Differential Equations Project 3 30 July 2002 Wesley Day

Fourier Series

Differential Equations Project 3 30 July 2002 Wesley Day

Fourier Series were developed as an alternate method of expressing functions. A Fourier Series is essentially an infinite sum of sine waves. Sine waves add by superposition and many different waveforms can be obtained by adding sine waves of various frequencies. Fourier Series were built on this principle. Sometimes it is easier to solve problems using an equivalent transcendental expression (Fourier Series) than to solve a problem in its given state. The first theorem presented in Hovanessian 76 on pages 155 and 156 is: Let f(t) be a function defined in the interval and outside this interval , then f(t) can be represented by a series [6.1] with and , where k is an element of the whole numbers [6.2]; which converges at every point t0 to [6.3]. Equation 6.3 at a point of discontinuity will give the average value of the function f(t). A similar definition is provided by Hearn and Metcalfe 96 on pages 92 and 93. Boyce and Diprima 2001 gives a slightly different presentation in section 10.2. The Fourier Theorem is stated in Mei 95 as: [5.2.3].

There are three standard forms in which Fourier Series are represented. The first and most common is the sine-cosine form. Cunningham and Stuller 95 provides definition 15.1 on page 542: . A Fourier Series expansion can be found for a square wave, for example, which is very common in electrical engineering. A square wave (periodic unit step function with oscillating sign) with amplitude 2A and period T has the Fourier Series representation, where k goes from 0 to ∞ [Cunningham and Stuller 95]. The second standard Fourier Series form is the amplitude-phase form. Definition 15.18 on page 550 of Cunningham and Stuller 95 states: ; where , , and , where n is defined in the set of natural numbers. The third Fourier Series form is the exponential form. Definition 15.22 in Cunningham and Stuller 95 states: . Vn is a phasor and f1 is equal to 1/T.

An important related basic theorem is the Parseval Theorem. The energy of a signal can be defined as the sum of all of the energies of all of the harmonics. The mathematical definition is given in Mei 95 as: [5.2.4]. Hearn and Metcalfe 96 provide a similar definition on page 94. Calculating energy and power is fundamental in many engineering disciplines.

Some functions may actually not be periodic, but we may still desire to represent them with a Fourier Series. This can be done if a function satisfies the Dirichlet Conditions. They are stated in Johnson and Johnson 82 on page 26 as follows: If f(x) is a periodic function of period 2L such that, for , f(x) is bounded and has a finite number of maxima and minima and a finite number of discontinuities, then the Fourier Series converges to , where [2.1.5]. L is essentially represented in the equations above as π, although some simplifications of broader definitions were made. These conditions allow us to write Fourier Series for many non-periodic functions, which is of great interest in solving mathematical problems. Often, being able to manipulate a differential equation into a Fourier Series form allows for its easier solving.

The world is full of functions that satisfy the Dirichlet Conditions. All of these functions can be analyzed using Fourier Series. Many differential equations can only be solved by expressing functions as Fourier Series. In mathematics and engineering, the ability to solve differential equations is often of paramount importance and a great deal of resources are often dedicated to solving these problems. Fourier Series and their various forms lead to solutions to common problems.

References

Boyce, William and Richard Diprima. Elementary Differential Equations and Boundary Value Problems. New York: John Wiley & Sons, Inc., 2001.

Cunningham, David and John Stuller. Circuit Analysis. Geneva: Houghton Mifflin, 1995.

Hearn, Grant and Andrew Metcalfe. Spectral Analysis in Engineering. New York: Halsted Press, 1996.

Hovanessian, S. A. Computational Mathematics in Engineering. Lexington: Lexington Books, 1976.

Johnson, David and Johnny Johnson. Mathematical Methods in Engineering and Physics. Englewood Cliffs: Prentice-Hall, Inc., 1982.

Mei, Chiang. Mathematical Analysis in Engineering. New York: Cambridge University Press, 1995.