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In this paper, an efficient space-time backward substitution method (STBSM) is presented for solving nonlinear transient heat conduction problems in 2D and 3D functionally graded materials (FGMs). To improve the computation efficiency of the traditional backward substitution method (BSM) for transient problems, the space-time radial basis function (STRBF) and space-time trigonometric basis function (STTBF) are introduced for interpolation in the proposed method. The time dimension is regarded as a pseudo-space dimension to generate the space-time Euler metric r T in STRBF, which can be applied to transform the d -dimensional transient governing equation into (d + 1) -dimensional steady-state governing equation. Benefiting from merits of the traditional BSM, an accurate numerical solution is available with only a few of computational points. Several examples are solved to verify the accuracy and efficiency of the proposed method and its merits and flaws are discussed in the end. • A space-time method is attempted for 2D and 3D heat conduction problems in FGMs. • Space-time domain and space-time radial basis function are introduced into backward substitution method. • By introducing pseudo-space-time dimension to transform the origin transient problem into steady-state problem. • The proposed method has excellent computational efficiency. [ABSTRACT FROM AUTHOR]

A meshless numerical scheme, named as space-time backward substitution method (STBSM), is proposed for 3D unsteady advection-diffusion equations (ADEs). In this method, the numerical solution is composed by approximation of boundary points and approximation of internal domain. Here, the space-time radial basis function (STRBF) is applied to construct internal approximation which satisfies homogeneous boundary condition. While, time dimension, as a new variable in radial coordinate, is incorporated into the global space-time basis function. Different from using other common global basis functions, the STBSM with STRBF does not require the temporal discretization, which makes it easy-to program and time saving. In addition, compactly supported space-time radial basis function (CS-STRBF) is introduced to reduce the original resultant full matrix to a sparse matrix, compared with global space-time radial basis function. It can overcome ill-conditioned matrix caused by an excessive condition number. Numerical results of two examples show that the proposed method provides accurate and stable solutions with low computational cost for three dimensional advection-diffusion equations. Moreover, comparison results indicate that the accuracy of the method is 2 − 4 orders of magnitude higher than the dimension splitting element-free Galerkin (DSEFG) [19] algorithm with less time expense. [ABSTRACT FROM AUTHOR]

In this paper, an efficient localized meshless method based on the space–time Gaussian radial basis functions is discussed. We aim to deal with the left Riemann–Liouville space fractional derivative wave and damped wave equation in high-dimensional space. These significant problems as anomalous models could arise in several research fields of science, engineering, and technology. Since an explicit solution to such equations often does not exist, the numerical approach to solve this problem is fascinating. We propose a novel scheme using the space–time radial basis function with advantages in time discretization. Moreover this approach produces the (n + 1)-dimensional spatial-temporal computational domain for n-dimensional problems. Therefore the local feature, as a remarkable and efficient property, leads to a sparse coefficient matrix, which could reduce the computational costs in high-dimensional problems. Some benchmark problems for wave models, both wave and damped, have been considered, highlighting the proposed method performances in terms of accuracy, efficiency, and speed-up. The obtained experimental results show the computational capabilities and advantages of the presented algorithm.

In this paper, an efficient localized meshless method based on the space–time Gaussian radial basis functions is discussed. We aim to deal with the left Riemann–Liouville space fractional derivative wave and damped wave equation in high-dimensional space. These significant problems as anomalous models could arise in several research fields of science, engineering, and technology. Since an explicit solution to such equations often does not exist, the numerical approach to solve this problem is fascinating. We propose a novel scheme using the space–time radial basis function with advantages in time discretization. Moreover this approach produces the (n + 1)-dimensional spatial-temporal computational domain for n-dimensional problems. Therefore the local feature, as a remarkable and efficient property, leads to a sparse coefficient matrix, which could reduce the computational costs in high-dimensional problems. Some benchmark problems for wave models, both wave and damped, have been considered, highlighting the proposed method performances in terms of accuracy, efficiency, and speed-up. The obtained experimental results show the computational capabilities and advantages of the presented algorithm. [ABSTRACT FROM AUTHOR]