Equations of Physics and Engineering

02/12/2015 - 18:31 von Basti05b | Report spam
Equations of Physics and Engineering

Hello:

How do you solve let say Bessel differential equations with order of an integer and half?

We know that in the textbooks, we approach it via infinite series solutions.

In Volume 1 of my book, we use Riccati theorems to handle classical equations of physics and engineering.

While some examples provided for Bessel and Associated Legendre differential equations, using Riccati equations, but treatment of such equations postponed to other volumes of author's series.

You will have a chance to examine all classical equations of physics, using Riccati theorems provided in the book, and advance further.

I suppose it takes a decade such that, we eventually transform solving of differential equations of classical type to Riccati DNA.

Applications are ranging from solving integrations to differential equations. For example, as noted in DNA of mathematics book, the following function:

S1: (t+2)/(t*sqrt((t+1)*(2*t^2+b*(t+1))));

Where b is a non-zero real number, has integration of all rational functions.
However Maple, and Mathematica, expresses them with Elliptic functions.

These are integrations squeezed between Riccati and polynomials, and presented normally as a class, depending on degree of polynomial.

Many such elliptic solving integrals with complete rational solutions will be presented in the future volumes.

Sincerely

Dr.Mehran Basti
Retired mathematician

PS: Volume 1 will be ready by Christmas.
 

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#1 Basti05b
14/12/2015 - 00:15 | Warnen spam
Hello:

In volume 1, we notice that if we inflate a polynomial, then integrals squeezed in its space are solvable. For example a degree 6 polynomial:

P1:=x^6+1/1296*b1^5*t*x^5+5/432*b1^4*t*x^4+5/54*b1^3*t*x^3+5/12*b1^2*t*x^2+b1*t*x+t;

This polynomial has a Riccati representation. Although you may be able to solve it with simple algebraic maneuvering.

If we now let b1-> b1(t), and t-> T(t), then differential equation representing the new polynomial produces integral which is thus solvable.

As polynomials and Riccati representations become more complex, we can handle elliptic integrals.

We do not in this new science consider a polynomial individually, but as a class.

The above class of polynomials are solvable for any degree with its Riccati representations (it is explained in Volume 1, coming up about 2 weeks).

Sincerely

Dr.Mehran Basti

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