论文标题
非线性PDE中变量的功能分离：一般方法，扩散型方程的新解
Functional separation of variables in nonlinear PDEs: General approach, new solutions of diffusion-type equations
论文作者
论文摘要
该研究简要概述了非线性PDE变量功能分离方法的现有修改。它提出了一种更通用的方法，以基于具有整体术语和广义分裂原理的特殊转换，以构建应用数学和数学物理学非线性方程的精确解决方案。这种方法的有效性由包含可变系数的反应和对流术语的非线性扩散型方程说明。重点是相当通用形式的方程，该方程取决于一个，两个或三个任意函数（这种非线性PDE最难分析并找到精确的解决方案）。描述了许多新的功能分离溶液和广义的行驶波解决方案（总共显示了30多个精确的解决方案）。结果表明，在某些情况下，变量的功能分离方法比（i）基于不变的表面条件的非古典对称性减少方法更有效，以及（ii）基于单个微分约束的微分约束方法。获得的确切解决方案可用于测试数学物理学和力学的各种数值和近似分析方法。
The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied mathematics and mathematical physics, based on a special transformation with an integral term and the generalized splitting principle. The effectiveness of this approach is illustrated by nonlinear diffusion-type equations that contain reaction and convective terms with variable coefficients. The focus is on equations of a fairly general form that depend on one, two or three arbitrary functions (such nonlinear PDEs are most difficult to analyze and find exact solutions). A number of new functional separable solutions and generalized traveling wave solutions are described (more than 30 exact solutions have been presented in total). It is shown that the method of functional separation of variables can, in certain cases, be more effective than (i) the nonclassical method of symmetry reductions based on an invariant surface condition, and (ii) the method of differential constraints based on a single differential constraint. The exact solutions obtained can be used to test various numerical and approximate analytical methods of mathematical physics and mechanics.
