Course Hero - We put you ahead of the curve!
You have requested the below document.
- Title: lecture01
- Type: Notes
- School: UNC
- Course: MATH 221
- Term: Fall
1 CHAPTER Overview of frequently encountered PDE s 1. PDE s in the natural sciences Ordinary and partial di erential equations (ODE s and PDE s henceforth) are frequently encountered in numerous areas of study. A knowledge of the basic scienti c background is necessary to write down equations of interest. Perhaps one should not be surprised that the same background knowledge is useful in devising solution methods. The typical procedures by which PDE s are derived should be known to researchers working on solution methods. 1.1. Conservation laws. A large number of PDE s arise from the physical principle of conservation. Physicists have always been interested in describing changes in the world surrounding us. By observation, theory and experiment certain concepts have been arrived at, among which the concept that one can de ne physical quantities that remain the same during some process. These quantities are said to be conserved. Typically a quantity is conserved in a hypothesized isolated system. In reality no system is truly isolated and the most interesting applications come about when we study the interaction of two or more systems. This leads to the question of how one can follow the changes in physical quantities of the separate systems. An extremely useful procedure is to set up an accounting procedure. To start with a mundane example, consider the physical quantity of interest to be the quantity of currency Q in a building B. If the building is a commonplace one, it is to be expected that when completely isolated, the amount of currency in the building is xed (1.1) Q = Q0 . Q0 is some constant. Eq. (1.1) is self-evident but not particularly illuminating of course the amount of money is constant if nothing goes in or out! Similarly in physics, statements such as the total mass-energy of the universe is constant are not terribly useful, though one should note this particular statement is not obviously true. Things get more interesting when we consider a more realistic scenario in which the system is not isolated. People might be coming and going from building B and some might actually have money in their pockets. In more leisurely economic times, one might be interested just in the amount of money in the building at the end of the day. Just a bit of thought leads to Qn = Qn 1 + Qn 1,n where Qn is the amount of money at the end of day n, Qn 1 that from the previous day and Qn 1,n the di erence between money received and that paid in the 3 4 1. OVERVIEW OF FREQUENTLY ENCOUNTERED PDE S building during day n Qn 1,n = Rn 1,n Pn 1,n . Keeping track of Rn 1,n and Pn 1,n separately, for instance in two distinct columns on a ledger, seems easier to people more inclined towards addition than subtraction, and this leads to double entry accounting, an important discovery of Renaissance Italy (see http://www.acaus.org/history/hs_pac.html). As economic activity picks up and we take building B to mean bank it becomes important to keep track of the money in the bank at all times, not just at the end of the day. It then makes sense to think of the rate at which money is moving in or out of the building so we can not only track the amount of currency at any given time, but also be able to make some future predictions. Since some time has passed in order for economic activity to pick, we can assume that addition and subtraction have become much more familiar and are actively taught to small children. We ll therefore use a single quantity F to denote the amount of money leaving or entering building B during time interval t with the understanding that positive values of F represent incomes and negative ones expenditures. Such understandings go by the name of sign conventions. They re not especially meaningful but it aids communication if we all stick to the same ones. The amount of currency in the building then changes in accordance to (1.2) Q(t + t) = Q(t) + F t . By the time such equations were being written out uid ow was a scienti c frontier investigated by the Bernoullis (see http://www.maths.tcd.ie/pub/ HistMath/People/Bernoullis/RouseBall/RB_Bernoullis.html) and F got to be referred to as a ux, the Latin term for ow. It is readily apparent that (1.2) is a good approximation for small intervals, but probably a bad one if t is large since economic activity might change from hour to hour. In order to better keep track of things one might think of F as being de ned at any given time t so we have F (t) the instantaneous ux of currency at time t. By the time people were thinking along these lines Newton and Leibniz had introduced calculus and su cient time has since passed that the notions of calculus are widely known at least among college students if not small children. We can therefore write t+ t (1.3) Q(t + t) = Q(t) + t F ( )d and get a suitably impressive statement which, form nonewithstanding, carries the same signi cance as (1.2). On the verge of the modern era economic activity might really expand and buildings become so large that it makes sense to keep track of the amount of money in individual rooms and also track in ows and out ows through individual doors. We can identify a room or door by its spatial position denoted by x = (x1 , x2 , x3 ) but we encounter a problem in that position vectors such as x refer to a single point and no matter how small we make the currency it still has to occupy some space. This conceptual di culty is overcome by introducing a ctitious density of currency at time t which we shall denote by q(x, t). The only real meaning we associate with this density is that if we sum up the values of q(x, t) in some volume 1. PDE S IN THE NATURAL SCIENCES 5 we obtain the amount of currency in that volume Q( , t) = q(x, t)dx . On afterthought, we might observe that the same sort of question should have arisen when we de ned Q(t) as being de ned at one instant in time. Ingrained psychological perspectives make Q(t) more plausible, but were we to live our lives such that quantum uctuations are observable, Q(t) would be much more questionable. If we have a spatial density for Q it seems natural to do the same for F and we de ne f (x, t) as being the instantaneous ux of currency in a small region around (x, t). A bit of thought suggests that the ux should be a vector quantity since we have three directions along which a density can be de ned. Along any given direction a scalar ux is obtained by a scalar product; in particular along the direction normal to a boundary n(x) the scalar ux is given by f (x, ) n(x). The relation between the total ux and the ux densities is given by (1.4) F ( ) = B f (x, ) n(x) dx . Careful observers will notice the appearance of B as de ning the integration domain. By this we mean that the integration is to be taken over the boundary of the domain B, or in everyday terms, the exterior walls and doors of building B. There is a bit of inconsistency in the sciences as to what we mean by ux . Sometimes it means the amount of some quantity passing through a nite region such as F above. Other times it actually means ux density such as f . This possibility of confusion shows the value of using the same conventions. Imposing such conventions is however a social activity and subject to historical iteration. In this course the convention ux =f shall be imposed by instructor at. Gathering together all the above we can write a much more sophisticated-looking statement (1.5) t+ t Q(B, t + t) = B q(x, t + t)dx = B q(x, t)dx + t E(B,t) B f (x, ) n(x) dx d which nonetheless is essentially the same as (1.2) or (1.3). There are some special cases in which additional events a ecting the balance of Q can occur. For instance if by B we mean a reserve bank, money might be (legally) printed and destroyed in the building. Again by analogy with uid dynamics when such an event occurs we say that there exist sources of E within B, much like a spring is a source of surface water. Let (t) be the total sources at time t. By now we know what to expect; (t) might actually be obtained by summing over several sources placed in a number of positions, for instance the separate printing presses and furnaces that exist in B. It is useful to introduce a spatial density of sources (x, t). Our conservation statement now becomes (1.6) t+ t t+ t q(x, t+ t)dx B B q(x, t)dx = t B f (x, ) n(x) dx d + t B (x, )dx d The statement above encompasses all physical conservation laws. It is however quite straightforward in interpretation: 6 1. OVERVIEW OF FREQUENTLY ENCOUNTERED PDE S change in money in B = net money coming in or going out of B + net money produced or destroyed in B. It should be emphasized that the above statement has true physical meaning and is referred to as an integral formulation of a conservation law. The key term is integral and refers to the fact that we are summing over some spatial domain. Remember that the densities were arti cial constructs that we introduced. Eq. (1.6) is useful and often applied directly in the analysis of physical systems. From an operational point of view it does have some inconveniences though. These have mainly to do with pesky integration domains B which typically are di cult to describe and over which it is di cult to perform integrations. To avoid this, mathematicians and physicists have gone one further step and imagined f (x, t) as being de ned everywhere not only on B (the doors and windows of B). These internal uxes can be shown to have a proper physical interpretation to which we shall come back later. For now let s see the implications of this extension. If we not only assume that f (x, t) is de ned everywhere, but also that it has nice properties such like enough smoothness to ensure di erentiability then we can apply the Gauss theorem and transform the integral over B into one over B (1.7) B f (x, ) n(x) dx = B f (x, ) dx . Here we encounter another convention problem in that some disciplines use outward normals pointing in which case (1.7) holds while other disciplines use an inward pointing normal in which case we have (1.8) B f (x, ) n(x) dx = B f (x, ) dx . Fluid dynamics uses the second convention which leads to (1.8) and this is the one we ll adopt since so many developments in numerical methods for PDE s initially arose from uid dynamics problems. Applying (1.8) to (1.6) leads to (1.9) t+ t t+ t q(x, t + t) q(x, t) + B t f (x, ) d dx = t B (x, )dx d . There was nothing special about the shape of the building B or the length of the time interval t that we used in deriving (1.9). We can therefore consider special, in nitesimal domains and intervals and obtain a di erential form q + f = , (1.10) t where, as is customary, the dependence of q, f , on space and time is understood but not written out explicitly. Eq. (1.10) is known as the local or di erential form of the conservation law for E. It is often easier to work with since there are no complications arising from the domain shape that appear directly in the statement of conservation. In physics the above scenario is encountered many times. Physicists have arrived at certain quantities which obey (1.6). In many situation it is permissible to speak of local quantities and use (1.10). Classical physics arrived at mass, momentum, energy and electrical charge as physical concepts that lead to quantities that satisfy conservation laws. Contemporary physics uni ed momentum and energy 1. PDE S IN THE NATURAL SCIENCES 7 in the theory of relativity and also gave new, microscopic quantities that satisfy conservation such as lepton number. 1.2. Special forms of conservation laws. 1.2.1. Newton s law. The full general form (1.10) often arises in real-world applications. But it also many times possible to carry out certain simpli cations that lead to equations that are easier to solve. As a simple example, consider the classic problem of dynamics of studying the motion of a point mass m. It has no internal structure and its motion is characterized by the second law of dynamics which is a statement of conservation of momentum d (mv) = F. (1.11) dt Here we have the correspondence q (mv), F with (1.10), hence the statement: external forces are sources of momentum . Instead of a PDE, the lack of internal structure has led to an ODE. 1.2.2. Advection equations. Other special forms of (1.10) are not quite so trivial. Often f , depend on q, that is we have f (q), (q). The speci c form of this dependence is given by physical analysis typically. But accounting for all physical e ects is so di cult that often simple approximations are used. For instance we can assume that f (e) is su ciently smooth to have a Taylor expansion (1.12) f (q) = f0 + f (q0 )(q q0 ) + . . . = and consider what happens when we use various truncations of the Taylor expansion. Typically we can take f0 = 0 since it doesn t a ect the PDE (1.10) anyway. Choosing a system of units such that q0 = 0 the simplest truncation is (1.13) and the = 0 form of (1.10) is q + (u q) = 0 . t If we consider that u does not depend on the spatial coordinates we obtain q +u e= 0 (1.15) t which goes by the name of the constant velocity advection equation. The name comes from its use in modeling the transport of some substance by a ow; this process is known as advection. Its one-dimensional form is the basis of much development in numerical methods for PDE s q q (1.16) +u =0, t x and we shall study it in detail. If u does depend on x we have q q (1.17) + (u q) = +q u+u q = 0 t t or q + u q = q u (1.18) t (1.14) f (q) = f (0)q = u q 8 1. OVERVIEW OF FREQUENTLY ENCOUNTERED PDE S known as the variable velocity advection equation. In very many cases the advection velocity eld u is divergence free (1.19) so we have the simpler form q +u q =0 . t 1.2.3. Di usion equations. Another widely encountered dependence of f on q is of the form (1.20) (1.21) and this leads to (1.22) q ( q) = (q) . t This is known as the heat equation or the di usion equation. If (the thermal di usivity) is a constant we have (1.23) q = 2 q + (q) , t a widely encountered form of the heat equation. For many problems time evolution is so slow that the q/ t derivative is negligible and (1.23) becomes 2 q = / 2 q = 0 f (q) = q u =0 , (1.24) (1.25) known as the Poisson equation. If = 0 we obtain the special form known as the Laplace or harmonic equation. 1.2.4. Advection-di usion equations. As might be expected, the physical ux dependence might combine the two forms (1.13), (1.21) encountered above (1.26) f (q) = u q q , from which we obtain q (1.27) + u q = (q) + ( q) q u , t known, naturally enough, as the advection-di usion equation. Again, in most applications u is divergence-free so (1.27) becomes q + u q = (q) + ( q) . t 1.2.5. Vector valued conservation laws. Up to now we have considered that the conserved quantity is a scalar q. Often it is more convenient to group scalars together as a vector, for instance when thinking of the momentum of a body. The generalization of the conservation law (1.10) is immediate (1.28) q(x, t) + f (q(x, t), (x, t)) = (q(x, t), (x, t)) . t Here the explicit dependence on space x and time t of q has been pointed out, as well as the possible dependence of the uxes f and sources on both space and time and the conserved variables q(x, t). Note that f has a di erent meaning in the present context. As a result of taking the divergence we should still obtain (1.29) 2. PDE S IN OTHER DISCIPLINES 9 a vector quantity for (1.29) to be consistent. This means that f is now a tensor of dimension n n where n is the number of components of e (and ). 1.2.6. Convection-di usion equations. In uid ow, among other applications, the velocity eld u is related to the conserved quantities (1.30) u = u (x, t, q) . This particular situation goes by the name of convection. Similar to (1.27) we can write a convection-di usion equation q (1.31) + u(q) q = (q) + ( q) q u t and its vector valued generalization q (1.32) + u(q) e = (q) + ( q) q u . t 1.3. Conservative and non-conservative forms. We have seen that a large class of di erential equations are derived from conservation laws. The basic form of a conservation law is: time change = -(di erence in outward uxes) + (sources). In mathematical terms we have the local, di erential formulation q = f (q) + . (1.33) t This is known as the conservative form of the law of conservation of q. The same principle of conservation might be stated di erently if f (q) is expanded. For instance, when f = u q we can derive from the conservative form q (1.34) = (u q) + t the mathematically equivalent form q (1.35) = q u u q + . t Eq. (1.35) is known as the non-conservative form of the conservation law for q. Though equivalent from the analytical point of view, the numerical solution procedures for the two forms show di erent characteristics as we shall see later on. 2. PDE s in other disciplines Historically, most of the e ort in studying PDE s has been directed at those suggested by mathematical physics and that somehow arise from a conservation law. Recently PDE s have been introduced in a number of other elds of study such as mathematical nance or ecology. For instance an important development in mathematical nance is the Black-Scholes equation describing the trading of European options (2.1) 2 V 1 V 2 V + rSt + 2 St rV = 0 t 2 s2 s with V the value of an option, t time and s an asset allocation. Though such equations arise from di erent fundamental principles it is striking that they have the same form as those arising in mathematical physics. The Black-Scholes equation can be described as a mixed di usion advection equation with a source term for example. We shall concentrate on a mathematical physics background in discussing 10 1. OVERVIEW OF FREQUENTLY ENCOUNTERED PDE S numerical solution of PDE s but keep in mind that the same methods are widely applicable. 3. Typical problems involving ODE s and PDE s Now that we have arrived at the general form of PDE s which are of interest in many applications we can turn to actually nding solutions. An important rst observation is that specifying the equation to be solved does not allow a unique solution. We must also specify additional boundary and/or initial conditions. Just as we have important special equations such as the advection or the Laplace equation, there are a number of important, archetypal problems involving ODE s and PDE s. 3.1. Initial value problem for ODE s. The simplest problem is the initial value problem for a rst order system of ODE s (3.1) q = f (t, q) q(t = 0) = q0 . This also encompasses initial value problems for ODE s of higher order since an ODE of order p can always be rewritten as a system of p ODE s of order 1. To exemplify, consider (3.2) q = g(t, q, q , q ) . By introducing the auxilliary functions (3.3) we obtain the system (3.4) which is of the form (3.1). q r d r= s dt s g(t, q, r, s) r = q , s = q 3.2. Boundary value problem for ODE s. For ODE s of order 2 or greater or for systems of two or more ODE s one can meaningfully impose boundary conditions at distinct points within the computational domain. The archetypal ODE boundary value problem is for a second order ODE with conditions at the end points of the computational domain q = f(t, q, q ) (3.5) q(a) = q1 . q(b) = q2 Instead of the function values, its derivatives might be speci ed such as in q = f(t, q, q ) q (a) = r1 . (3.6) q (b) = r2
Find millions of documents here - Study Guides, Homework Solutions, Papers, Exam Answer Keys and more.
Course Hero has millions of course related materials that will enable you to learn better, faster and get an A in all your courses.
Below is a small sample set of documents:
lecture01.pdf
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MATH >> 221 Fall, 2008
Path: UNC >> MED >> 1 Fall, 2008
Path: UNC >> MED >> 1 Fall, 2008
Path: UNC >> COMP >> 775 Fall, 2008
Path: UNC >> ENST >> 202 Fall, 2008
Path: UNC >> ENST >> 202 Fall, 2008
Path: UNC >> ENST >> 202 Fall, 2008
Path: UNC >> ENST >> 202 Fall, 2008
Path: UNC >> ENST >> 202 Fall, 2008
Path: UNC >> ENST >> 201 Fall, 2008
Path: UNC >> ENST >> 201 Fall, 2008
Path: UNC >> ENST >> 201 Fall, 2008
Path: UNC >> COMP >> 535 Fall, 2008
Path: UNC >> COMP >> 535 Fall, 2008