Partial differential equations (PDEs) are widely used in mechanics, control processes, ecological and economic systems, chemical cycling systems, and epidemiology. Although there are some numerical methods for solving PDEs, simple and efficient methods have always been the direction that scholars strive to pursue. Based on this problem, we give the meshless barycentric interpolation collocation method (MBICM) for solving a class of PDEs. Four numerical experiments are carried out and compared with other methods; the accuracy of the numerical solution obtained by the present method is obviously improved.

Hyperchaotic system, as an important topic, has become an active research subject in nonlinear science. Over the past two decades, hyperchaotic system between nonlinear systems has been extensively studied. Although many kinds of numerical methods of the system have been announced, simple and efficient methods have always been the direction that scholars strive to pursue. Based on this problem, this paper introduces another novel numerical method to solve a class of hyperchaotic system. Barycentric Lagrange interpolation collocation method is given and illustrated with hyperchaotic system () as examples. Numerical simulations are used to verify the effectiveness of the present method.

This paper numerically simulates three‐dimensional Kolmogorov model with chaotic dynamic behavior by barycentric Lagrange interpolation collocation method. Some numerical examples are studied for finding some new chaotic behaviors and demonstrating some existing chaotic dynamic behaviors of the Kolmogorov model. Results obtained by the present method indicate that the method has merits of small operations and good numerical stability.

Although some numerical methods of the Lorenz system have been announced, simple and efficient methods have always been the direction that scholars strive to pursue. Based on this problem, this paper introduces a novel numerical method to solve the Lorenz-type chaotic system which is based on barycentric Lagrange interpolation collocation method (BLICM). The system (1) is adopted as an example to elucidate the solution process. Numerical simulations are used to verify the effectiveness of the present method.

The reaction-diffusion process always behaves extremely magically, and any a differential model cannot reveal all of its mechanism. Here we show the patterns behavior can be described well by the fractional reaction-diffusion model (FRDM), which has unique properties that the integer model does not have. Numerical simulation is carried out to elucidate the attractive properties of the fractional (3+1)-dimensional Gray-Scott model, which is to model a chemical reaction with oscillation. The Fourier transform for spatial discretization and fourth-order Runge-Kutta method for time discretization are employed to illustrate the fractal reaction-diffusion process.

The fractional-order Benjamin-Bona-Mahony-Burgers (BBMB) equation is a generalization of the classical BBMB equation. It’s dynamic behaviors is much more complex than that of the corresponding integer-order BBMB equation. The main purpose of this paper is to explore the dynamic behaviors of the fractional-order BBMB equations by using the Fourier spectral method. Firstly, the numerical solution is compared with the exact solution. It is proved that the proposed method is effective and high precision for solving the spatial fractional order BBMB equation. Then, some dynamical behaviors of fractional order BBMB equations are obtained by using the present method, and some novel fractal waves of the the fractional-order BBMB equation on unbounded domain are shown.