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This 2 page Class Notes was uploaded by Ara Hartmann on Tuesday October 13, 2015. The Class Notes belongs to CE421 at Lafayette College taught by DavidBrandes in Fall. Since its upload, it has received 29 views. For similar materials see /class/222308/ce421-lafayette-college in Civil Engineering at Lafayette College.
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Date Created: 10/13/15
A Review of Some Useful Mathematics CE 424 Groundwater Hydrology Firstorder Ordina Differential Eguations ODEsl Firstorder means the highest derivative is one ordinary indicates there is only one independent variable and the d derivatives look like t 1 Separable dy f tgy dt dy 7 I f0 dt C onst Cansl depends on the initial or boundary condition gy Note Boundary conditions refer to constraints at the physical boundary of the domain eg h 0 at x 0 Initial conditions refer to time constraints on the solution which are usually specified at I 0 or at I 00 Ex firstorder reaction from CE 221 dC 7 kC With C Cg t 0 an initial condition I C t j 7 j kdt CH C 0 C 1n 7 kt CO C C0 87kt exponential decay of C 2 Linear dy 7 t t dt fy g mod 2 em To solve multiply by an integrating factor IF e e g am my e g0 d Ft Ft 7 e e t dt y l g yew J39eFm gtdt Const Ex linear storage model from CE 421 This model is based on the continuity equation with groundwater out ow Q proportional to storage S by a factorK i Q a K i Q a QL IFe K 611 611 611 K K QetKetK dt ketKL ietK1 Q i l Kl Q starts at 0 and slowly approaches 139 Secondorder ODEs eg wave equations don39t show up much in groundwater hydrology Yes you should be aware of nonlinear ODEs with weird and bizarre solutions like chaos but they don39t show up in this course either Partial Differential Eguations gPDEs You will notice that some chapters of the text are full of these squiggly things If an equation has a partial derivative this means that there are several independent variables involved for groundwater these are usually the spatial coordinates x y z and time I Partial differential equations can be of several distinct types the most comm on in groundwater hydrology are described brie y below 1 Laplace39s equation which describes steady groundwater ow in a homogenous aquifer 62h 62h 72 72 0 h is hydraulrc head 6x By This equation requires boundary conditions in x w y for solution For simple geometry and boundary conditions the equation can be solved analytically but for most problems of interest to us it is easier to approximate the solution using numerical approximation methods We will cover the nitedifference method in this course Essentially the way this works is that differential equations are replaced by di erence equations which are solved algebraically 2 Diffusion equations such as this one describing unsteady groundwater flow in a horn ogenous aquifer 62h 62h SS 6 6x2 6y2 K at and the solute transport equation here in one spatial dimension x 62C ac 6C 72 u 7 7 C is the concentration of a solute in groundwater 6x 6x 6t Solution of these equations requires boundary conditions ml an initial condition for I For simple geometries the solutions can be worked out using Laplace transforms ugh we won39t do that in this course Again we can also use numerical approximation methods