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:

    Will women find themselves in the same position they have always been? Or do we see liberation as solving the conditions of women in our society?... If we continue to shy away from this problem we will not be able to solve it after independence. But if we can say that our first priority is the emancipation of women, we will become free as members of an oppressed community.
    Ruth Mompati (b. 1925)

    If we parents accept that problems are an essential part of life’s challenges, rather than reacting to every problem as if something has gone wrong with universe that’s supposed to be perfect, we can demonstrate serenity and confidence in problem solving for our kids....By telling them that we know they have a problem and we know they can solve it, we can pass on a realistic attitude as well as empower our children with self-confidence and a sense of their own worth.
    Barbara Coloroso (20th century)

    Why does man freeze to death trying to reach the North Pole? Why does man drive himself to suffer the steam and heat of the Amazon? Why does he stagger his mind with the mathematics of the sky? Once the question mark has arisen in the human brain the answer must be found, if it takes a hundred years. A thousand years.
    Walter Reisch (1903–1963)

    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)

    Through a series of gradual power losses, the modern parent is in danger of losing sight of her own child, as well as her own vision and style. It’s a very big price to pay emotionally. Too bad it’s often accompanied by an equally huge price financially.
    Sonia Taitz (20th century)