulness of this method, we employ it in some cases. For error analysis of the method, the… More > Graphic Abstract

Zhuowan Fan¹, Yancheng Liu¹, Anyu Hong^1,*, Fugang Xu^1,*, Fuzhang Wang²

CMES-Computer Modeling in Engineering & Sciences, Vol.132, No.1, pp. 341-355, 2022, DOI:10.32604/cmes.2022.019715

Abstract In this work, the localized method of fundamental solution (LMFS) is extended to Signorini problem. Unlike the traditional fundamental solution (MFS), the LMFS approximates the field quantity at each node by using the field quantities at the adjacent nodes. The idea of the LMFS is similar to the localized domain type method. The fictitious boundary nodes are proposed to impose the boundary condition and governing equations at each node to formulate a sparse matrix. The inequality boundary condition of Signorini problem is solved indirectly by introducing nonlinear complementarity problem function (NCP-function). Numerical examples are carried out to validate the reliability… More >