This thesis is concerned with a mathematical study of several problems involving infiltration from irrigation channels into an unsaturated homogeneous soil. All the problems considered are two dimensional and are solved numerically by employing boundary integral equation techniques. In the first chapter I introduce some of the literature and ideas surrounding my thesis. Some background information is stated followed by an outline of the thesis and a list of author’s published works that support the material in the thesis. Full descriptions of the fundamental equations used throughout the thesis are provided in chapter 2. Chapter 3 contains the first problem considered in this thesis which is infiltration from various shapes of single and periodic irrigation channels. Specifically strip, semi-circular, rectangular and v shaped channels. The solutions are obtained using the boundary element technique. The solutions are then compared with the results obtained by Batu [14] for single and periodic strip sources. In chapter 4 a boundary integral equation method is adopted for the solution of flow from single and periodic semi-circular channels into a soil containing impermeable inclusions. The impermeable inclusions considered are of rectangular, circular and square shapes. The aim is to observe how the various shapes of inclusions can affect the direction of the flow particularly in the region adjacent to the zone where plant roots would be located. Chapter 5 solves the problem of infiltration from single and periodic semicircular irrigation channels into a soil containing impermeable layers. A modification is made to the boundary integral equation in order to include the impermeable layers with the integration over the layers involving Hadamard finite-part integrals. The objective of the work is to investigate how the number and the depth of the impermeable layers affects the flow. Chapter 6 employs a particular Green’s function in the boundary integral equation. The Green’s function is useful for flow from a single channel since it removes the need to evaluate the boundary integral along the soil surface outside the irrigation channel. A time dependent infiltration problem is considered in chapter 7. The Laplace transform is applied to the governing equations and the boundary integral equation technique is used to solve the resulting partial differential equation. The Laplace transform is then inverted numerically to obtain the time dependent values of the matric flux potential. / Thesis (Ph.D.) - University of Adelaide, School of Mathematical Sciences, 2008

It is generally recognized that one of the most important factors limiting crop production is soil nitrogen. Farmers today are using more nitrogen fertilizers than ever before. All indications show that this usage will probably continue to increase. With the expanded use of nitrogen, it has become desirable to increase the efficiency from nitrogen fertilizers. One factor which will affect the efficiency is the position of the nitrogen in the soil in relation to the plant roots.