小波Galerkin法在非线性分岔问题求解中的应用

Application of the wavelet Galerkin method to solution of nonlinear bifurcation problems was studied through a typical Bratu problem. Firstly, 1D and 2D Bratu equations were discretized with the Coiflet based wavelet Galerkin method, then both the pseudo arc-length scheme for tracing solution curves and the extended equations for calculating limit bifurcation points were derived in the case of 1-parameter Bratu problems, similarly both the pseudo arc-length scheme for tracing solution surfaces and the extended equations for solving cusp bifurcation points were also derived in the case of 2-parameter Bratu problems. Numerical results show that, the wavelet Galerkin method not only has higher accuracy during bifurcation point calculation, but also is capable of capturing fold lines and cusp catastrophe quantitatively in the case of 2-parameter bifurcation problems. This example exhibits the specific procedure of numerical bifurcation analysis based on the wavelet Galerkin method and demonstrates its potential for capturing complex bifurcation behaviors of multi-parameter problems.