Abstract
There are a lot of problems in civil and structure engineering science that can be described by ordinary differential equation (ODEs). Especially the vibration problems of beam and column are important issues, which can be written as a Fourth-order differential equation. Structure elements like beam or column with their structure bearings will compose the Boundary Value Problems(BVPs).Group Preserving Schemes (GPS) and Lie-group SL(4,R) shooting methods are numerical methods which can easily solve the linear and nonlinear ODEs. Now we try to use these methods to solve the engineering numerical problem like beam vibration or deflection problems. Furthermore, there are some eigenvalue problems in these ODEs that we will also try to solve. In the past, it took a lot of time to solve these ODEs. We will try a better Lie algebra in Lie-group shooting method and make our numerical analysis more efficiency. Finally, we will observe that these numerical methods have good results in solving these boundary value problems, and we hope it can be applied on the development of the civil engineering or other science in the future.
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