Bifurcation Theory and Applications of Nonlinear Elliptic Equations

This direction primarily focuses on the qualitative behavior of solutions to nonlinear elliptic partial differential equations, especially the bifurcation phenomena that occur in the structure of solutions when parameters vary. By employing methods such as topology, variational and functional analysis, it explores the existence, multiplicity, and stability of solutions, and applies these findings to fields such as mathematical physics, geometric analysis, and biological modeling, revealing critical patterns and structural transitions in nonlinear phenomena.

Boundary value problem of differential equation

This direction focuses on the theory of solving ordinary and partial differential equations under given boundary conditions and the analysis of the properties of solutions. It primarily studies the existence, uniqueness, regularity, and continuous dependence on boundary data of solutions. By combining function space theory, integral equations, and the method of upper and lower solutions, it provides a mathematical foundation for steady-state processes and spatial distribution problems in physics and engineering.

Difference equation theory and numerical solution of differential equations

This field encompasses the qualitative analysis of discrete dynamical systems (difference equations) and numerical computation methods for differential equations. It includes the construction, stability, convergence, and error analysis of difference schemes, with a focus on the theory and implementation of algorithms such as finite difference methods, finite element methods, and spectral methods. The aim is to provide efficient and reliable computational approaches for computer simulation of continuous models, which are applied in fields such as scientific computing and engineering simulation.