Eigenfunction expansions

if the eigenvalues Aj, A2,. .., and corresponding eigenfunctions i(x), 4 2(x), have been found, one can sometimes expand a given function/(x) in a series of these eigenfunctions. That is 

The series in the above equation is called an eigenfunction expansion of f x) on [a, b]. In order to determine the c s, property 2 of Theorem 4.1 will be used in the following way  

These are the so-called Fourier coefficients of/(x) with respect to the eigenfunctions of the given Sturm-Liouville problem. A more comprehensive discussion on Eourier coefficients will be given in the next chapter. Also, the conditions that a function must satisfy in order to have a series expansion, as given in Equation 4.33 will be discussed in Chapter 5. 

Following is an example demonstrating the computation of Fourier coefficients. 

eigenfunction expansion will be useful in solving certain types of boundary value problems to be discussed in Chapter 6. 

---

On inserting this ansatz into Eq. (123), the solution can be determined in the form of an eigenfunction expansion, as shown by Walker [163]. The parameter controlling the number of terms of this expansion having to be taken into account is Pe h/L, which is usually of the order of 0(0.01 - 1) in micro reactors. For this reason, often only the first term contributes. With the entrance condition cj, (f) = 1, the axial dependence can then be written as... 

The above self-similar velocity profiles exists only for a Re number smaller than a critical value (e.g. 4.6 for a circular pipe). The self-similar velocity profiles must be found from the solution of the Navier-Stokes equations. Then they have to be substituted in Eq. (25) which must be solved to compute the local Nusselt number Nu z). The asymptotic Nusselt number 7Vm is for a pipe flow and constant temperature boundary condition is given by Kinney (1968) as a function of Rew and Prandtl (Pr) numbers. The complete Nu(z) curve for the pipe and slit geometries and constant temperature or constant flux boundary conditions were given by Raithby (1971). This author gave /Vm is as a function of Rew and fluid thermal Peclet (PeT) number. Both authors solved Eq. (25) via an eigenfunction expansion. 

For another example of a result obtained by means of the eigenfunction expansion consider the expression (7.16) for the spectral density of the fluctuations. By a slight change of notation it can be cast in the form... 

Exercise. Derive (7.18) without using the eigenfunction expansion. [Hint Apply successive partial integrations to (III.3.4) and use (IV.3.7).]... 

For a symmetric bistable potential as in fig. 37 another treatment is possible, based on the eigenfunction expansion of (1.8) as given in V.7. The successive eigenfunctions [Pg.335]

From Eq. (C.2) we conclude that the square integrable solution / contains the independent solution spectral function p(E) to be used in the completeness relation and the eigenfunction expansion. The former gives... 

E.C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Qaredon, Oxford, Vol. 1,1946,1962 Vol. 2,1958. 

To evaluate the sum in brackets we used the eigenfunction expansion (27) for the left contact. 

The time evolution of the wave packet given by Eq. (27) can be evaluated by an eigenfunction expansion or by a split-operator technique. With the latter technique, the wave packet X t + St) after a small increment of time St can be expanded approximately as... 

Table 1 Convergence behavior of eigenfunction expansion for the dimensionless velocity and comparison with routine NDSolve [41] (parallel plates, periodic flow, j5 =0.1).

Eigenfunction expansions as used in Ref. 168 are not accurate near the critical point. Instead, we developed a shooting point method in order to make a direct numerical integration of Eq. (110) with the condition Eq. (112). Real energies (bound and virtual) were found by bisection methods, and for complex energies it was necessary to combine the Newton-Raphson and grid methods. 

The fact that K is time independent opens the possibility to work with eigenfunction expansions. We consider the case in which K has pure point spectrum, i.e. no continuum. This is always the case for /V-level models with periodic time dependent fields. Further remarks on other cases are given in Appendix A. 

These problems are chosen because they illustrate important ideas and concepts in addition to simply solving problems. However, the analysis in this chapter is completely based on classical eigenfunction expansions. 

Although the problem defined by (3-95) and (3-96) is time dependent, it is linear in uJ and confined to the bounded spatial domain, 0 < r < 1. Thus it can be solved by the method of separation of variables. In this method we first find a set of eigensolutions that satisfy the DE (3-95) and the boundary condition at r = 1 then we determine the particular sum of those eigensolutions that also satisfies the initial condition at 7 = 0. The problem (3-95) and (3-96) comprises one example of the general class of so-called Sturm-Louiville problems for which an extensive theory is available that ensures the existence and uniqueness of solutions constructed by means of eigenfunction expansions by the method of separation of variables.14 It is assumed that the reader is familiar with the basic technique, and the solution of (3-95) and (3-96) is simply outlined without detailed proofs. We begin with the basic hypothesis that a solution of (3-95) exists in the separable form... 

D. TWO-DIMENSIONAL CREEPING FLOWS SOLUTIONS BY MEANS OF EIGENFUNCTION EXPANSIONS (SEPARATION OF VARIABLES)... 

The solution of these equations by means of standard eigenfunction expansions can be carried out for any curvilinear, orthogonal coordinate system for which the Laplacian operator V2 is separable. Of course, the most appropriate coordinate system for a particular application will depend on the boundary geometry. In this section we briefly consider the most common cases for 2D flows of Cartesian and circular cylindrical coordinates. 

I. General Eigenfunction Expansions in Cartesian and Cylindrical Coordinates... 

---