Such problems are called sturmliouville problems and their solutions have a rich structure as we shall see. Sturmliouville eigenvalue problem 8, 910 is called regular if the coe. It follows that hyjlyiis real, and therefore itself is real. Introduction to sturmliouville theory trinity university. Sturmliouville theorem, prehistory of the laplacian eigenvalue problem in rd lecturer. After nding the eigenvalues nand eigenfunctions y nof the appropriate sturmliouville eigenvalue problem, equation 2 is solved yielding t nt.
Sturmliouville problems 55 this has nontrivial solution for the pair a, b if and only if. The heat flow in a nonuniform rod is modeled by the partial differential equation c. The differential equation is said to be in sturmliouville form or selfadjoint form. The socalled sturmliouville problems define a class of eigenvalue problems, which include many of the previous problems as special cases. The eigenvalues of a sturm liouville problem are all of multiplicity one. The initial distribution of the temperature needs to be speci ed as well.
The eigenvalues of sturmliouville sl problems depend not only continuously but smoothly on the problem. Variational techniques for sturmliouville eigenvalue problems. We were able to find the eigenvalues of problems 14 explicitly because in each problem the coefficients in the boundary conditions satisfy \\alpha\beta0\ and \\rho\delta. The fourier cosine transform of a function fx will be denoted by. The proof of the conclusions 1 and 2 of the theorem is quite simple and is a generalization of the proof of the corresponding theorem for eigenvalues and eigenvectors of a symmetric matrix which is proved in many physics courses and. Proposition 4 let be an eigenvalue of a regular or periodic sturm liouville problem. All secondorder linear ordinary differential equations can be recast in the form on the lefthand side of by multiplying both sides of the equation by an appropriate integrating factor although the same is not true of secondorder partial differential equations, or if y is a vector. Sturmliouville theory the proof shows that if px 6qx then z b a z2p.
In the simplest of cases all coefficients are continuous on the finite closed interval a, b. Pdf this paper is devoted to studying a qanalogue of sturmliouville eigenvalue problems. A classical sturmliouville equation, is a real secondorder linear differential. The eigenvalues of a sturmliouville problem are all of multiplicity one. We are interested in the sturmliouville eigenvalue problem slep with dirichlet boundary conditions. In sturmliouville theory, we say that the multiplicity of an eigenvalue of a sturm liouville problem l. We introduce and present the general solution of three twoterm fractional differential equations of mixed caputoriemannliouville type. We then solve a dirichlet type sturmliouville eigenvalue problem for a fractional differential equation derived from a special composition of a caputo and a riemannliouville operator on a finite interval where the. Sturm liouville operator is selfadjoint operator on h. The result is trivially true if is a simple eigenvalue. The sturmliouville eigenvalue problem and npcomplete. Sturmliouville problem, all of the following theorems are valid. Ryan walker a gentle introduction to sturm liouville problems introduction the nonsingular problem the singular problem references sturm liouville theory 1 mathematicians have.
Most of our proofs are adapted from 1 and are given using variational methods. Thus, we can solve them using the algorithms of 20 for solving the sturm liouville eigenvalue problem. The eigenvalues of a sturmliouville problem are the values of. An eigenvalue of the the slproblem 1 is a value of for which a nontrivial solution exist. Introduction to sturmliouville theory and the theory of. A classical sturmliouville equation, is a real secondorder linear differential equation.
The nontrivial solution is called an eigenfunction. Then the subspace spanned by the eigenfunctions corresponding to admits an orthonormal basis of realvalued functions. Eigenfunctions of a regular sturmliouville problem satisfy an orthogonality property, just like the eigenfunctions in section 4. If not, then just as before we could derive by dividing by yx and the boundary term in would vanish since yb 0, and we would obtain z b a yz0. Eigenvalue problems, sturmliouville boundary value problems.
For any regular sturmliouville problem, the following theorems are valid. Its proof is very similar to the analogous theorem 4. In all examples we have analyzed so far the boundary value problem that. We mostly deal with the general 2ndorder ode in selfadjoint form. Note that sl differential equation is essentially an eigenvalue problem since. The eigenvalues of a sturm liouville problem are the values of. We then solve a dirichlet type sturmliouville eigenvalue problem for a fractional differential equation derived from a special composition of a caputo and a riemannliouville operator on a finite interval where the boundary conditions are induced by. Adkins master of science graduate department of mathematics university of toronto 2014 a basic introduction into sturm liouville theory. The theory of this type of eigenproblem is called sturm liouville theory. There is not a largest eigenvalue and 5 00 as n s 00.
We formulate a selfadjoint qdifference operator in a. Pdf fractional sturmliouville eigenvalue problems, i. Inner product and norm to construct an orthonormal basis in a vector space we must be able to measure lengths and angles. Properties of sturmliouville eigenfunctions and eigenvalues. The eigenvalues of sturm liouville sl problems depend not only continuously but smoothly on the problem.
An expression for the derivative of the nth eigenvalue with respect to a given parameter. Then the sl di erential equation becomes an eigenaluev equation in the space h ly y. Corresponding to each eigenvalue there is an eigenfunction. The eigenfunctions of a sturm liouville problem can be chosen to be real. Many computational problems including the discrete problems mentioned above can be recasted as this weighted integration problem.
Integral equations and greens functions ronald b guenther and john w lee, partial di. The problem of finding a complex number if any, such that the bvp 6. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Proposition 4 let be an eigenvalue of a regular or periodic sturmliouville problem. Fractional sturmliouville eigenvalue problems, i springerlink. Proposition 6 the set of eigenvalues of a regular sturm liouville problem is countably in nite, and is a monotonically increasing sequence 0 sturm liouville problem, except that the sequence is monotonically nondecreasing. After nding the eigenvalues nand eigenfunctions y nof the appropriate sturm liouville eigenvalue problem, equation 2 is solved yielding t nt. Vibrations of a one dimensional string in lecture 1, for the problem of vibration of 1d string, depending on the type of boundary condition bc, we consider the following. This theory began with the original work of sturm from 1829 to 1836 and then followed by the short but signi cant joint paper of sturm and liouville in 1837, on secondorder linear ordinary di erential equations with an eigenvalue parameter. Eigenvalue problems, sturmliouville boundary value. Sturm liouville theorem, prehistory of the laplacian eigenvalue problem in rd lecturer.
The eigenfunctions of a sturmliouville problem can be chosen to be real. For most values of the parameter, problem 1 has only the trivial solution. Problem 5 isnt, although some authors use a definition of sturmliouville problem that does include it. Suppose lis hermitian with respect to an appropriate scalar product, and that yx is an eigenfunction of lwith corresponding eigenvalue.
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