For those who search for the historical past of Fourier evaluation, you’ll see that Jean-Baptiste Joseph Fourier formalized the collection that might bear his title whereas engaged on the warmth move downside.

A Fourier collection represents a periodic sign as a sum of sinusoids whose frequencies are integer a number of of the elemental frequency.

We intuitively know {that a} sizzling spot in a conductive medium will unfold warmth in all instructions till the temperature is uniform. There is no such thing as a seen oscillatory habits on this phenomenon, neither in house nor time. Why then introduce a collection of sinusoids?

The preliminary temperature profile, the governing differential equation, and the boundary situations decide the evolution of the temperature operate u(x, t) in the issue of a one-dimensional conductive medium reminiscent of a skinny steel bar. Because it seems, the spatial frequency parts of the preliminary temperature profile will probably be damped by a decaying exponential over time, with an exponential issue that grows just like the sq. of the spatial frequency. In different phrases, excessive frequencies within the preliminary temperature profile decay a lot quicker than the low frequencies, which explains the smoothing of the temperature distribution.

On this story, **we’ll evaluation the fundamentals of Fourier collection for a operate outlined on a finite area**. We’ll forged the issue such that the ensuing Fourier collection has some fascinating properties on the area boundaries. This strategy will repay after we apply the Fourier collection to resolve an issue involving differential equations with some constraints on the boundaries.

## Fourier collection: a software to signify periodic capabilities

Fourier collection can approximate periodic capabilities. Let g(x) be a periodic operate with interval 2L.

Why a interval of 2L?

We’re curious about capabilities outlined on the finite area [0, L]. We are able to assemble a periodic operate g(x) whose interval is 2L from the operate f(x) outlined over [0, L] with some padding chosen to have fascinating properties. We’ll get again up to now later.

Assuming a Fourier collection exists, we are able to write g(x) as:

For instance, let’s take into account the next periodic operate g(x), with interval 2L = 0.6:

Making use of equations (2), (3), (4) and utilizing Simpson numerical integration offers the next values for a₀, aₙ, and bₙ:

These values, the Fourier coefficients, permit us to construct an approximation of g(x) with equation (1). The extra phrases we embrace within the summation, the extra exact would be the approximation. Determine 2 exhibits a number of approximations with numerous numbers of phrases from the summation in equation (1).

We are able to already formulate a number of observations:

- Finite discontinuities within the sign are tolerable, however they generate wiggling within the reconstructed approximation. We refer to those oscillations within the neighborhood of discontinuities because the Gibbs phenomenon.
- The Fourier collection is the sum of an infinite variety of phrases, however we are able to truncate the summation and nonetheless have an inexpensive approximation of the unique operate.
- The unique sign may very well be a pattern of discrete factors. The Fourier collection can interpolate the operate anyplace on the x-axis.

## Capabilities outlined on a finite area

In engineering issues, we regularly encounter capabilities outlined on a finite area. For instance, within the case of the one-dimensional temperature distribution of a conductive medium, the temperature operate is outlined over the [0, L] vary, the place L is the size of the skinny steel bar. How can the Fourier collection be used on this setting?

To reply this query, we first acknowledge that any periodic operate g(x) that coincides with the operate on curiosity f(x) over the vary [0, L] is a sound candidate for a Fourier collection illustration of f(x). In any case, we don’t care how the Fourier collection behaves outdoors the [0, L] vary.

## The naive periodic replication of f(x)

Probably the most simple option to construct g(x) is to copy f(x) within the interval [-L, 0], as in determine 3:

The Fourier integration for the naive periodic replication of f(x) yields equations (5) to (7):

By inserting (5), (6), (7) in equation (1) to f(x) from Determine 3, we receive the Fourier collection reconstruction proven in Determine 4:

The Fourier collection intently matches the unique sign, besides on the vary boundaries, the place the reconstruction oscillates and jumps. Since we explicitly constructed a periodic sign of interval L, the Fourier collection interprets the transitions at x=0 and x=L as finite discontinuities.

Finite discontinuities are allowed by the Fourier collection, however the Gibbs phenomenon degrades the reconstruction across the discontinuities.

For a lot of engineering instances, that is problematic. For instance, within the case of warmth switch in a skinny steel bar, what occurs on the bar extremities (a.ok.a. the boundary situations) is an intrinsic a part of the issue description. We might have an remoted bar, which suggests the temperature gradient should be 0 at each ends. Alternatively, we might have arbitrary set temperatures at x=0 and x=L. In these frequent situations, we can not use the naive periodic replication of f(x) as a result of the Gibbs phenomenon corrupts the sign on the ends of the vary.

## Even half-range growth

As a substitute of replicating f(x), we might have a flipped model of f(x) within the vary [-L, 0], like in Determine 5:

This strategy eliminates the discontinuities at x=0 and x=L. The Fourier integration for the even half-range growth of f(x) yields equations (8) to (10):

Determine 6 exhibits the Fourier collection reconstruction of f(x):

A characteristic of the even half-range growth is the truth that g(x) being even, all bₙ coefficients (Cf. equation (10)) are 0, and thus its Fourier collection is solely fabricated from cosine phrases. As a consequence, **the spinoff of the Fourier collection is zero at x=0 and x=L**. You’ll be able to confirm this by differentiating equation (1) with respect to x, with all bₙ phrases set to 0.

That’s what we would like in a state of affairs the place, for instance, the steel bar is remoted, so there isn’t a warmth leakage on the extremities.

## Odd half-range growth

What if we created an odd operate as a substitute? This may be achieved by pasting a rotated model of f(x) within the interval [-L, 0], like in Determine 7:

The Fourier integration for the odd half-range growth of f(x) yields equations (11) to (13):

Determine 8 exhibits the Fourier collection reconstruction of f(x):

g(x) being odd, the Fourier collection is made solely of sine phrases. Because of this, **the Fourier collection is zero at x=0 and x=L**. This property may be exploited, for instance, after we simulate the form of an oscillating guitar string. The string peak is constrained to 0 at x=0 and x=L, so we’d naturally mannequin the preliminary situation with odd half-expansion.

## Even quarter-range growth

We may be much more artistic and design a periodic operate with a interval of 4L. If we would like a spinoff of precisely 0 at x=0 and a clean transition, each in worth and in spinoff, at x=L, we are able to append a rotated copy of f(x) within the [L, 2L] interval and make this operate even. Determine 9 exhibits an instance:

The Fourier integration for the even quarter-range growth of f(x) yields equations (14) to (16):

Determine 10 exhibits the Fourier collection reconstruction of f(x):

Though it isn’t seen from the determine, the spinoff of the Fourier collection reconstruction is 0 at x=0 and an identical to the unique sign at x=L.

## Odd quarter-range growth

The final case we’ll take into account is after we desire a worth of 0 at x=0 and a spinoff of 0 at x=L. We construct g(x) by appending a flipped model of f(x) within the [L, 2L] vary and make this operate odd.

The Fourier integration for the odd quarter-range growth of f(x) yields equations (17) to (19):

Determine 12 exhibits the Fourier collection reconstruction of f(x):

We are able to see that the reconstruction goes by 0 at x=0. The spinoff is zero at x=L, even when the unique sign spinoff just isn’t.

## Conclusion

We thought-about the issue of discovering an acceptable Fourier collection growth for a sign f(x) outlined over the finite interval [0, L]. Fourier collection apply to periodic capabilities, so we needed to construct a periodic operate that matches f(x) over the outlined area. We noticed 4 strategies to outline the periodic operate g(x). Every ensures particular properties on the vary boundaries:

- Even half-range growth: The Fourier collection has a spinoff of 0 at x=0 and x=L
- Odd half-range growth: The Fourier collection has a worth of 0 at x=0 and x=L
- Even quarter-range growth: The Fourier collection has a spinoff of 0 at x=0 and clean worth and spinoff at x=L
- Odd quarter-range growth: The Fourier collection has a worth of 0 at x=0 and a spinoff of 0 at x=L

In a future story, we’ll study how warmth is transferred in a skinny steel bar. The answer entails changing the preliminary temperature profile to a Fourier collection. We’ll observe that the selection for the kind of Fourier collection growth is of course dictated by the boundary situations (e.g., the bar is remoted at x=0 and held to a hard and fast temperature at x=L). The seemingly arbitrary periodic capabilities we created on this put up will out of the blue make sense!