What is a 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.
How do you solve 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 boundary value problem in differential equations?
A Boundary value problem is a system of ordinary differential equations with solution and derivative values specified at more than one point. Most commonly, the solution and derivatives are specified at just two points (the boundaries) defining a two-point boundary value problem.
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.
Is Sturm Liouville operator 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.
What is Sturm Liouville system?
What is Sturm-Liouville form?
A Sturm-Liouville equation is a second order linear differential. equation that can be written in the form. (p(x)y′)′ + (q(x) + λr(x))y = 0. Such an equation is said to be in Sturm-Liouville form.