# Symmetry methods applied to Richard's equations and problems of infiltration.

• Manal El-Kafri

Student thesis: Doctoral Thesis

### Abstract

Water resources development around the world has taken many different forms and directions since the dawn of civilization. Water shortage in arid and semiarid regions has encouraged the search for additional sources currently not exploited intensively. Hence, knowledge of the infiltration process is a requirement for understanding water management.

The main aim here is to solve the one-dimensional nonlinear time-dependent Richard's equation for water flow in an unsaturated uniform soil. The main theory of soil infiltration is introduced using a mathematical-physical approach to describe water movement in unsaturated soils. This gives rise to Richard's flow equation; which is presented for both unsaturated and also saturated soil.

Methods for solving Richard's equation by both analytical and numerical techniques are then introduced. This gives rise to a discussion of the similarity methods first used by Philip to determine analytical solutions of Richard's equation in an unsaturated soil.

This is then generalised to determine a broader class of solutions using the Lie (classical) symmetry approach. The non-classical symmetries of Bluman and Cole are also determined. Although these group methods provide the most widely applicable technique to find solutions of ordinary and partial differential equations, a large number of tedious calculations are involved. With the help of computer algebra it is shown that the determining equations for the non-classical case lead to four new highly non-linear equations which are solved in five particular cases. Each case of classical and non-classical solutions is then reduced to an ordinary differential equation and explicit solutions are produced when possible.

The potential classical and non-classical method, first suggested by Bluman, Reid and Kumei, is also discussed and presented. The potential non-classical method produced new results, which the potential classical method did not. The solution is useful as a tool by which to judge the quality of numerical methods.

A practical solution of classical (Lie/ potential) and non-classical symmetry of Richard's equation is presented. Finally, conclusions and suggestions for further work are discussed.
Date of Award Mar 2006 English

### Keywords

• Soil infiltration rate

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