Solution Sturm

The most general solution to a Sturm-Liouville problem is a function... 

To determine the interplay between the spectral properties, both boundary conditions, we return to Weyl s theory [32]. The key quantity in Weyl s extension of the Sturm-Liouville problem to the singular case is the m-function or ra-matrix [32-36]. To define this quantity, we need the so-called Green s formula that essentially relates the volume integral over the product of two general solutions of Eq. (1), u and v with eigenvalue X and the Wronskian between the two solutions for more details, see Appendix C. The formulas are derived so that it immediately conforms to appropriate coordinate separation into the... 

Chatwin, R.A. and Purcell, J.E. (1971). Approximate solution of a Sturm-Liouville system using nonorthogonal expansions Application to a-a nuclear scattering,... 

Volumes have been written about the red herring known as Schrodinger s cat. Without science writers looking for sensation, it is difficult to see how such nonsense could ever become a topic for serious scientific discussion. Any linear differential equation has an infinity of solutions and a linear combination of any two of these is another solution. To describe situations of physical interest such an equation is correctly prepared by the specification of appropriate boundary conditions, which eliminate the bulk of all possible solutions as irrelevant. Schrodinger s equation is a linear differential equation of the Sturm-Liouville type. It has solutions, known as eigenfunctions, the sum total of which constitutes a state function or wave function, which carries... 

Coming back once more to the alcoholic solutions After the alcohol has evaporated, there are still attraction forces of the dipole-dipole or hydrogen bonding kind at work that slow down the evaporation of the perfume material. These originate in part from interactions between the molecules of the different materials contained in the perfume. Perfumers may use materials of low volatility in their perfumes intending thereby to slow down the evaporation of more volatile perfume components. This is the practice commonly referred to as fixation (Sturm and Mansfeld 1976 Jellinek 1978). 

Equation (8.79) is of the Sturm-Liouville type, and with boundary conditions Eq. (8.84), solutions may be written as R (r ), cylindrical eigenfunctions or Bessel functions. Therefore, the solution to Eq. (8.75) may be written as... 

The boundary value problem posed by the differential equation (2.166) and the two boundary conditions (2.168) and (2.169) leads to the class of Sturm-Liouville eigenvalue problems for which a series of general theorems are valid. As we will soon show the solution function F only satisfies the boundary conditions with certain discrete values /q of the separation parameter. These special values /q are called eigenvalues of the boundary value problem, and the accompanying solution functions Fi are known as eigenfunctions. The most important rules from the theory of Sturm-Liouville eigenvalue problems are, cf. e.g. K. Janich [2.33] ... 

The formal solution of the fractional rotational diffusion equation is obtained just as that of Eq. (135) from the Sturm-Liouville representation [7,57]... 

The Sturm-Liouville representation (136) is a formal solution as a knowledge of all eigenfunctions )( 4>), and corresponding eigenvalues Xp is required. However, this representation is very useful because it allows one to obtain a formal solution for the longitudinal complex susceptibility = x i ) According to... 

The formal step-off transient solution of Eq. (172) for t > 0 is obtained from the Sturm-Liouville representation... 

Our objective is to ascertain how anomalous diffusion modihes the dielectric relaxation in a bistable potential with two nonequivalent wells, Eq. (195). The formal step-off transient solution of Eq. (172) for t > 0 is obtained from the Sturm-Liouville representation, Eq. (179), with the initial (equilibrium) distribution function... 

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... 

The physical description of the functional derivative Vee (r) requires knowledge of the wavefunction 4 for the determination of the electron-interaction component W e(r) = Wnlr) -i- W (r), and knowledge of both the wavefunction P and the Kohn-Sham orbitals < i(x) for the correlation-kinetic-energy component W, (r). The corresponding Kohn-Sham wavefunction is then a single Slater determinant. It has, however, also been proposed [42,52,53] that the wavefunction V be determined by solution of the Sturm-Liouville equation... 

We can now finish the reactor problem using the fact that the cylindrical equation was of Sturm-Liouville type with homogeneous boundary conditions. When this is the case, the solutions

orthogonal functions with respect to the weighting function r(x). Since we identified the weighting function for the reactor as r( ) = f, we can write when n is different fix)m m... 

Because they depend on the Sturm-Liouville equation, the separation of variables method and the integral transform yield exactly the same solution, as you would expect. But the advantage of the integral transform is the simplicity of handling coupled PDEs, for which other methods are unwieldy. Moreover, in applying the finite integral transform, the boundary conditions need not be homogeneous (See Section 11.2.3). 

We could have made the RHS of Eq. 11.11 a positive number rather than negative, but it can be proved that the negative number is the only option that will yield physically admissible solutions. This will be proved later in dealing with a general Sturm-Liouville system (see Section 11.2.2). 

One can see that the integral transform indeed facilitates the resolution of ODE boundary value problems and also partial differential equations comprised of Sturm-Liouville operators (e.g., Eq. 11.45). The simplicity of such operational methods lead to algebraic solutions and also give a clearer view on how the solution is represented in Hilbert space. Moreover, students may find that the Sturm-Liouville integral transform is a faster and fail-safe way of getting the solution. Thus, Eq. 11.52 represents the solution to an almost infinite variety of ordinary differential equations, as we see in more detail in the homework section. 

The solution methodology for the Sturm-Liouville integral transform is now quite straightforward and of course yields the same results as the separation of variables method, as the reader can verify. 

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