What is 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. As an illustration we analyse small planar oscillations of hanging chain.

What is the point of Sturm-Liouville?

The two-body system equation describes the evolution of a two-body system under the influence of torque. The Sturm-Liouville form of the equation helps understand the spectral of two-body system.

What is Sturm-Liouville problem explain?

Sturm-Liouville problem, or eigenvalue problem, in mathematics, a certain class of partial differential equations (PDEs) subject to extra constraints, known as boundary values, on the solutions.

What is Laguerre and Legendre function?

The Legendre, Laguerre, and Hermite equations are all homogeneous second order Sturm-Liouville equations. In solving these equations explicit solutions cannot be found. That is solutions in in terms of elementary functions cannot be found. In many cases it is easier to find a numerical or series solution.

How do you solve the Sturm-Liouville problem?

These equations give a regular Sturm-Liouville problem. Identify p,q,r,αj,βj in the example above. y(x)=Acos(√λx)+Bsin(√λx)if λ>0,y(x)=Ax+Bif λ=0. Let us see if λ=0 is an eigenvalue: We must satisfy 0=hB−A and A=0, hence B=0 (as h>0), therefore, 0 is not an eigenvalue (no nonzero solution, so no eigenfunction).

What is fourth order SL theory?

A discrete fourth-order elliptic theory on a one-dimensional interval is constructed. It is based on ‘Hermitian. derivatives’ and compact higher-order finite difference operators, and is shown to possess the analogues. of the standard elliptic theory such as coercivity and compactness.

Are Legendre polynomials orthogonal?

In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applications.

How do you know if a polynomial is orthogonal?

(c) A polynomial p \= 0 is an orthogonal polynomial if and only if (p,q) = 0 for any polynomial q with deg q < deg p. p(x)q(x)dx. Note that (xn,xm) = 0 if m + n is odd.

Where are Legendre polynomials used?

For example, Legendre and Associate Legendre polynomials are widely used in the determination of wave functions of electrons in the orbits of an atom [3], [4] and in the determination of potential functions in the spherically symmetric geometry [5], etc.

How do you find a Legendre polynomial?

Legendre polynomials The simplest representatives of the associated Legendre functions are the Legendre polynomials, which are functions of zero order: p n ( μ ) = p n 0 ( μ ) . The Legendre polynomials appear, for example, as expansion coefficients of the following generating function: (2.1.33)

Are Legendre polynomials Orthonormal?