In this paper, a Legendre-collocation spectral method is developed for the second order Volterra integro-differential equation with pantograph delay. We provide a rigorous error analysis for the proposed method. The spectral rate of convergence for the proposed method is established in both L2-norm and L∞-norm.

This paper is concerned with the Hopf bifurcation analysis of tumor-immune system competition model with two delays. First, we discuss the stability of state points with different kinds of delays. Then, a sufficient condition to the existence of the Hopf bifurcation is derived with parameters at different points. Furthermore, under this condition, the stability and direction of bifurcation are determined by applying the normal form method and the center manifold theory. Finally, a kind of Runge-Kutta methods is given out to simulate the periodic solutions numerically. At last, some numerical experiments are given to match well with the main conclusion of this paper.

A kind of compressible miscible displacement problems which include molecular diffusion and dispersion in porous media are investigated. The mixed finite element method is applied to the flow equation, and the transport one is solved by the symmetric interior penalty discontinuous Galerkin method. Based on a duality argument, employing projection estimates and approximation properties, a posteriori residual-type hp error estimates for the coupled system are presented, which is often used for guiding adaptivity. Comparing with the error analysis carried out by Yang (Int. J. Numer. Meth. Fluids, 65(7) (2011), pp. 781–797), the current work is more complicated and challenging.

In this paper, we construct semi-discrete two-grid finite element schemes and full-discrete two-grid finite element schemes for the two-dimensional time-dependent Schrödinger equation. The semi-discrete schemes are proved to be convergent with an optimal convergence order and the full-discrete schemes, verified by a numerical example, work well and are more efficient than the standard finite element method.

This work presents a moving mesh methodology based on the solution of a pseudo flow problem. The mesh motion is modeled as a pseudo Stokes problem solved by an explicit finite element projection method. The mesh quality requirements are satisfied by employing a null divergent velocity condition. This methodology is applied to triangular unstructured meshes and compared to well known approaches such as the ones based on diffusion and pseudo structural problems. One of the test cases is an airfoil with a fully meshed domain. A specific rotation velocity is imposed as the airfoil boundary condition. The other test is a set of two cylinders that move toward each other. A mesh quality criteria is employed to identify critically distorted elements and to evaluate the performance of each mesh motion approach. The results obtained for each test case show that the pseudo-flow methodology produces satisfactory meshes during the moving process.

Invariant solutions of two-dimensional elastodynamics in linear homogeneous isotropic materials are considered via the group theoretical method. The second order partial differential equations of elastodynamics are reduced to ordinary differential equations under the infinitesimal operators. Three invariant solutions are constructed. Their graphical figures are presented and physical meanings are elucidated in some cases.

This paper is concerned with the development of an efficient algorithm for the analytic solutions of nonlinear fractional differential equations. The proposed algorithm Laplace homotopy analysis method (LHAM) is a combined form of the Laplace transform method with the homotopy analysis method. The biggest advantage the LHAM has over the existing standard analytical techniques is that it overcomes the difficulty arising in calculating complicated terms. Moreover, the solution procedure is easier, more effective and straightforward. Numerical examples are examined to demonstrate the accuracy and efficiency of the proposed algorithm.

The scattering of the open cavity filled with the inhomogeneous media is studied. The problem is discretized with a fourth order finite difference scheme and the immersed interface method, resulting in a linear system of equations with the high order accurate solutions in the whole computational domain. To solve the system of equations, we design an efficient iterative solver, which is based on the fast Fourier transformation, and provides an ideal preconditioner for Krylov subspace method. Numerical experiments demonstrate the capability of the proposed fast high order iterative solver.

This paper presents a physical investigation and mathematical analysis on mechanical behavior of the regular jugged discontinuity. In particular, we focus on the creep property of structural plane with various slope angles under different normal stress through shear creep tests of structural plane under shear stresses. According to the test results, the shear creep property of structural plane was described and the creep velocity and long-term strength of the structural plane during shear creep were also investigated. An empirical formula is finally established to evaluate shear strength of discontinuity and a modified Burger model was proposed to represent the shear deformation property during creep.