1.1: Introduction Chapter 1: Partial Differential Equations Definition: A differential equations whose dependent variable varies with respect to more than one independent variable is called a partial differential equation 1
Physical laws of heat flow: 1. Heat flow or conduction occurs in the presence of a temperature gradient. In particular, the rate of heat flow per unit cross sectional area is proportional to the temperature gradient. k(xa u x 2. he direction of heat flow is from points of higher temperature to points of lower temperature 3. he specific heat capacity determines the change in temperature of an object based on the amount of heat introduced. u = 1 amount of heat introduced = E c(x mass of object mc(x Putting this into the geometry of a wire: 2
We have derived where β = k ρc u t (x,t = u β 2 x + P(x,t 2 is called the diffusion coefficient and P(x,t = Q(x,t ρc. Note: Here we have one time and two space derivatives, so we need one initial condition and two boundary conditions. he standard initial condition is an initial temperature distribution given as a function of x: u(x, = f(x, < x < L For now we will be looking at a wire whose ends are always kept at C: u(,t =, u(l,t =, t > For now we will also assume there are no internal heat sources: P(x,t = Putting it all together, we have the heat equation: u (x,t = β 2 u < x < L,t > t x 2 u(x, = f(x < x < L u(,t = u(l,t = t > (1 Rule of thumb: We need as many initial/boundary conditions as we have time/space derivatives. But, just having these does not guarantee the problem is solvable. 3
1.2: Method of separation of variables Idea: Assume we can express a solution of the heat equation as u(x,t = X(x(t his reduces solving the PDE to solving two separate (but related ODE problems. When we make this assumption: 4
Summary of steps to solve the heat equation: Step 1 Suppose u(x,t = X(x(t and determine corresponding ODEs Step 2 Solve the time and space problems: (t βk(t = X (x KX(x =, X( = X(L = Step 3 Put them back together as u(x,t = X(x(t and use u(x, = f(x to solve for unknowns Definition: For the boundary value problem X (x KX(x =, X( = X(L = the values of K that give a non-trivial solution are called eigenvalues and the corresponding solutions are called eigenfunctions. For the general heat equation we have ( nπ 2 eigenvalues: K = L( nπ eigenfunctions: X n (x = a n sin L x For arbitrary boundary conditions, the problem may not have a solution. Example: 5
1.3: Fourier series Here we see how to approximate functions using trig terms. Definition: A function f(x is even if f( x = f(x hat is, f(x is symmetric with respect to the y axis. If f(x is piecewise continuous, we have a a f(xdx = 2 Definition: A function f(x is odd if a f( x = f(x f(xdx hat is, f(x is symmetric with respect to the origin. If f(x is piecewise continuous, we have a a f(xdx = Note: A function must be even, odd or neither. 6
Definition: Let f be a piecewise continuous function on the interval [, ]. he Fourier series of f is the trig series f(x a 2 + [ ] a n cos + b n sin where Notes: a n = 1 b n = 1 f(x cos f(x sin dx, n =, 1, 2,... dx, n = 1, 2, 3,... 1. We write f(x since we can build the Fourier series based on f(x but it may not converge 2. he first term is a 2 to make it easier to remember 3. While Fourier series can represent arbitrary functions, they are especially useful for approximating periodic functions 4. Fourier series have an advantage over aylor series since the Fourier series (if it converges will approximate the function of the whole domain, not just around a single point. Where do the coefficients a n and b n come from? 7
Notation: Limit from the right, or right-hand limit: f(x + = lim f(x + h h + Limit from the left, or left-hand limit: f(x = lim f(x h h + Pointwise convergence of Fourier series: If f and f are piecewise continuous on [,] then for any x in (, 1 [ f(x + + f(x ] = a 2 2 + [ ] a n cos + b n sin where a n and b n are defined by the Euler formulas. For x = ±, the series converges to 1 [ f( + + f( ] 2 Note: his says that if f and f are piecewise continuous on the interval, then the Fourier series converges to f(x wherever f is continuous, and converges to the average wherever f is discontinuous. Uniform convergence of Fourier series Let f be a continuous function on (, and periodic with period 2. If f is piecewise continuous on [,], then the Fourier series for f converges uniformly to f on [,]. hat is, for each ǫ >, there exists an integer N (that depends on ǫ such that f(x [ a 2 + [ a n cos ] ] + b n sin < ǫ for all N N and all x (,. hat is, for some large but finite N, if we take at least that many terms in the Fourier series, we re guaranteed to have our series solution within some small distance ǫ of the true solution. 8
Differentiation of Fourier series Let f(x be a continuous function on (, and 2-periodic. Let f (x and f (x be piecewise continuous on [,], and then f(x = a 2 + [ ] a n cos + b n sin f (x πn [ a n sin ] + b n cos Note: Here we not only need to satisfy the condition for uniform convergence, but we also need f (x to be piecewise continuous before we can differentiate a Fourier series. Integration of Fourier series Let f(x be piecewise continuous on [,] with Fourier series f(x a 2 + [ ] a n cos + b n sin hen for any x in [,] we have x f(tdt = x a 2 dt + x [ ( ( ] nπt nπt a n cos + b n sin dt Note: For integration we don t even need pointwise convergence, and there is no requirement on f. 9
1.4: Fourier cosine and sine series Definition: If f(x is defined on (,L then the odd 2L-periodic extension of f(x is { f(x < x < L f o (x = f( x L < x < and the even 2L-periodic extension of f(x is { f(x < x < L f e (x = f( x L < x < and the L-periodic extension of f(x is f(x = { f(x < x < L f(x + L L < x < Note: hese aren t the only ways to extend f(x to L < x < L but only these will give you even or odd functions. 1
Definition: Let f(x be piecewise continuous on [,]. he Fourier cosine series of f(x on [,] is where a n = 2 f(x a 2 + a n cos he Fourier sine series of f(x on [,] is where b n = 2 f(x cos dx, n =, 1,... f(x b n sin f(x sin dx, n = 1, 2,... erminology: he Fourier cosine and sine series are called half-range expansions for f(x 11
1.5: he heat equation Here we deal with solving variations of the heat equation. See the text for derivations in the general cases. 12
1.6: he wave equation Definition: he equation of motion for a taut vibrating string is given by the wave equation: 2 u = α 2 2 u t 2 x2, < x < L,t > u(,t = u(l,t =, t > u(x, = f(x, < x < L u (x, = g(x, < x < L t We use separation of variables to solve the wave equation, and the formal solution is given by u(x,t = [ ( nπα ( nπα ] a n cos L t + b n sin L t sin where a n and b n are determined from the Fourier sine series f(x = g(x = a n sin L nπα b n L sin L L 13