Variational techniques for sturmliouville eigenvalue problems. The sturmliouville eigenvalue problem and npcomplete. Pdf this paper is devoted to studying a qanalogue of sturmliouville eigenvalue problems. The differential equation is said to be in sturmliouville form or selfadjoint form. We mostly deal with the general 2ndorder ode in selfadjoint form. Sturmliouville problem, all of the following theorems are valid. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. We are interested in the sturmliouville eigenvalue problem slep with dirichlet boundary conditions. We use a formula relating the sturm liouville eigenvalue problem to a weighted integration problem, see 20. 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. Fractional sturmliouville eigenvalue problems, i springerlink.
Sturmliouville eigenvalue problem 8, 910 is called regular if the coe. The eigenvalues of sturm liouville sl problems depend not only continuously but smoothly on the problem. Its proof is very similar to the analogous theorem 4. It follows that hyjlyiis real, and therefore itself is real. An eigenvalue of the the slproblem 1 is a value of for which a nontrivial solution exist. Properties of sturmliouville eigenfunctions and eigenvalues. 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. Sturmliouville theorem, prehistory of the laplacian eigenvalue problem in rd lecturer. In sturmliouville theory, we say that the multiplicity of an eigenvalue of a sturmliouville problem l. Corresponding to each eigenvalue there is an eigenfunction. Note that sl differential equation is essentially an eigenvalue problem since.
Values of for which 1,2 has a nontrivial solution are called eigenvalues and a nontrivial solution u corresponding to is called an eigenfunction. Introduction to sturmliouville theory and the theory of. We introduce and present the general solution of three twoterm fractional differential equations of mixed caputoriemannliouville type. We formulate a selfadjoint qdifference operator in a. Proposition 4 let be an eigenvalue of a regular or periodic sturmliouville problem. The eigenfunctions of a sturm liouville problem can be chosen to be real. In all examples we have analyzed so far the boundary value problem that. Problem 5 isnt, although some authors use a definition of sturmliouville problem that does include it.
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. 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. 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. The result is trivially true if is a simple eigenvalue. For most values of the parameter, problem 1 has only the trivial solution. 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. The eigenvalues of a sturmliouville problem are the values of. 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. Pdf fractional sturmliouville eigenvalue problems, i. Eigenfunctions of a regular sturmliouville problem satisfy an orthogonality property, just like the eigenfunctions in section 4. The appropriate sturm liouville problem is solved, nding the eigenvalues nand the corresponding eigenfunctions u nx. The socalled sturmliouville problems define a class of eigenvalue problems, which include many of the previous problems as special cases. After nding the eigenvalues nand eigenfunctions y nof the appropriate sturmliouville eigenvalue problem, equation 2 is solved yielding t nt. Thus, we can solve them using the algorithms of 20 for solving the sturm liouville eigenvalue problem.
A catalogue of sturmliouville di erential equations. A nonzero function y that solves the sturmliouville problem pxy. Sturm liouville theorem, prehistory of the laplacian eigenvalue problem in rd lecturer. In sturmliouville theory, we say that the multiplicity of an eigenvalue of a sturm liouville problem l. The problem of finding a complex number if any, such that the bvp 6. Suppose lis hermitian with respect to an appropriate scalar product, and that yx is an eigenfunction of lwith corresponding eigenvalue. 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. Sturmliouville theory the proof shows that if px 6qx then z b a z2p.
We will merely list some of the important facts and focus on a few of the properties. Inner product and norm to construct an orthonormal basis in a vector space we must be able to measure lengths and angles. Sturmliouville problems arise directly as eigenvalue problems in one space. Proposition 4 let be an eigenvalue of a regular or periodic sturm liouville problem. Then the sl di erential equation becomes an eigenaluev equation in the space h ly y. The eigenvalues of a sturmliouville problem are all of multiplicity one. Integral equations and greens functions ronald b guenther and john w lee, partial di. Eigenvalue problems, sturmliouville boundary value. The heat flow in a nonuniform rod is modeled by the partial differential equation c. 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. A nonzero function y that solves the sturm liouville problem pxy. The fourier cosine transform of a function fx will be denoted by. The eigenvalues of sturmliouville sl problems depend not only continuously but smoothly on the problem.
The eigenvalues of a sturm liouville problem are all of multiplicity one. Sturm liouville operator is selfadjoint operator on h. Eigenvalue problems, sturmliouville boundary value problems. The initial distribution of the temperature needs to be speci ed as well.
Conversion of a linear second order differential equation to sturm liouville form. The eigenfunctions of a sturmliouville problem can be chosen to be real. Many computational problems including the discrete problems mentioned above can be recasted as this weighted integration problem. The nontrivial solution is called an eigenfunction. Then the subspace spanned by the eigenfunctions corresponding to admits an orthonormal basis of realvalued functions. Such an eigenvalue problem is called a regular sturmliouville problem. Ryan walker a gentle introduction to sturm liouville problems introduction the nonsingular problem the singular problem references sturm liouville theory 1 mathematicians have. An expression for the derivative of the nth eigenvalue with respect to a given parameter. After nding the eigenvalues nand eigenfunctions y nof the appropriate sturm liouville eigenvalue problem, equation 2 is solved yielding t nt.
The theory of this type of eigenproblem is called sturm liouville theory. The eigenvalues of a sturm liouville problem are the values of. A classical sturmliouville equation, is a real secondorder linear differential. In the simplest of cases all coefficients are continuous on the finite closed interval a, b. Adkins master of science graduate department of mathematics university of toronto 2014 a basic introduction into sturm liouville theory. Introduction to sturmliouville theory trinity university. Most of our proofs are adapted from 1 and are given using variational methods. For any regular sturmliouville problem, the following theorems are valid. There is not a largest eigenvalue and 5 00 as n s 00. Sturmliouville problems 55 this has nontrivial solution for the pair a, b if and only if.