- #1
mano0or
- 3
- 0
Show that the Legendre equation as well as the Bessel equation for n=integer are Sturm Liouville equations and thus their solutions are orthogonal. How I can proove that ..?
:(
Last edited:
Legendre equation , the Bessel equation and Sturm Liouville equation
- Thread starter mano0or
- Start date
In summary, the Legendre equation and the Bessel equation for n=integer are both examples of Sturm Liouville equations. This means that their solutions are orthogonal, a well-known property of Sturm Liouville problems. The proof for this can be found in any mathematical methods book.
Show that the Legendre equation as well as the Bessel equation for n=integer are Sturm Liouville equations and thus their solutions are orthogonal. How I can proove that ..?
:(
- #2
Jorriss
- 1,083
- 26
Can you put it in sturm-liouville form?
- #3
mano0or
- 3
- 0
Yes ...
see . this is the form of Sturm Liouville
and this table help me
i found the mathematical explain for bessel .
but i don't know how i can prove their solutions are orthogonal ..!
can you help me
see . this is the form of Sturm Liouville
and this table help me
i found the mathematical explain for bessel .
but i don't know how i can prove their solutions are orthogonal ..!
can you help me
Last edited:
- #4
Jorriss
- 1,083
- 26
Did you post the problem exactly? If so, as I interpret it, you do not need to prove that yourself. It is well known that the solutions to sturm-liouville problems are orthogonal and the proof can be found in any mathematical methods book.
- #5
mano0or
- 3
- 0
yea
i understand that hour before .. i wasn't have enough information about sturm-liouville
or maybe my brain stopped :tongue:
thanks :)
i understand that hour before .. i wasn't have enough information about sturm-liouville
or maybe my brain stopped :tongue:
thanks :)
1. What are the Legendre, Bessel, and Sturm-Liouville equations?
The Legendre, Bessel, and Sturm-Liouville equations are three types of differential equations that are commonly used in mathematical physics and engineering. Each equation has its own specific form and is used to solve different types of problems.
2. What is the purpose of the Legendre equation?
The Legendre equation, named after French mathematician Adrien-Marie Legendre, is used to solve problems involving spherical harmonics, which are important in quantum mechanics and electromagnetic theory. It is also used in the study of heat conduction, fluid mechanics, and other areas of physics and engineering.
3. How is the Bessel equation used in real-world applications?
The Bessel equation, named after German mathematician Friedrich Bessel, is used to model phenomena that exhibit cylindrical or spherical symmetry, such as heat conduction in a circular or spherical object, vibrations of a circular drum, and electromagnetic fields in a cylindrical or spherical region. It is also used in signal processing and image analysis.
4. What is the significance of the Sturm-Liouville equation?
The Sturm-Liouville equation, named after French mathematician Jacques Charles François Sturm and French mathematician Joseph Liouville, is a special type of second-order linear differential equation that is used to solve boundary value problems. It has wide-ranging applications in physics, engineering, and mathematics, including in quantum mechanics, fluid dynamics, and signal processing.
5. How are these equations related to each other?
The Legendre, Bessel, and Sturm-Liouville equations are all examples of special types of second-order linear differential equations. They have similar forms and can be solved using similar techniques, such as series solutions and integral transforms. They also have applications in similar areas of mathematics and physics, making them important tools for scientists and engineers.
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