Pde involving airy function

In summary, the conversation discusses using Fourier transform methods to derive an expression for the solution of a partial differential equation involving the Airy function. The approach involves interchanging the order of integration and using the Fourier transform of the product of two functions to obtain the desired form. Small corrections to the original approach are also mentioned.
  • #1
Qyzren
44
0
pde involving airy function!

If u(x,t) satisfies ∂u/∂t + ∂³u/∂x³ = 0, with u(x,0) = f(x), and u, ∂u/∂x, ∂²u/∂x² -> 0 as |x| -> ∞, use Fourier transform methods to show that u(x,t) = (3t)^(-1/3) ∫f(y) Ai[(x-y)/((3t)^(-1/3))] dy (integral from -∞ to ∞), where Ai(x) is the Airy function, for which Ai(x) = 1/π ∫cos(ω³/3 + ωx) dω (integral from 0 to ∞).


Attempt:
all my integrals are from now on are -∞ to ∞
F{u(x,t)} = U(ω,t) = 1/(2π) ∫u(x,t) e^[iωx] dx
F{u_t} = ∂U/∂t(ω,t)
F{u_xxx} = (-iω)³U(ω,t)

=> ∂U/∂t + iω³U = 0
∂U/U = -iω³∂t
(i'm going to use w for ω for more convience)
U(w,t) = c(w)*e^[-iw³t] where c(w) arb fn of w.
=> u(x,t) = ∫c(w) e^[-iw³t] e^[-iwx] dw
The IC gives
u(x,0) = f(x) = ∫c(w) e^[-iwx] dw
=> c(w) = F{f(x)} = 1/(2π) ∫f(x)e^[iwx] dx
u(x,t) = ∫1/(2π) ∫f(y) e^[iwy] dy e^[-iw³t] e^[-iwx] dw
interchanging order of integration
u(x,t) = 1/(2π)∫f(y)[∫e^[-iw(x-y)] e^[-iw³t] dw] dy

and I'm stuck...
did i make a mistake somewhere? if not how do i get it to the form they want?
ie u(x,t) = (3t)^(-1/3) ∫f(y) Ai[(x-y)/((3t)^(-1/3))] dy

Thank you for your help
 
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  • #2
!



Thank you for your post. Your approach so far seems correct, but there are a few small mistakes that I would like to point out. First, in your expression for U(w,t), it should be e^(-iw³t) instead of e^(-iw³t). This will affect the final result.

Next, when you interchange the order of integration, you should also change the limits of integration for w to be from 0 to ∞, since the Airy function is only defined for positive values of ω.

Finally, to get the desired form of u(x,t), you need to use the identity for the Fourier transform of the product of two functions. This will give you the integral in the form of the Airy function, with the appropriate scaling factor of (3t)^(-1/3).

I hope this helps you complete your solution. Good luck with your research!
 

What is a PDE involving Airy function?

A PDE (Partial Differential Equation) involving Airy function is a type of mathematical equation that involves the Airy function, which is a special function used in many areas of physics and engineering. It is a solution to the differential equation known as the Airy equation, and it is often used to describe wave phenomena.

What is the Airy function?

The Airy function is a special function that is used to solve differential equations in physics and engineering. It is named after the British astronomer George Biddell Airy who first studied it in the 19th century. The Airy function can be described using a complex number, and it has many applications in fields such as optics, acoustics, and fluid mechanics.

How is the Airy function related to PDEs?

The Airy function is used to solve PDEs involving wave phenomena, such as the wave equation, the heat equation, and the Schrödinger equation. It is a solution to the Airy equation, which is a second-order linear differential equation. By using the Airy function, scientists and engineers can find solutions to these types of PDEs and better understand the behavior of waves in different systems.

What are some applications of PDEs involving Airy function?

PDEs involving Airy function have many applications in various fields of science and engineering. For example, they are used in optics to describe the behavior of light waves passing through lenses and other optical devices. In acoustics, they can be used to model the propagation of sound waves in different media. PDEs involving Airy function also have applications in fluid mechanics to study the behavior of water waves and air flow.

What are the challenges of working with PDEs involving Airy function?

One of the main challenges of working with PDEs involving Airy function is that they can be difficult to solve analytically. This means that finding exact solutions to these equations can be a complex and time-consuming process. As a result, scientists and engineers often use numerical methods to approximate solutions for these types of PDEs. Additionally, the behavior of the Airy function can be quite complex, which can make it challenging to interpret the results of PDEs involving this function.

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