Modeling Tools for Environmental Engineers and Scientists

it can be simplified by dividing throughout by α(t). Often, in many environmentalsystems, θ(t) does not change with t. Further, if G is linear in z(t), u(t),and their derivatives, then the equation is linear, and the principle of superpositioningcan be applied.The general standard form of a PDE with two independent variables x andt is as follows:whereG[u, u x , u t , u xx , u xt , u tt , (x,t), f(x,t)] = 0 (3.3)2u x = ∂ u; u t = ∂ u; u xx = ∂ u∂x∂t∂t2; u xt = ∂ u∂t2; u tt = ∂ u∂t2, (x,t)is a parameter and f(x,t) is a known function. In the context of a model, x cancorrespond to a spatial coordinate, t to time, f(x,t) to the input, θ(x,t) to thesystem parameter, and u(x,t) to the system outputs. The formulation shouldalso define the problem domain, i.e., ranges for x and t.223.2 MATHEMATICAL ANALYSISSome of the formulations identified above are tractable to an analyticalmethod of analysis, while many require a computational (also referred to asnumerical) method of analysis. Both methods of analysis can form the basisof computer-based mathematical modeling.3.2.1 ANALYTICAL METHODSIn analytical methods, the solution to a formulation is found as an expressionconsisting of the parameters and the independent variables in terms ofthe symbols. Sometimes this method of solution is referred to as parameterizedsolutions. The solution can be exact or approximate. Only for a limitedclass of formulations is it possible to find an exact analytical solution. Anapproximate solution has to be sought in other cases, such as in the modelsfor large environmental systems.For example, consider Equation (3.1), where y is the unknown, x is aknown variable, and θ is a parameter. The solution for y might or might notexist; if it does, it might not be unique. If Equation (3.1) could be rearrangedto the formA(x,)y B(x,) = 0 (3.4)a solution can be found by inverting the matrix A(x,θ) if and only if A(x,θ) isa nonsingular n × n matrix. If y appears nonlinearly, multiple solutions may© 2002 by CRC Press LLC