Narrow Results

In this paper, we introduce a renormalization (TR) method based on Taylor expansion, which is used to solve nonlinear vibration problems and soliton equations. It provides a new mathematical tool for obtaining the asymptotic analytic solutions of these equations. By the TR method, we solve the Van der Pol equation and the regularized long-wave equation, which obtain the corresponding asymptotic solutions, respectively. For the first equation, compared to some existing perturbation methods, the TR method can obtain its solution without relying on physical intuition and prior assumptions. For the second equation, due to the lack of small parameter, other perturbation methods are difficult to solve for it, and the process of solving its exact solutions is relatively complicated. However, the TR method can easily and quickly obtain its asymptotic solution. The numerical comparison between the asymptotic solution obtained and the solution obtained by other methods verified the effectiveness and accuracy of the proposed method. The results show that the TR method is a simple and powerful mathematical tool for solving nonlinear vibration problems and soliton equations, which has important application value.

In this paper we apply the homotopy analysis method (HAM) to study the van der Pol equation with a linear delayed feedback. The paper focuses on the calculation of periodic solutions and associated bifurcations, Hopf, double Hopf, Neimark–Sacker, etc. In particular, we discuss the behavior of the systems in the neighborhoods of double Hopf points. We demonstrate the applicability of HAM to the analysis of bifurcation and stability.

In this paper, we investigate the dynamics of the periodic motion for switched van der Pol equation with impulsive effect, utilizing the theory of mapping dynamics in switching systems. For the optimized problem, we consider such impulsive dynamical model as switched system and analyze its features from a discontinuous point of view. Then, conceptions of switching sets as well as discrete mappings are briefly reviewed. By constructing generic mappings, we analyze the flow's periodic behaviors from the perspective of mapping structures. Finally, we apply our analysis and criterion to a specific impulsive model at fixed points and the periodic motions with impulse to the boundary are illustrated.

In this paper, we study a new numerical method of order 4 in space and time based on half-step discretization for the solution of three-space dimensional quasi-linear hyperbolic equation of the form $u_{tt}=A(x,y,z,t,u)u_{xx}+B(x,y,z,t,u)u_{yy}+C(x,y,z,t,u)u_{zz}+f(x,y,z,t,u,u_x,u_y,u_z,u_t)$, $A>0$, $B>0$, $C>0$, $0<x$, $y$, $z<1$, $t>0$ subject to prescribed appropriate initial and Dirichlet boundary conditions. The scheme is compact and requires nine evaluations of the function $f.$ For the derivation of the method, we use two inner half-step points each in $x$-, $y$-, $z$- and $t$-directions. The proposed method is directly applicable to three-dimensional (3D) hyperbolic equations with singular coefficients, which is main attraction of our work. We do not require extra grid points for computation. The proposed method when applied to 3D damped wave equation is shown to be unconditionally stable. Operator splitting method is used to solve 3D linear hyperbolic equations. Few benchmark problems are solved and numerical results are provided to support the theory which is discussed in this paper.

In this paper, we propose a new high accuracy discretization based on the ideas given by Chawla and Shivakumar for the solution of two-space dimensional nonlinear hyperbolic partial differential equation of the form u_tt = A(x, y, t)u_xx + B(x, y, t)u_yy + g(x, y, t, u, u_x, u_y, u_t), 0 < x, y < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions. We use only five evaluations of the function g and do not require any fictitious points to discretize the differential equation. The proposed method is directly applicable to wave equation in polar coordinates and when applied to a linear telegraphic hyperbolic equation is shown to be unconditionally stable. Numerical results are provided to illustrate the usefulness of the proposed method.