How do you find eigenvalues and eigenfunctions of Sturm-Liouville problem?

How do you find eigenvalues and eigenfunctions of Sturm-Liouville problem?

(p(x)y′)′ + (q(x) + λr(x))y = 0, a < x < b, (plus boundary conditions), is called an eigenfunction, and the corresponding value of λ is called its eigenvalue. The eigenvalues of a Sturm-Liouville problem are the values of λ for which nonzero solutions exist.

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 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.

Which of the following is Eigen function of D DX?

The function eax is an eigenfunction of the operator d/dx because (d/dx)eax ¼ aeax, which is a constant (a) multiplying the original function. the function satisfies an eigenvalue equation.

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.

Is Sturm-Liouville self adjoint?

Sturm–Liouville equations as self-adjoint differential operators. In this space L is defined on sufficiently smooth functions which satisfy the above regular boundary conditions. Moreover, L is a self-adjoint operator: with the same eigenfunctions.

How is the Sturm-Liouville problem formulated in operator form?

Although a Sturm–Liouville problem can be formulated in operator form as L [ y ] = λ y similarly to matrix eigenvalue problem Ax = λ x, the operator L is usually an unbounded differential operator and y is a smooth function.

What are the eigenvalues of a SL boundary value problem?

A SL boundary value problem (BVP) consists of equation (2.1) together with BC (2.4)-(2.6). With conditions (2.2) and (2.3) it is well known that this problem is a regular self-adjoint SL problem which has an in\\fnite but countable number of only real eigenvalues.

When do you call a Sturm-Liouville problem classical?

There are two kinds of Sturm–Liouville problems. One is called classical or regular when p ( x) > 0 and w ( x) > 0 for all points from the closed interval x ∈ [0,ℓ]. These assumptions are necessary to render the theory as simple as possible while retaining considerable generality.

Which is a generalization of Sturm-Liouville theory?

Sturm–Liouville theory is actually a generalization for infinite dimensional case the famous eigenvalue/eigenvector problems for finite square matrices that we discussed in Part I of this tutorial.