一类四阶非线性薄膜方程的对称约化及数值模拟

Numerical simulations and symmetry reductions for a class of fourth-order nonlinear thin film equations

  • 摘要: 为深入理解一类四阶非线性薄膜方程的特性,并揭示其解的变化规律,求解了其精确解.采用条件Lie-Bäcklund对称进行了对称分析与约化,获得了方程的广义分离变量解;运用Maple软件进行了符号计算及数值求解,并利用MATLAB软件实现了解的可视化表达,得到了该方程的精确解,并绘制了相应的二维和三维图,直观展现了解的动态行为与变化特征.通过所得的精确解可准确描述四阶非线性薄膜方程的变化规律,以期为该类方程的进一步研究提供参考.

     

    Abstract: Abstract: To deeply understand the characteristics of a class of fourth - order nonlinear thin - film equations and to reveal the evolutionary behavior of their solutions, exact solutions of the equations were derived. The conditional Lie - Bäcklund symmetry was adopted for symmetry analysis and reduction, and the generalized separation variable solutions of the equations were obtained. Maple software was used for symbolic computation and numerical solutions, and MATLAB software was utilized to realize the visual representation of the solutions. The exact solutions of the equations were obtained, and the corresponding two - dimensional and three - dimensional graphs were plotted, which intuitively showed the dynamic behaviors and evolutionary characteristics of the solutions. The obtained exact solutions can accurately describe the evolutionary laws of the fourth - order nonlinear thin - film equations, aiming to provide a reference for the further research of this type of equations.

     

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