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Selasa, 30 Oktober 2012

Bessel functions

Cylinder functions of the first kind. A Bessel function of order can be defined as the series
(*)
which converges throughout the plane. A Bessel function of order is the solution of the corresponding Bessel equation. If the argument and the order are real numbers, the Bessel function is real, and its graph has the form of a damped vibration (Fig.); if the order is even, the Bessel function is even, if odd, it is odd.

Figure: b015840a
Graphs of the functions and .
The behaviour of a Bessel function in a neighbourhood of zero is given by the first term of the series (*); for large , the asymptotic representation
holds. The zeros of a Bessel function (i.e. the roots of the equation ) are simple, and the zeros of are situated between the zeros of . Bessel functions of "half-integral" order are expressible by trigonometric functions; in particular
The Bessel functions (where are the positive zeros of , ) form an orthogonal system with weight in the interval . Under certain conditions the following expansion is valid:
In an infinite interval this expansion is replaced by the Fourier–Bessel integral
The following formulas play an important role in the theory of Bessel functions and their applications:
1) the integral representation
2) the generating function
3) the addition theorem for Bessel functions of order zero
4) the recurrence formulas
For references, see Cylinder functions.

Senin, 22 Oktober 2012

 



Orthogonal Trajectories



We have seen before (see separable equations for example) that the solutions of a differential equation may be given by an implicit equation with a parameter something like
displaymath190
This is an equation describing a family of curves. Whenever we fix the parameter C we get one curve and vice-versa. For example, consider the families of curves
displaymath194
where m and C are parameters. Clearly, we may change the names of the variables and still have the same geometric curves. For example, the above families define the same geometric object as
displaymath200
Note that the first family describes all the lines passing by the origin (0,0) while the second the family describes all the circles centered at the origin (including the limit case when the radius 0 which reduces to the single point (0,0)) (see the pictures below).

and
In this page, we will only use the variables x and y. Any family of curves will be written as
displaymath210
One may ask whether any family of curves may be generated from a differential equation? In general, the answer is no. Let us see how to proceed if the answer were to be yes. First differentiate with respect to x, and get a new equation involving in general x, y, tex2html_wrap_inline218 , and C. Using the original equation, we may able to eliminate the parameter C from the new equation.

Example. Find the differential equation satisfied by the family
displaymath224
Answer. We differentiate with respect to x, to get
displaymath228
Since we have
displaymath230
then we get
displaymath232
You may want to do some algebra to make the new equation easy to read. The next step is to rewrite this equation in the explicit form
displaymath234
this is the desired differential equation.

Example. Find the differential equation (in the explicit form) satisfied by the family
displaymath224
Answer. We have already found the differential equation in the implicit form
displaymath232
Algebraic manipulations give
displaymath240
Let us reconsider the example of the two families
displaymath200
If we draw the two families together on the same graph we get

As we see here something amazing happened. Indeed, it is clear that whenever one line intersects one circle, the tangent line to the circle (at the point of intersection) and the line are perpendicular or orthogonal. We say the two curves are orthogonal at the point of intersection.

Definition. Consider two families of curves tex2html_wrap_inline244 and tex2html_wrap_inline246 . We say that tex2html_wrap_inline244 and tex2html_wrap_inline246 are orthogonal whenever any curve from tex2html_wrap_inline244 intersects any curve from tex2html_wrap_inline246 , the two curves are orthogonal at the point of intersection.

For example, we have seen that the families y = m x and tex2html_wrap_inline258 are orthogonal. One may then ask the following natural question:

Given a family of curves tex2html_wrap_inline260 , is it possible to find a family of curves which is orthogonal to tex2html_wrap_inline260 ?
The answer to this question has many implications in many areas such as physics, fluid-dynamics, etc... In general this question is very difficult. But in some cases, we may be able to carry on the calculations and find the orthogonal family. Let us show how.

Consider the family of curves tex2html_wrap_inline260 . We assume that an associated differential equation may be found, say
displaymath234
We know that for any curve from the family passing by the point (x,y), the slope of the tangent at this point is f(x,y). Hence the slope of the line perpendicular (or orthogonal) to this tangent is tex2html_wrap_inline272 which happens to be the slope of the tangent line to the orthogonal curve passing by the point (x,y). In other words, the family of orthogonal curves are solutions to the differential equation
displaymath276
From this we see what we have to do. Indeed consider a family of curves tex2html_wrap_inline260 . In order to find the orthogonal family, we use the following practical steps
Step 1. Find the associated differential equation.
Step 2. Rewrite this differential equation in the explicit form displaymath234
Step 3. Write down the differential equation associated to the orthogonal family displaymath276
Step 4. Solve the new equation. The solutions are exactly the family of orthogonal curves.
Step 5. You may be asked to give a geometric view of the two families. Also you may be asked to find a specific curve from the orthogonal family (something like an IVP).
Example. Find the orthogonal family to the family of circles
displaymath284
Answer. First, we look for the differential equation satisfied by the circles. We differentiate with respect to the variable x to get
displaymath288
We rewrite this equation in the explicit form
displaymath290
Next we write down the equation for the orthogonal family
displaymath292
This is a linear as well as a separable equation. If we use the technique of linear equations, we get the integrating factor
displaymath294
which gives
displaymath296
We recognize the family of lines and we confirm our earlier observation (that the two families are indeed orthogonal).

This example is somehow easy and was given here to illustrate the technique.

Example. Find the orthogonal family to the family of circles
displaymath298
Answer. We have seen before that the explicit differential equation associated to the family of circles is
displaymath300
Hence the differential equation for the orthogonal family is
displaymath302
We recognize an homogeneous equation. Let us use the technique developed to solve this kind of equations. Consider the new variable tex2html_wrap_inline304 (or equivalently y = x z). Then we have
displaymath308
and
displaymath310
Hence we have
displaymath312
Algebraic manipulations imply
displaymath314
This is a separable equation. The constant solutions are given by
displaymath316
which gives z=0. The non-constant solutions are found once we separate the variables
displaymath320
and then we integrate
displaymath322
Before we perform the integration for the left-hand side, we need to use partial decomposition technique. We have
displaymath324
We will leave the details to you to show that A = 1, B=-2, and C=0. Hence we have
displaymath332
Hence
displaymath334
which is equivalent to
displaymath336
where tex2html_wrap_inline338 . Putting all the solutions together we get
displaymath340
Going back to the variable y, we get
displaymath344
which is equivalent to
displaymath346
We recognize a family of circles centered on the y-axis and the line y=0 (the x-axis which was easy to guess, isn't it?)

If we put both families together, we appreciate better the orthogonality of the curves (see the picture below).


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Mohamed Amine Khamsi