Study On Analytic Approximate Solutions For Differential Systems With Symbolic Computation

The study on methods of solving nonlinear differential equations is a significant part of research on nonlinear science. Adomian decomposition method is an efficient way to construct analytic approximate solutions of nonlinear differential equations, and it is widely used thanks to its simple idea. However, the analytic approximate solution-s obtained by using the Adomian decomposition method are usually convergent in a limited region because of the rapid expansion of middle expressions in symbolic compu-tation. Recently some scholars combine the Laplace transformation method with the Adomian decomposition method and call it Laplace decomposition method. Laplace decomposition method is more efficient than the Adomian decomposition method. This dissertation extends the application of Laplace decomposition method to non-linear partial differential equations, and proposes a modified Laplace decomposition method to overcome a flaw of original Laplace decomposition method. Besides, a software named LDMP is developed to automatically deliver analytic approximate so-lutions of nonlinear differential systems based on those two methods. The main points of this dissertation are summarized bellow.In chapter 1, research background related to this dissertation is introduced, the development of methods for solving nonlinear differential systems is reviewed, and the achievements and state of the art in this field at home and abroad are concluded briefly.In chapter 2, the Laplace decomposition method and its modified algorithm are introduced. Firstly, the idea and process of the directly expanded Laplace decomposi-tion method are presented. Then, a flaw of Laplace decomposition method is analyzed through the specific examples, and the modified Laplace decomposition method is pro-posed. By solving different types of equations, the scope of application and advantages and disadvantages of the algorithms are compared. It can be seen that Laplace de-composition method has a good effect on broadening the convergence region of series solutions and improving the accuracy of series solutions.In chapter 3, the software LDMP is introduced. The interfaces of software and functions and implementation skills of main modules are explained. By solving several different types of equations, the software and its algorithms are validated to be effective. LDMP is user-friendly and convenient, and the idea of parallelization is adopted in part of the programming. One just needs to input the equations and possible initial and boundary conditions in correct format, then LDMP can output the obtained analytic approximate solutions automatically and give the comparison curves between solutions at different orders and their error curves, which furtherly evidence the validity of obtained results.