Characteristics of Sturm-Liouville Problems P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Identification of Mothers for Orthogonal Series Functions (Eigen Functions)….

Special Solutions of Linear SO-ODEs

Orthogonal Series : Inner Product and Norm The value of scalar product of two vectors will determine, whether these vectors are orthogonal or not. A generalized definition of scalar product of any two functions is defined as an inner product. Consider a Function Space consisting of functions f (x) and g(x) defined on the interval [a, b] (for some a, b > 0) together with a positive weight-function r(x). The generalized concept of scalar product (inner product) is expressed as: Similarly the norm is defined as:

Inner Product as A Measure of Angle between A Pair of Functions The angle between these functions is defined as: Functions f and g are orthogonal on [a, b] with respect to the weight r if The inner product and orthogonality depend on the choice of a, b and r. If orthogonality is achieved with r(x) ≡ 1, these definitions reduce to the “ordinary orthogonal functions”. The distance between these functions is defined as:

Examples of Orthogonal Series The functions fn(x) = sin(nx) (n = 1, 2, . . .) are pairwise orthogonal on [0, π] relative to the weight function r(x) ≡ 1. The functions are pairwise orthogonal on [−1, 1] relative to the weight function r(x) = SQRT(1 − x2). They are examples of Chebyshev polynomials of the second kind. Hermite polynomial weight function

Examples of Orthogonal Series Lagurre polynomial weight function Let Jm be the Bessel function of the first kind of order m. Any set of functions fn(x) = Jm(αmnx/a) with αmn denote its nth positive zero. are pairwise orthogonal on [0, a] with respect to the weight function w(x) = x.

First Few Roots of Jm(x) 1 2.4048 3.8317 5.1356 6.3802 7.5883 8.7715 2 5.5201 7.0156 8.4172 9.7610 11.0647 12.3386 3 8.6537 10.1735 11.6198 13.0152 14.3725 15.7002 4 11.7915 13.3237 14.7960 16.2235 17.6160 18.9801 5 14.9309 16.4706 17.9598 19.4094 20.8269 22.2178

Hypergeometric Series A generalized hypergeometric series pFq is defined by where ()k denotes the Pochammer symbol

Theorem : Series Expansions Suppose that {f1, f2, f3, . . .} is an orthogonal set of functions on [a, b] with respect to the weight function r. The piecewise continuous function f on [a, b] is generated as then the coefficients an are given by The series expansion above is called a generalized Fourier series for f . an are the generalized Fourier coefficients. Regular Sturm-Liouville Problems are generators of Orthogonal series.

Sturm-Liouville DE : A Mother of Orthogonal Series A nonzero function y that solves the Sturm-Liouville problem Boundary conditions: is found to be an Eigen function, 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.

Sturm-Liouville Boundary Value Problem A SL-BVP with p, q and are specified such that p(x) > 0 and (x)  0  x [a,b]. where is called as a SL-EVP, if there exists a non-trivial solution for any = , where  is a complex number. Such a value μ is called an eigenvalue and the corresponding non-trivial solutions y(.; μ) are called Eigen functions.

Example: Euler like SV-BVP Solve With boundary conditions Solution: Take Ansaz as x. The characteristic equation is Solutions: Three different cases are possible:

Example: Euler like SV-BVP : case 1 Solve With boundary conditions Solution:

Example: Euler like SV-BVP : Case 2 Solve With boundary conditions Now we get a double root Solution:

Example: Euler like SV-BVP : Case 3 Solve With boundary conditions The two complex roots Solution:

Series as Solution of SVP Hence  must satisfy for some positive integer n. This generates the eigenvalues The corresponding eigenfunctions

Theorem 1 A nonzero function y that solves the Sturm-Liouville problem Boundary conditions: Theorem 1: The eigenvalues form an increasing sequence of real numbers with The eigenfunction yn corresponding to λn is unique and has exactly n−1 zeros in the interval a < x < b. Daileda

History of Fourier Series The Fourier series is named in honour of Jean-Baptiste Joseph Fourier (1768–1830). He made an important contributions to the study of trigonometric series. Fourier introduced the series for the purpose of solving the heat equation in a metal plate, publishing his initial results in his 1807. Mémoire sur la propagation de la chaleur dans les corps solides (Treatise on the propagation of heat in solid bodies) Published his second article, Théorie analytique de la chaleur (Analytical theory of heat) in 1822.

The Heat Equation The heat equation is a partial differential equation. Prior to Fourier's work, no solution to the heat equation was known in the general case. Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series. From a modern mathematics point of view, Fourier's results are somewhat informal.

Theorem 2 : Identification of Ancestral Mother A nonzero function y that solves the Sturm-Liouville problem Boundary conditions: Generalized Fourier Series Theorem 2: Suppose that yj and yk are eigen functions corresponding to distinct eigenvalues λj and λk . Then yj and yk are orthogonal on [a, b] with respect to the weight function (x) = r (x). That is

The Mother of Generalized Fourier Series SVT can generalize the concept of Fourier series from the usual trigonometric basis functions. SVT identifies Fourier series as an orthonormal series consisting of Eigen functions to a special Sturm-Liouville problem. Assume that we have an infinite linear combination where yn is orthogonal to ym for n  m. Then the inner product of f and ym

Real Thermofluid Problems/Sources as A Series Functions Let f be an arbitrary function (source) on [0, l]. A generalized Fourier series for f is identified as where are the generalized Fourier coefficients. Let y1,y2, . . . be a set of orthogonal Eigen functions of a regular Sturm-Liouville problem, and let f be a piece-wise smooth function in [0, l]. Then, for each x in [0, l] 

Classification of SL-EVPs An SL-EVP is called a regular SL-EVP, if p > 0 and r > 0 on [a, b]. An SL-EVP is called a singular SL-EVP, if (i) p > 0 on (a, b) and p(a) = 0 = p(b), and (ii) r  0 on [a, b]. An SL-EVP is called a periodic SL-EVP, if p(a) = p(b), p > 0 and r > 0 on [a, b], p, q, r are continuous functions on [a, b], coupled with boundary conditions