An approximate analytical solution to the Boussinesq equation is presented here. Uniform initial conditions and a step function increase of piezometric head on the boundary are assumed. The one‐dimensional problem is reduced to an ordinary differential equation through Boltzmann's transformation, and a technique exploiting some basic characteristics of the exact solution leads to an approximate polynomial solution of the problem. The technique presented here can also be applied to one‐dimensional nonlinear diffusion problems. A numerical procedure incorporating some of the analytical characteristics of the exact solution is presented. A discussion concludes the paper.

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A groundwater management problem is presented involving pumping cost minimization with both well discharges and well locations as decision variables. A grid of candidate well locations is set up and optimal arrangements of wells are sought within this discrete space. A genetic algorithm approach is presented with the following particular features: (a) A suitable scaling is applied to the objective function in order to alleviate its regionally flat behavior. (b) No penalty functions are involved in constraint handling. Instead, the feasible region is transformed into a rectangular domain. The transformation introduced is proved to be bijective. (c) A binary representation of well configurations is presented and compared to a combinatorial one. The binary representation necessitates the introduction of specially designed genetic operators. Besides purely genetic algorithms, the concept of cellular automaton is introduced as the basis of an alternative formulation of the optimization problem. The lattice of the cellular automaton provides the discrete set of candidate well positions. The well configuration is represented by a group of agents occupying an equal number of lattice sites. The agents change positions as dictated by the structure of the automaton and, also, by an associated genetic algorithm, which directs the evolution of the whole scheme toward an optimal configuration. An improved performance of this approach is noted and discussed in comparison to the purely genetic algorithm schemes of the present work. A simulated annealing approach is also applied to the same problem for comparison purposes. Finally, a new and more efficient hybrid annealing-genetic approach is introduced and discussed.

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The Boussinesq equation, supplemented with Dirichlet‐type boundary conditions, is used to describe one‐dimensional groundwater flow. After a number of transformations a nonlinear ordinary differential equation is obtained with flux as the dependent variable. Through perturbation the slope of the water profile at the origin is expressed in the form of a rapidly converging series and can, therefore, be calculated to the desired accuracy. Thus, the corresponding two‐point boundary problem is solved without iteration. Also, quantities related to physical aspects of the problem are easily calculated. Furthermore, the same treatment is directly applicable to a number of other physical and engineering situations governed by the same equations.