A fourth-order diffusion model is presented with a nonlinear reaction term to simulate some special chemical and biological phenomenon. To obtain the solutions to those problems, the nonlinear Galerkin finite element method under the framework of the Hermite polynomial function for the spatial domain is utilized. The Euler backward difference method is used to solve the equation in the temporal domain. Subject to the Dirichlet and Navier boundary conditions, the numerical experiments for Bi-flux Fisher–Kolmogorov model present excellent convergence, accuracy and acceleration behavior. Also, the numerical solutions to the Bi-flux Gray–Scott model, subject to no flux boundary conditions, show excellent convergence, accuracy and symmetry.

提出了带有非线性反应项的四阶扩散模型，以模拟某些特殊的化学和生物现象。为了获得这些问题的解决方案，利用了Hermite多项式函数框架下的非线性Galerkin有限元方法。欧拉向后差分法用于在时域中求解方程。受Dirichlet和Navier边界条件的影响，Bi-flux Fisher-Kolmogorov模型的数值实验显示了出色的收敛性，准确性和加速行为。同样，在没有磁通边界条件的情况下，双磁通灰度-斯科特模型的数值解显示出极好的收敛性，准确性和对称性。