Are the Singular Points of the Chebyshev Equation Regular?

Click For Summary
SUMMARY

The singular points of the Chebyshev equation, specifically at x = 1 and x = -1, have been determined to be regular singular points. This conclusion is reached by evaluating the limits of the functions p(x) and q(x) as defined in the Chebyshev differential equation. The conditions for regular singular points are satisfied, confirming that both x = 1 and x = -1 are indeed regular singular points.

PREREQUISITES
  • Understanding of Chebyshev differential equations
  • Familiarity with singular points in differential equations
  • Knowledge of limit evaluation in calculus
  • Basic concepts of regular and irregular singular points
NEXT STEPS
  • Study the properties of Chebyshev polynomials
  • Learn about the Frobenius method for solving differential equations
  • Explore the implications of regular singular points in differential equations
  • Investigate the applications of Chebyshev equations in numerical analysis
USEFUL FOR

Mathematicians, students of differential equations, and researchers interested in the properties of Chebyshev equations and their applications in mathematical modeling.

Poirot1
Messages
243
Reaction score
0
I have computed the singular points of Chebyshev equation to be x= 1, -1. What is the best way to find whether they are regular? Thanks.
 
Physics news on Phys.org
The Chebisheff DE is...

$\displaystyle y^{\ ''} - \frac{x}{1-x^{2}}\ y^{\ '} + \frac{\alpha^{2}}{1-x^{2}}\ y= y^{\ ''} + p(x)\ y^{\ '} + q(x)\ y=0$ (1)

If $x_{0}$ is a singularity of p(x) and q(x) and both the limits...

$\displaystyle \lim_{x \rightarrow x_{0}} (x-x_{0})\ p(x)$

$\displaystyle \lim_{x \rightarrow x_{0}} (x-x_{0})^{2}\ q(x)$ (2)

... exist finite, then $x_{0}$ is a regular singular point. You can verify that $x_{0}=1$ and $x_{0}=-1$ are both regular singular points...

Kind regards

$\chi$ $\sigma$
 
Thanks
 

Similar threads

Undergrad Question about the roots of Chebyshev polynomials
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Improper boundary in non-linear ODE (pseudospectral methods)
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Converting Second Order ODE to Hypergeometric Function
  • · Replies 7 ·
Replies
7
Views
3K
Graduate Searching for radial part solution of Klein-Gordon type equation
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Techniques for Solving Equations with Irregular Singular Points
  • · Replies 2 ·
Replies
2
Views
5K
MHB What Are the Singular Points of the Differential Equation w'' + z*w' + kw = 0?
  • · Replies 1 ·
Replies
1
Views
2K
Book recommendations about singular points of algebraic curves
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Isn't Ordinary Point Just a Specific Case of Regular Singular Point?
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Boundary conditions in the time evolution of Spectral Method in PDE
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Chebyshev Differentiation Matrix
  • · Replies 2 ·
Replies
2
Views
2K