International Journal of Computational Methods

In this paper, a novel approximation to the Caputo time fractional derivative of variable order (VO) $\nu (\xi ,\eta )$ ($0<\nu (\xi ,\eta )<1$) is established. Since time fractional derivatives are integral, with a weakly singular kernel the discretization on the uniform mesh may not lead to satisfactory accuracy. So, numerical approximation is constructed on nonuniform meshes based on Newton interpolation polynomial. As a result, the novel approximation can be viewed as the modification of the existing method [Shen, S., Liu, F., Chen, J., Turner, I. and Anh, V. [2012] “Numerical techniques for the variable order time fractional diffusion equation,” Appl. Math. Comput. 218(22), 10861–10870]. The truncation errors and some significant properties of coefficients are analyzed. Furthermore, we derived Scheme-I and Scheme-II with second-order accuracy in spatial direction to solve the mobile–immobile advection–dispersion model (MIMADM) based on the VO Caputo derivative. The designed scheme converts the considered problems into a linear system of equations. Moreover, the stability properties of Scheme-I are discussed in detail. Ultimately, the proposed scheme is examined on several text examples demonstrating the high precision and effectiveness of the scheme. Also, comparative study between the outcomes achieved by implementing the suggested scheme with other numerical scheme in the existing literature is provided and also the acquired numerical results of these applications indicate that the suggested technique performs extremely well in terms of reliability and capability.

Physics-informed neural network (PINN) has recently gained increasing interest in computational mechanics. This work extends the PINN to computational solid mechanics problems. Our focus will be on the investigation of various formulation and programming techniques, when governing equations of solid mechanics are implemented. Two prevailingly used physics-informed loss functions for PINN-based computational solid mechanics are implemented and examined. Numerical examples ranging from 1D to 3D solid problems are presented to show the performance of PINN-based computational solid mechanics. The programs are built via Python with TensorFlow library with step-by-step explanations and can be extended for more challenging applications. This work aims to help the researchers who are interested in the PINN-based solid mechanics solver to have a clear insight into this emerging area. The programs for all the numerical examples presented in this work are available at https://github.com/JinshuaiBai/PINN_Comp_Mech.

The radial basis function neural network (RBFNN) is a widely used tool for interpolation and prediction problems. In this paper, we propose to improve the traditional RBFNN by automatically identifying core neurons in the hidden layer, based on the $l_1$ regularization. Our proposed approach will greatly reduce the number of neurons required, which will save the memory and also the computational cost. To determine the radial parameter $𝜖$ in the RBFs, we propose to use the $K$-fold cross-validation method. Moreover, the principal component analysis (PCA) method is used to reconstruct the distance between samples for high-dimensional data sets. Numerical experiments are provided to demonstrate the effectiveness of the proposed approach.

This paper adopts the Moving Kriging (MK) interpolation meshless method to analyze the static and dynamic behaviors of stiffened functionally graded material (FGM) plate in thermal environment based on the physical neutral surface. The ribbed FGM plate is regarded as a composite structure of a FGM plate and ribs. The displacement transformation relationship between stiffeners and FGM plates is obtained through the displacement compatible conditions and MK interpolation. The meshfree model for ribbed FGM plate is obtained by superimposing the total energy of the FGM plate and the stiffeners based on the first-order shear deformation theory (FSDT) and physical neutral surface. The nonlinear temperature field along thickness direction is introduced into the meshless model of stiffened FGM plate. The equations governing the bending and free vibration of the ribbed FGM plate in thermal environment are obtained according to the principle of Minimum Potential Energy and Hamilton’s Principle. Thereafter, several ribbed FGM plate examples in different temperatures and with different locations of ribs are calculated. The results are compared with those given by the ABAQUS and literature. The results show that the effectiveness and accuracy of the proposed method in analyzing the ribbed FGM plate in thermal environment.

Mathematical models for measuring pollutants, by predicting the amount of air quality elements, play an important role to protect the human health. As one of these models, delay Volterra integral equations are applied to simulate a network of sensors with past memory to evaluate the emissions of pollutants in the air. This paper presents a computational method to solve these types of delay integral equations using the discrete Galerkin scheme together with the moving least squares (MLS) approach as basis. The MLS is an effective technique to estimate an unknown function which includes a locally weighted least squares polynomial fitting over a small set of all points. The composite Gauss–Legendre quadrature formula is utilized to compute integrals appearing in the proposed method. Since the scheme is constructed on a local scattered data approximation, its algorithm is attractive and easy to run on a computer with normal features. The error estimation and convergence rate of the method are provided. Finally, numerical examples illustrate the efficiency and accuracy of the new technique and confirm the theoretical results obtained in the error analysis.

The paper presents a uniformly convergent finite difference method based on the defect correction technique to solve parabolic singular perturbation problems with discontinuous convection coefficient and source. The solution to the problem exhibits interior layer across the discontinuity and demonstrates turning point behavior. The simultaneous presence of perturbation parameters and discontinuity makes the problem stiff. A higher-order method is developed using an implicit difference scheme in time on a uniform mesh and a combination of the upwind difference method and the central difference scheme over an adaptive mesh in space. The method involves iteratively solving increasingly accurate discrete problems by computing and using the defect to correct the approximate solution. Parameter uniform error estimates show uniform convergence of first-order in time and second-order in space. Numerical experiments confirm the accuracy of the proposed scheme and support the theoretical analysis.