Differential Equations And Boundary Value Problems

[Gauthier Zitnik]

Inthe study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions.[1] A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.

To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equations is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact well-posed.

Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions (solutions to Laplace's equation); the solution was given by the Dirichlet's principle.

Boundary value problems are similar to initial value problems. A boundary value problem has conditions specified at the extremes ("boundaries") of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term "initial" value). A boundary value is a data value that corresponds to a minimum or maximum input, internal, or output value specified for a system or component.[2]

A boundary condition which specifies the value of the function itself is a Dirichlet boundary condition, or first-type boundary condition. For example, if one end of an iron rod is held at absolute zero, then the value of the problem would be known at that point in space.

A boundary condition which specifies the value of the normal derivative of the function is a Neumann boundary condition, or second-type boundary condition. For example, if there is a heater at one end of an iron rod, then energy would be added at a constant rate but the actual temperature would not be known.

Aside from the boundary condition, boundary value problems are also classified according to the type of differential operator involved. For an elliptic operator, one discusses elliptic boundary value problems. For a hyperbolic operator, one discusses hyperbolic boundary value problems. These categories are further subdivided into linear and various nonlinear types.

In electrostatics, a common problem is to find a function which describes the electric potential of a given region. If the region does not contain charge, the potential must be a solution to Laplace's equation (a so-called harmonic function). The boundary conditions in this case are the Interface conditions for electromagnetic fields. If there is no current density in the region, it is also possible to define a magnetic scalar potential using a similar procedure.

Written in a clear and accurate language that students can understand, Trench's new book minimizes the number of explicitly stated theorems and definitions. Instead, he deals with concepts in a conversational style that engages students. He includes more than 250 illustrated, worked examples for easy reading and comprehension. One of the book's many strengths is its problems, which are of consistently high quality. Trench includes a thorough treatment of boundary-value problems and partial differential equations and has organized the book to allow instructors to select the level of technology desired. This has been simplified by using symbols, C and L, to designate the level of technology. C problems call for computations and/or graphics, while L problems are laboratory exercises that require extensive use of technology. Informal advice on the use of technology is included in several sections and instructors who prefer not to emphasize technology can ignore these exercises without interrupting the flow of material. (From the 1st edition)

This book was published previously by Brooks/Cole Thomson Learning, 2001. This free edition is made available in the hope that it will be useful as a textbook or reference. The book is offered subject to the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License -nc-sa/3.0/deed.en_G.

Before we start off this section we need to make it very clear that we are only going to scratch the surface of the topic of boundary value problems. There is enough material in the topic of boundary value problems that we could devote a whole class to it. The intent of this section is to give a brief (and we mean very brief) look at the idea of boundary value problems and to give enough information to allow us to do some basic partial differential equations in the next chapter.

Now, with that out of the way, the first thing that we need to do is to define just what we mean by a boundary value problem (BVP for short). With initial value problems we had a differential equation and we specified the value of the solution and an appropriate number of derivatives at the same point (collectively called initial conditions). For instance, for a second order differential equation the initial conditions are,

It is important to now remember that when we say homogeneous (or nonhomogeneous) we are saying something not only about the differential equation itself but also about the boundary conditions as well.

We mentioned above that some boundary value problems can have no solutions or infinite solutions we had better do a couple of examples of those as well here. This next set of examples will also show just how small of a change to the BVP it takes to move into these other possibilities.

In this case we have a set of boundary conditions each of which requires a different value of \(c_1\) in order to be satisfied. This, however, is not possible and so in this case have no solution.

So, with Examples 2 and 3 we can see that only a small change to the boundary conditions, in relation to each other and to Example 1, can completely change the nature of the solution. All three of these examples used the same differential equation and yet a different set of initial conditions yielded, no solutions, one solution, or infinitely many solutions.

The answers to these questions are fairly simple. First, this differential equation is most definitely not the only one used in boundary value problems. It does however exhibit all of the behavior that we wanted to talk about here and has the added bonus of being very easy to solve. So, by using this differential equation almost exclusively we can see and discuss the important behavior that we need to discuss and frees us up from lots of potentially messy solution details and or messy solutions. We will, on occasion, look at other differential equations in the rest of this chapter, but we will still be working almost exclusively with this one.

A Boundary value problem is a system of ordinary differential equations with solution and derivative values specified at more than one point. Most commonly, the solution and derivatives are specified at just two points (the boundaries) defining a two-point boundary value problem.

Here, \(y,f,g \in R^n\) and the systemis called explicit because the derivative \(y^\prime\) appearsexplicitly. The \(n\) boundary conditions defined by \(g\) must beindependent; that is, they cannot be expressed in terms of eachother (if \(g\) is linear the boundary conditions must be linearlyindependent).

In practice, most BVPs do not arise directly in the form(1) but instead as a combination of equations definingvarious orders of derivatives of the variables which sum to \(n\ .\)In an explicit BVP system, the boundary conditions and the righthand sides of the ordinary differential equations (ODEs) caninvolve the derivatives of each solution variable up to an orderone less than the highest derivative of that variable appearing onthe left hand side of the ODE defining the variable. To write ageneral system of ODEs of different orders in the form(1), we can define \(y\) as a vector made up of all thesolution variables and their derivatives up to one less than thehighest derivative of each variable, then add trivial ODEs todefine these derivatives. See the section on initial value problems for an example of how this is achieved. See also Ascheret al.(1995) who show techniques for rewriting boundaryvalue problems of various orders as first order systems. Suchrewritten systems may not be unique and do not necessarily providethe most efficient approach for computational solution.

The words two-point refer to the fact that the boundarycondition function \(g\) is evaluated at the solution at the twointerval endpoints \(a\) and \(b\) unlike for initial value problems(IVPs) where the \(n\) initial conditions are all evaluated at asingle point. Occasionally, problems arise where the function \(g\)is also evaluated at the solution at other points in \((a,b)\ .\) Inthese cases, we have a multipoint BVP. As shown in Ascher et al. (1995), amultipoint problem may be converted to a two-point problem bydefining separate sets of variables for each subinterval betweenthe points and adding boundary conditions which ensure continuityof the variables across the whole interval. Like rewriting theoriginal BVP in the compact form (1), rewriting amultipoint problem as a two-point problem may not lead to aproblem with the most efficient computational solution.

Most practically arising two-point BVPs have separated boundaryconditions where the function \(g\) may be split into two parts (onefor each endpoint):\[g_a(y(a))=0,\quad g_b(y(b))=0.\] Here, \(g_a\in R^s\) and \(g_b\inR^n-s\) for some value \(s\) with \(1

For linear BVPs, where the ODEs and boundary conditions are bothlinear, the equation \(g(s, y(b;s))=0\) is a linear system ofalgebraic equations. Hence, generally there will be none, one oran infinite number of solutions, analogously to the situation withsystems of linear algebraic equations.

In addition to the possibilities for linear problems, nonlinearproblems can also have a finite number of solutions. Consider thefollowing simple model of the motion of a projectile with airresistance:\[\tag4\beginarrayrcly^\prime&=&\tan(\phi),\\v^\prime&=&-\fracgv\tan(\phi) - \nu v\sec(\phi),\\\phi^\prime&=&-\fracgv^2.\endarray\]

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