Heat Equation - Solving The Heat Equation Using Fourier Series

Solving The Heat Equation Using Fourier Series

The following solution technique for the heat equation was proposed by Joseph Fourier in his treatise Théorie analytique de la chaleur, published in 1822. Let us consider the heat equation for one space variable. This could be used to model heat conduction in a rod. The equation is

(1)

where u = u(x, t) is a function of two variables x and t. Here

  • x is the space variable, so x ∈, where L is the length of the rod.
  • t is the time variable, so t ≥ 0.

We assume the initial condition

(2)

where the function f is given, and the boundary conditions

.

(3)

Let us attempt to find a solution of (1) which is not identically zero satisfying the boundary conditions (3) but with the following property: u is a product in which the dependence of u on x, t is separated, that is:

(4)

This solution technique is called separation of variables. Substituting u back into equation (1),

Since the right hand side depends only on x and the left hand side only on t, both sides are equal to some constant value −λ. Thus:

(5)

and

(6)

We will now show that nontrivial solutions for (6) for values of λ ≤ 0 cannot occur:

  1. Suppose that λ < 0. Then there exist real numbers B, C such that
    From (3) we get
    and therefore B = 0 = C which implies u is identically 0.
  2. Suppose that λ = 0. Then there exist real numbers B, C such that
    From equation (3) we conclude in the same manner as in 1 that u is identically 0.
  3. Therefore, it must be the case that λ > 0. Then there exist real numbers A, B, C such that
    and
    From (3) we get C = 0 and that for some positive integer n,

This solves the heat equation in the special case that the dependence of u has the special form (4).

In general, the sum of solutions to (1) which satisfy the boundary conditions (3) also satisfies (1) and (3). We can show that the solution to (1), (2) and (3) is given by

where

Read more about this topic:  Heat Equation

Famous quotes containing the words solving the, solving, heat, equation and/or series:

    “You are right to demand that an artist engage his work consciously, but you confuse two different things: solving the problem and correctly posing the question.”
    —Anton Pavlovich Chekhov (1860–1904)

    “Certainly, young children can begin to practice making letters and numbers and solving problems, but this should be done without workbooks. Young children need to learn initiative, autonomy, industry, and competence before they learn that answers can be right or wrong.”
    —David Elkind (20th century)

    “I remember my youth and the feeling that will never come back any more—the feeling that I could last for ever, outlast the sea, the earth, and all men; the deceitful feeling that lures us on to joys, to perils, to love, to vain effort—to death; the triumphant conviction of strength, the heat of life in the handful of dust, the glow in the heart that with every year grows dim, grows cold, grows small, and expires—and expires, too soon, too soon—before life itself.”
    —Joseph Conrad (1857–1924)

    “Jail sentences have many functions, but one is surely to send a message about what our society abhors and what it values. This week, the equation was twofold: female infidelity twice as bad as male abuse, the life of a woman half as valuable as that of a man. The killing of the woman taken in adultery has a long history and survives today in many cultures. One of those is our own.”
    —Anna Quindlen (b. 1952)

    “Autobiography is only to be trusted when it reveals something disgraceful. A man who gives a good account of himself is probably lying, since any life when viewed from the inside is simply a series of defeats.”
    —George Orwell (1903–1950)