In the frame of the traditional wavelet-Galerkin method based on thecompactly supported wavelets, it is important to calculate the so-calledconnection coefficients that are some integrals whose integrands involveproducts of wavelets, their derivatives as well as some known coefficients inconsidered differential equations. However, even for linear differentialequations with non-constant coefficient, the computation of connectcoefficients becomes rather time-consuming and often even impossible. In thispaper, we propose a generalized wavelet-Galerkin method based on the compactlysupported wavelets, which is computationally very efficient even fordifferential equations with non-constant coefficients, no matter linear ornonlinear problems. Some related mathematical theorems are proved, based onwhich the basic ideas of the generalized wavelet-Galerkin method are describedin details. In addition, some examples are used to illustrate its validity andhigh efficiency. A nonlinear example shows that the generalizedwavelet-Galerkin method is not only valid to solve nonlinear problems, but alsopossesses the ability to find new solutions of multi-solution problems. Thismethod can be widely applied to various types of both linear and nonlineardifferential equations in science and engineering.
展开▼
