Second-Order Linear ODEs (Textbook, Chap 2)
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1 Second-Order Linear ODEs (Textbook, Chap ) Motivation Recall from notes, pp , the second example of a DE that we introduced there. d φ 1 1 φ = φ 0 dx λ λ Q w ' (a1) This equation represents conservation of water mass (actually conservation of water volume; the model assumes an essentially constant water density) in a confined, but leaky aquifer. The unknown is denoted by the letter φ [L], representing hydraulic head and the independent variable by the letter x [L], representing location. It is a second order ordinary differential equation (ODE). The symbols λ, φ 0, and Q w denote known parameters. Respectively, they represent aquifer transmissivity divided by a leakage coefficient (λ [L ] =TB /K, see p. 59 and the paragraph on dimensional homogeneity), the head [L] in a vertically adjacent (overlying or underlying) aquifer, and the pumping per unit area [L/t]. All terms in (a) have units of [1/L]. To avoid confusion with the notation in the text, which uses the symbol λ for something else, let s replace λ with the symbol β (= λ), or d φ 1 1 φ = φ 0 dx β β Q w ' (a) Another DE presented on p. 46 represents solute mass balance in a flowing river, with the time rate of change of mass in storage balanced by advective and dispersive fluxes, or C t C + v x C D x = 0 (b) Here the unknown is denoted by the letter C, representing solute concentration. There are two independent variables, denoted by the letters x (space) and t (time). As there are now two (or more) independent variables, it is a partial differential equation (PDE). The highest order derivative in x defines it as a second order equation in space, while the highest order derivative in t defines it as a first order equation in time. The symbols v and D denote parameters, the mean river velocity and a dispersion coefficient. Suppose that there is a continuous, unchanging source of solute, and that the flow rate in the river is constant. Then C/ t=0, the solute concentration is in steady-state, and (b) becomes a second order ODE in x. dc d C v D = 0 (c) dx dx The first term represents solute advection and the second dispersion, which are in balance. Notice that the partial differences ( ) have become ordinary differences d( ) s, as there is now 10
2 only one independent variable, x. Suppose further that the solute is organic and that it can biotransformed by in situ reactions. The simplest model of this is that the solute concentration decays exponentially with time, which in (c) is modeled by adding a so-called first-order decay process, dc v dx d C D dx = α C (d1) where α [1/T] is the decay constant. Here advection and dispersion are balance by decay. We can write this in mathematical standard form by dividing by the quantity ( D) and gathering all terms on the left-hand side. d C dx v dc D dx α C = 0 D (d) This is second-order ODE in spatial location x. These are two common examples of second-order ODEs encountered in hydrology, but there are many others. These equations are also linear, if their parameters are independent of the dependent variable. They can vary with location x, however. Introduction (Textbook, Section.1) These two examples belong to the general class of linear second-order ODEs which your book writes as (1) y ' ' + p( x) y' + q( x) y = r( x) where, r is the forcing and, for a linear model, parameters p, q, and r do NOT depend on the dependent variable, y, but can depend on the independent variable, x. How do the two examples above fit this model? Model Translation to eqn. (1) d φ 1 1 φ = φ 0 dx β β Q w ' 1 1 y = φ x = x, p = 0, q = = q, r = φ Q β β, 0 w ' d C dx v dc D dx α C = 0 D v α y = C, x = x, p =, q =, r = 0 D D Note that p,q, and r can vary with x, and the models remain linear. Thus β, φ 0, and Q w in the first model, and v, D, and α, in the second model, can also vary in x, for linear models. 11
3 Then, from the text, Chapter. Not in the text s Chapter : For a second-order homogeneous linear ODE () a boundary value problem consists of equation () taken over a finite interval, a x b, with two boundary conditions (BCs), one at each end of the interval. The boundary conditions can be written as k1 y( a) + k y' ( a) = Ka (e) l y( b) + l y' ( b) = K 1 b 1
4 where the k s and l s are parameters, weighing the relative contribution of y and y at each boundary, and the K s are values. For example, if you know the value of y at a, then k 1 =1, k =0, and K a is the value y(a). In PDEs this is called a first type or Dirichlet boundary. If you know the gradient of y at b, then then l 1 =0, l =1, and K b is the value y (b). In PDEs this is called a second type or Neumann boundary. Finally, at a, if both k 1 and k are non-zero it is called a thirdtype or Cauchy boundary in PDEs. All three boundary condition types are used to solve equations of the type given in (1). However, Chapter of your textbook focuses on initial value problems for nd order ODEs. In hydrology these problems are less common than nd order BVPs, but they are very common in other areas of mathematical physics, like geophysics. Back to the text * * Or for the BVP, the conditions in eqn. (e) on the previous page are used to determine the two constants, c 1 and c. Two conditions are needed as this is a nd order equation. In the IVPs these two initial conditions (4) are taken at the beginning of the domain, and in BVPs these two boundary conditions (e) are taken at each end of the domain. In either event, the general solution is given by (5). 13
5 Two Reduction of Order (Section.1) 14
6 Second-Order Linear ODEs with Constant Coefficients (Section.1) 15
7 Two distinct real roots 16
8 Real double root 17
9 Two complex roots Summary nd order ODE with constant coefficients: ODE y ' ' + ay ' + by = 0 Characteristic eqn. λ + a λ + bλ = 0 a where ω = b, and A, B, c 1, and c are constants. 4 18
10 Nonhomogeneous ODEs (Section.7) To solve the nonhomogeneous ODE (1), or an IVP or BVP for (1), we have to solve the homogeneous ODE () for a solution, and then find any solution y p of (1), to obtain a general solution (3) of equation (1). See the book for details and proofs of why this works. The method of undetermined coefficients There are more general methods to find y p, but we will focus on a simple method that works for most problems that you are likely to encounter. The more general method is Variation of Parameters, and is discussed in.0 (not covered in lecture). The simpler, but restricted method is called the Method of Undetermined Coefficients. The basic idea is to guess all possible terms in the solution y p, then solve for their coefficients, term by term, to satisfy (1). The result is y p. The guess consists of sums of terms like those in r(x). This works only if r(x) is composed of terms involving simple functions, in particular powers of x, trigonometric functions, and exponentials, or their products: 19
11 Your book appears to neglect products, but you can handle products of the basic functions in the first column of Table.1. You just apply the modification and sum rules to the product. 130
12 131
13 Equidimensional ODEs (Section.5) 13
14 Series Solutions (Chap. 5) So far we ve examined the general class of linear second-order ODEs which your book writes as (1) y ' ' + p( x) y' + q( x) y = r( x) where r is the forcing, and parameters p, q, and r do NOT depend on the dependent variable, y, but can depend on the independent variable, x. The methods we ve examined have considered relatively simple and special functions for the parameters, but what happens if the parameters have more complicated form? If r(x) is more complicated than can be handled by the Method of Undetermined Coefficients (.7) then we typically resort to the Method of Variation of Parameters (.10), which we have not covered in glass. But there is another method that works very well in this case called LaPlace Transforms (Chap 6) which we use on 1 st and nd order linear equations, and which is very versatile. It even gives the particular solution right away without solving for the homogeneous solutions first as an intermediate step. We ll come back to it later, when we study PDEs. There is another method we use if p(x) and q(x) have more complex functional form, and its particularly valuable for BVPs. In this approach we resort to series solutions. We know that we can approximate functions by series. The idea is to assume that the solution is written as a power series, to plug that series into the equation, and to equate sums of coefficients of equal power to zero, solving for the coefficients in the power series. For a second order equation each coefficient can be written terms of another coefficient, resulting in a recurrence relation, for the coefficients, and leaving only two underdetermined coefficients to be resolved by boundary or initial conditions. Often the series solutions are represented by special functions, with new functional names, such as the Bessel Function of first type of order zero. Certain forms of nd order ODEs occur often in mathematical physics, engineering, and hydrology. The series solutions for these problems have been worked out in detail, and are typically represented by the special functions. It is useful to learn to recognize them or, at least, to know that you should look to see if the equation you are dealing with is of one of these types. For example, if you have a problem in radial flow, such as in groundwater well hydraulics, Bessel Equations are common, while Legendre Equations are common in spherical flow or solute diffusion within a particle, like a spherical microporous feldspar grain. Later we ll see that the methods can be combined to solve PDEs. For example, consider a transient groundwater flow problem that is one-dimension in space. It can be LaPlace Transformed in time, to remove time and reduce it to an ODE in space, which is solved by one of the methods we have been studying. An example is radial flow to a pumping well in an infinite, leaky confined aquifer. After the LaPlace Transform in time, the resulting ODE (in LaPlace Space) is a Bessel Equation, for which one can look up and apply the solution. 133
15 The series solution approach, background from the text (p. 168): The text then goes on in 5. to talk about theory. If the series converges everywhere, the function is said to be analytic. If it diverges in certain locations those are called singularities. A simple example is a divide by zero. The function and its derivatives must converge to be analytic. Convergence is defined in the usual ways, e.g., by comparing successive terms in the series. Series solutions for analytical functions are straight-forward, it is all the special cases involving singularities that make solving a problem difficult. For certain of these more complicated equations the Method of Frobenius ( 5.4) is particularly useful. In any event, we won t visit these cases or this method. You need to take a course in ODEs, although you could also self-learn from the text, which is very clear on these subjects. Let s take a closer look at the special case of Bessel s Equation, which is well known and often encountered in hydrology. From the text, p. 189 ( 5.5): 134
16 Bessel s equation can be solved by the Method of Frobenius (see text, p. 189) Etc, leading to (p. 0, 5.6): 135
17 136
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