#### Year

2002

#### Degree Name

Doctor of Philosophy

#### Department

School of Mathematics and Applied Statistics

#### Recommended Citation

Roman-Miller, Lance, Exact integration of nonlinear differential equations, Doctor of Philosophy thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2002. http://ro.uow.edu.au/theses/2047

#### Abstract

This thesis has developed the theory of integration of ordinary differential equations by the method of integrating factors using the Picard method and the linearity of the defining equations of the integrating factors.

Work on series investigations in this thesis was in part aided by computations performed using Waterloo Maple Software. In this context and in the context of future research it is interesting to note the comments of Dahlquist [29] on the merits of such investigations:

"Series expansions are a very important aid in numerical calculations, especially for quick estimates made in hand calculation—for example, in evaluating functions, integrals, or derivatives. Solutions to differential equations can often be expressed in terms of series expansions. Since the advent of computers, it has, however, become more common to treat differential equations directly, using difference approximations instead of series expansions. But in connection with the development of automatic methods for formula manipulation, one can anticipate renewed interest for series methods. These methods have some advantages, especially in multidimensional problems."

The following topics have been considered in this thesis:

1. Abel's contribution to the theory of integrating factors.

2. The works of A R Forsyth and E L Ince and their relevance to the exact solution of nonlinear differential equations.

3. The method of exact solution of nonlinear differential equations by integrating factors has been examined by considering the solution of the Van der Pohl equation.

4. Picard's Method has been examined in great detail together with a brief examination of some of the historical analysis of the method associated with Picard's method and classical methods of solution.

5. The method of solution of differential equations by integrating factors has been used to construct integrating factors for systems of nonlinear differential equations. The solution method relies on the linearity of the equations for the integrating factor to recursively generate solutions, this property was used by Abel in his work on nonlinear differential equations [1]. In many cases the large volume of calculations has meant that details of computations have been placed in the appendices (see Appendix B).

6. Consideration of group theory is undertaken for the purpose of noting the link between integrating factors and group methods and also by way of noting that the Picard method can also provide further insight into the structure of the non classical group.

7. Series solution techniques in this thesis have been supplemented using series acceleration techniques such Pade approximants and Shanks transformations.

8. A wide variety of ordinary differential equations which include the Lorenz equation, the Thomas-Fermi equation, the Blasius equation, equations of the Emden type, Langmuir's equation, Hille's equation, Frommer's equation, the One Dimensional Poisson Boltzmann equation and Volterra's equation. In addition to these equations a wide variety of systems of nonlinear ordinary differential equations taken from the study of transport processes has also been solved and a summary of the results of their solutions is contained in Appendix B. Roman-Miller and Broadbridge [78] apply the techniques to the solution of the reduced Fisher equation, Blasius equation and the Lorenz model (see Appendix A).