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ry’s Equation

MATH 2640, Fall 2012Dr. Saber Hashim

Krista Sexton & Bongsu Kim

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iry function The Airy function is namedafter the British astronomerand physicist George Biddell

Airy (1801 – 1892), whoencountered it in his earlystudy of optics in physics (Airy1838). The notation Ai(x) wasintroduced by Harold Jeffreys.

Airy had become the British Astronomer Royal in 1835, andhe held that post until hisretirement in 1881.

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iry function Theory

Theory behind Airy’s equation is thetheory of electromagnetic diffractionand radio wave propagation. Airy

functions play an important role in thetheory of asymptotic representations ofvarious special functions. They havediverse applications in mathematicalphysics — for example, in the theory ofthe diffraction of radio waves at theearth’s surface. The Airy function alsounderlies the form of the intensitynear an optical directional caustic,such as that of the rainbow.Historically, this was themathematical problem that led Airy todevelop this special function. The Airyfunction is also important inmicroscopy and astronomy; thisdescribes the pattern, due todiffraction and interference, producedby a point source of light.

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What kind of problems is the method well-suited for?

y + (t ) y + q (t ) y =0

• The series solutions method is usedprimarily, when the coefficients p (t )

or q (t ) are non-constant.• Airy functions are prominently used in

problems defined by nonlinear equations.

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The easiest example of Airy’s equation

y - t y =0

This equation is used in physicsto model the diffraction of light.We want to find powerseries solutions for thissecond-order linear

differential equa tion.

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Continued…

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Continued…

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Useful for the followingcertain class of applications

– Classical physics• Physical optics,

Electromagnetism, Radiativetransfer

–

y ''- xy =0• Fluid Mechanics

– Continuity equation

– Momentum Equation in x and ydirections

– Quantum physics• Nonlinear wave propagation

– WKB method» Solves certain differential

equations

» , where

F(x) is smooth andpositive on the O(1) scale[1].

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The Airy’s equation works well when trying to solve Power Series

Radius of Convergence

Example:

Apply Ratio Test,which gives us

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Continued…

Analyzing the resultthe ratio test tells us

that the powerseries convergesonly when

or | x + 3 | < 4

The radius of convergence is R = 4

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