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HARVARD MATH 21B - DIFFERENTIAL EQUATIONS

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MATH 21b DIFFERENTIAL EQUATIONS Spring 20062 10. Differential equations The goal of this the final part of the course is to introduce a great extension of the ideas from linear algebra to the realm of vector spaces with an infinite number of dimensions. I describe momentarily a hypothetical application to provide some inkling of the ‘raison d’etre’ for infinite dimensional linear algebra. This particular application concerns a differential equation; an equation for a function that involves constraints on its derivatives. Additional examples of differential equations appear in these notes. Be forwarned that there are myriad scientific applications of infinite dimensional linear algebra; it is a tool worth knowing something about. To start the story on the promised application, imagine a mercury thermometer of the kind that your folks might have used to take your temperature when you were a wee babe. The thermometer can be thought of as a long, thin bar of mercury, with most of it insulated. The part near one end is heated by inserting it under one’s tongue. Now imagine briefly heating this part of the thermometer and then removing the heat source. After such a momentary heating, the temperature will be different at different points along the thermometer, and these temperatures give the values of a function, τ(x), where x is a coordinate along the bar. Now I am left to wonder how the temperature at each point changes as a function of the time elapsed after the heat source is removed. I expect that the initial temperature disparities will decrease as time goes on. In any event, if I want to say something quantitative about the time and position dependence of the temperature, I am defacto searching for a function of the variables (t, x) that is defined for times t ≥ 0 and points x whose value at any given (t, x) tells me the temperature of the point x on the thermometer at time t. I call this sought after function T(t, x). Here is a challenge: Find T(t, x) given the time t = 0 temperature profile τ(x). According to what was just said, T(0, x) is equal to τ(x). As argued in Section 10.4 below, this function T(t, x) is constrained to obey the differential equation !!tT = µ!2!x2T . Here, µ is a positive constant whose value is determined by various properties of the element mercury and by the units that are used to measure the position along the thermometer. As you can see, this equation says that the manner in which the temperature changes in time is determined by its x dependence at that time. In particular, it asserts that the derivative of T with respect to time is equal to µ times the second derivative of T with respect to x. This particular differential equation has many names, one being the ‘heat equation’. My challenge to find T(t, x) amounts to solving the heat equation with the time 0 constraint T(0, x) = τ(x).3 Where to begin? We do have experience in this course with solving equations for a time dependent vector in Rn. I refer here to an equation such as ddt !v = A !v where t → !v(t) is the sought after vector function of time and A is a constant, n×n matrix. Our techniques for solving this last sort of equation may be of some use to solving the heat equation provided that we justify the following analogy: The function T can play the role of the time dependent vector !v(t), and the operation that sends T → µ!2!x2T can play the role of matrix multiplication, !v → A !v. Humor me for a bit so that I can pursue this analogy. We know how to solve the vector equation when the matrix A is diagonalizable. Let me remind you how this is done: I first find a basis, { !ua}a=1,2,…,n, for Rn where each basis vector is an eigenvector of the matrix A. This is to say that A !ua = λa !ua where λa is a real or complex number. I then write the time t = 0 version of !v using this basis as !v(0) = v1 !u1 + v1 !u2 + ··· + vn !un . The corresponding solution to the vector differential equation is !v(t) = v1et!1 !u1 + v2et!2 !u2 + ··· + vnet!n !un . If I am to pursue this analogy to obtain our heat equation solution, then I must, perforce, obtain answers to the five questions that follow. Here is the first: • How can I view a function of x as a member of a vector space? If I can view functions of x in this way, then the assignment t → T(t, ·) can be viewed as a ‘time dependent’ vector in this vector space of functions. Here is the second question: • How can I view the operation of taking derivatives as that of a matrix or linear transformation acting on the vector space of functions?4 If I can view derivatives in this way, then I can view the assignment T → !2!x2T as the result of acting on a time dependent vector in my vector space of functions by a linear transformation. What follows is the third question. • Granted that the assignment of d2dx2f to a function f(x) can be interpreted as a linear transformation, what are its ‘eigenvectors’, the functions that obey d2dx2f = λ f where λ is a real or complex number? If I can answer this third question, I am then faced with the fourth: • What is a basis for an infinite dimensional vector space? In particular, is there a basis whose elements obeyd2dx2f = λ f with λ a real or complex number? If there is such a basis, then I can solve the heat equation once I answer this last question: • How do I write the initial temperature profile τ(x) as a linear combination of this basis of eigenvectors? When I come to terms with all of the above, then I can write down the desired function T(t, x). Answers to these five questions are part of the readings that follow. Take note, however, that the infinite dimensional linear algebra notions that are introduced in the ensuing discussion are used for much more than just the heat equation. They are tools that are wielded in all sorts of applications to the sciences. 10.1 Vector spaces whose elements are functions Many of the ideas of linear algebra which we have studied in the context of Rn are applicable in a much wider context. Mathematicians introduced the abstract notion of a ‘vector space’, or what is a synonym, a ‘linear space’, to describe this greater context. Rather than look at linear spaces in the abstract, the discussion in this and the next two sections look specifically at examples that have applications to differential equations. The


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HARVARD MATH 21B - DIFFERENTIAL EQUATIONS

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