Select the one which behaves properly on the boundaries

For those who search for the historical past of Fourier evaluation, you’ll see that Jean-Baptiste Joseph Fourier formalized the sequence that will bear his identify whereas engaged on the warmth stream downside.

A Fourier sequence 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} scorching spot in a conductive medium will unfold warmth in all instructions till the temperature is uniform. There isn’t a seen oscillatory habits on this phenomenon, neither in house nor time. Why then introduce a sequence of sinusoids?

The preliminary temperature profile, the governing differential equation, and the boundary circumstances decide the evolution of the temperature perform u(x, t) in the issue of a one-dimensional conductive medium comparable to a skinny steel bar. Because it seems, the spatial frequency elements 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 are going to overview the fundamentals of Fourier sequence for a perform outlined on a finite area. We’ll forged the issue such that the ensuing Fourier sequence has some fascinating properties on the area boundaries. This strategy will repay once we apply the Fourier sequence to resolve an issue involving differential equations with some constraints on the boundaries.

Fourier sequence: a instrument to characterize periodic features

Fourier sequence can approximate periodic features. Let g(x) be a periodic perform with interval 2L.

Why a interval of 2L?

We’re focused on features outlined on the finite area [0, L]. We will assemble a periodic perform g(x) whose interval is 2L from the perform f(x) outlined over [0, L] with some padding chosen to have fascinating properties. We’ll get again thus far later.

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

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

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

These values, the Fourier coefficients, enable 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 will 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 sequence is the sum of an infinite variety of phrases, however we are able to truncate the summation and nonetheless have an affordable approximation of the unique perform.
• The unique sign might be a pattern of discrete factors. The Fourier sequence can interpolate the perform wherever on the x-axis.

Capabilities outlined on a finite area

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

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

The naive periodic replication of f(x)

Probably the most simple technique to construct g(x) is to duplicate 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 sequence reconstruction proven in Determine 4:

The Fourier sequence 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 sequence interprets the transitions at x=0 and x=L as finite discontinuities.

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

For a lot of engineering circumstances, that is problematic. For instance, within the case of warmth switch in a skinny steel bar, what occurs on the bar extremities (a.okay.a. the boundary circumstances) is an intrinsic a part of the issue description. We might have an remoted bar, which means 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’t 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 an alternative 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 sequence reconstruction of f(x):

A function 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 sequence is completely fabricated from cosine phrases. As a consequence, the spinoff of the Fourier sequence is zero at x=0 and x=L. You may confirm this by differentiating equation (1) with respect to x, with all bₙ phrases set to 0.

That’s what we wish in a state of affairs the place, for instance, the steel bar is remoted, so there is no such thing as a warmth leakage on the extremities.

Odd half-range growth

What if we created an odd perform as a substitute? This may be completed 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 sequence reconstruction of f(x):

g(x) being odd, the Fourier sequence is made completely of sine phrases. For that reason, the Fourier sequence is zero at x=0 and x=L. This property could be exploited, for instance, once we simulate the form of an oscillating guitar string. The string peak is constrained to 0 at x=0 and x=L, so we might naturally mannequin the preliminary situation with odd half-expansion.

Even quarter-range growth

We could be much more inventive and design a periodic perform with a interval of 4L. If we wish 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 perform 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 sequence reconstruction of f(x):

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

Odd quarter-range growth

The final case we’ll take into account is once we need 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 perform odd.

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

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

We will see that the reconstruction goes by way of 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 appropriate Fourier sequence growth for a sign f(x) outlined over the finite interval [0, L]. Fourier sequence apply to periodic features, so we needed to construct a periodic perform that matches f(x) over the outlined area. We noticed 4 strategies to outline the periodic perform g(x). Every ensures particular properties on the vary boundaries:

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

In a future story, we are going to study how warmth is transferred in a skinny steel bar. The answer includes changing the preliminary temperature profile to a Fourier sequence. We’ll observe that the selection for the kind of Fourier sequence growth is of course dictated by the boundary circumstances (e.g., the bar is remoted at x=0 and held to a set temperature at x=L). The seemingly arbitrary periodic features we created on this put up will all of a sudden make sense!