What is a Sturm-Liouville system?
Sturm-Liouville systems are second-order linear differential equations with boundary conditions of a particular type, and they usually arise from separation of variables in partial differential equations which represent physical systems.
What is the fourth order Chebyshev polynomial?
Definition 5 The fourth kind of Chebyshev polynomial in of degree is denoted by W n ∗ and is defined by. For x ∈ [ a , b ] , if we put s = 2 x − ( a + b ) b − a , then s ∈ [ − 1 , 1 ] and W n ∗ ( x ) = W n ( s ) .
What is Sturm Liouville eigenvalue problem?
The problem of finding a complex number µ if any, such that the BVP (6.2)-(6.3) with λ = µ, has a non-trivial solution is called a Sturm-Liouville Eigen Value Problem (SL-EVP). Such a value µ is called an eigenvalue and the corresponding non-trivial solutions y(.; µ) are called eigenfunctions.
Is Sturm Liouville operator Hermitian?
3 Hermitian Sturm Liouville operators. In mathematical physics the domain is often delimited by points a and b where p(a)=p(b)=0. If we then add a boundary condition that w(x)p(x) and w (x)p(x) are finite (or a specific finite number) as x→a b for all solutions w(x), the operator is Hermitian.
Is the Sturm-Liouville operator Hermitian?
What is chebyshev differential equation?
Chebyshev’s differential equation is (1 − x2)y′′ − xy′ + α2y = 0, where α is a constant. (c) Find a polynomial solution for each of the cases α = n = 0,1,2,3.
Which is the correct equation for the Sturm Liouville problem?
A Sturm-Liouville problem consists of A Sturm-Liouville equation on an interval: (p(x)y′)′ +(q(x) +λr(x))y = 0, a < x < b, (1) together with Boundary conditions, i.e. specified behavior of y at x = a and x = b. We will assume that p, p′, q and r are continuous and p > 0 on (at least) the open interval a < x < b.
Which is the solution of the Chebyshev’s differential equation?
The solutions of the Chebyshev’s differential equation with is called Chebyshev Polynomials which form a complete orthogonal set on the interval with respect to . ( Further detail; see plots .)
Which is a general feature of the Chebyshev polynomials?
This follows from the fact that the Chebyshev polynomials solve the Chebyshev differential equations which are Sturm–Liouville differential equations. It is a general feature of such differential equations that there is a distinguished orthonormal set of solutions.
How are Chebyshev polynomials related to sine and cosine functions?
The Chebyshev polynomials are two sequences of polynomials related to the sine and cosine functions, notated as Tn(x) and Un(x) . They can be defined several ways that have the same end result; in this article the polynomials are defined by starting with trigonometric functions :