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 might bear his identify whereas engaged on the warmth circulate downside.
A Fourier sequence represents a periodic sign as a sum of sinusoids whose frequencies are integer a number of of the basic frequency.
We intuitively know {that a} sizzling spot in a conductive medium will unfold warmth in all instructions till the temperature is uniform. There isn’t any 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 operate u(x, t) in the issue of a one-dimensional conductive medium similar to a skinny steel bar. Because it seems, the spatial frequency elements of the preliminary temperature profile shall 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 sooner 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 operate outlined on a finite area. We’ll solid the issue such that the ensuing Fourier sequence has some fascinating properties on the area boundaries. This method will repay once we apply the Fourier sequence to resolve an issue involving differential equations with some constraints on the boundaries.
Fourier sequence: a software to characterize periodic capabilities
Fourier sequence can approximate periodic capabilities. Let g(x) be a periodic operate with interval 2L.
Why a interval of 2L?
We’re serious about capabilities outlined on the finite area [0, L]. We will 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 thus far later.
Assuming a Fourier sequence exists, we will write g(x) as:
For example, 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 provides 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 embody within the summation, the extra exact would be the approximation. Determine 2 reveals 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 will truncate the summation and nonetheless have an affordable approximation of the unique operate.
- The unique sign could possibly be a pattern of discrete factors. The Fourier sequence can interpolate the operate wherever on the x-axis.
Features 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 sequence 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 sequence illustration of f(x). In any case, we don’t care how the Fourier sequence behaves outdoors the [0, L] vary.
The naive periodic replication of f(x)
Probably the most easy option 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 get hold of the Fourier sequence reconstruction proven in Determine 4:
The Fourier sequence carefully 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 suggests the temperature gradient have to be 0 at each ends. Alternatively, we might have arbitrary set temperatures at x=0 and x=L. In these widespread 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 method 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 reveals the Fourier sequence 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 sequence is completely manufactured from cosine phrases. As a consequence, the spinoff of the Fourier sequence 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 situation the place, for instance, the steel bar is remoted, so there isn’t any warmth leakage on the extremities.
Odd half-range growth
What if we created an odd operate 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 reveals the Fourier sequence reconstruction of f(x):
g(x) being odd, the Fourier sequence is made completely of sine phrases. Because of this, the Fourier sequence is zero at x=0 and x=L. This property may be exploited, for instance, once we simulate the form of an oscillating guitar string. The string top 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 easy transition, each in worth and in spinoff, at x=L, we will append a rotated copy of f(x) within the [L, 2L] interval and make this operate even. Determine 9 reveals an instance:
The Fourier integration for the even quarter-range growth of f(x) yields equations (14) to (16):
Determine 10 reveals 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 operate odd.
The Fourier integration for the odd quarter-range growth of f(x) yields equations (17) to (19):
Determine 12 reveals the Fourier sequence reconstruction of f(x):
We will see that the reconstruction goes via 0 at x=0. The spinoff is zero at x=L, even when the unique sign spinoff is not.
Conclusion
We thought of the issue of discovering an acceptable Fourier sequence growth for a sign f(x) outlined over the finite interval [0, L]. Fourier sequence 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 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 easy 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 look at 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 hard and fast temperature at x=L). The seemingly arbitrary periodic capabilities we created on this submit will out of the blue make sense!
References
(R1) Superior Engineering Arithmetic, Erwin Kreyszig, John Wiley and Sons, 1988
Assorted Flavors of Fourier Sequence on a Finite Area was initially revealed in In the direction of Information Science on Medium, the place individuals are persevering with the dialog by highlighting and responding to this story.