Equation. Conditional Sturm

It is generally true that the normalized eigenfunctions of an Hermitian operator such as the Schrodinger Ti constitute a complete orthonormal set in the relevant Hilbert space. A completeness theorem is required in principle for each particular choice of v(r) and of boundary conditions. To exemplify such a proof, it is helpful to review classical Sturm-Liouville theory [74] as applied to a homogeneous differential equation of the form... 

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... 

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] ... 

By applying the Sturm-Liouville theorem, the coefficient A for partial differential equations with nonhomogeneous boundary conditions is obtained as ... 

The weighting function for this particular orthogonality condition defined with reference to the Sturm-Liouville equation is... 

This appears to be a formidable task. To accomplish it, we shall need some new tools. Under certain conditions, it may be possible to compute C without a trial-and-error basis. To do this, we shall need to study a class of ODE with homogeneous boundary conditions called the Sturm-Liouville equation. We shall return to the coated-wall reactor after we gamer knowledge of the properties of orthogonal functions. 

Orthogonal Functions and Sturm-Liouville Conditions 427 and the second by

[Pg.427]

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... 

The strategy for using Sturm-Liouville transforms is, first, to carefully lay out the algebraic rules for this class of operator. Obviously, the defining equation and boundary conditions must be of the Sturm-Liouville type, as discussed in Chapter 10. 

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). 

This new set of equations for Y now can be readily solved by either the method of separation of variables or the Sturm-Liouville integral transform method. We must also find u(x), but this is simply described by an elementary ODE (Lu = 0), so the Inhomogeneous boundary conditions (11.74) are not a serious impediment. ... 

We recall from Section 10.5.1 in Chapter 10, that solutions of the Sturm-LiouvUle equation, along with suitable Sturm-Uouville boundary conditions, always produced orthogonal functions. Thus, the functions... 

Sturm-Liouville problems are categorized according to the type of boundary conditions that the differential equation must satisfy. 

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. 

The / -equation and the accompanying boundary conditions constitute a singular Sturm-Liouville boundary value problem. Also the / -equation is Bessel s equation of order zero. 

This results from the fact that the separation leads to a Sturm-Liouville equation for each coordinate, for which the theorem mentioned above is valid. Also in the cases where the boundary conditions do not require 4 to vanish, but only require that 4 remain finite, there are no significant changes. 

A special type of two point boundary value problem arises in many areas of engineering. Such problems are frequently referred to as flie Sturm-Liouville problem after the two mathematicians who made the first extensive study of the problem and published results in 1836. A typical formulation of die problem is the following second order differential equation with associated boundary conditions ... 

The boundary conditions are in general of the mixed type involving a combination of the function value and derivative at the two boundaries taken here to occur tx = a andx = b. Special cases of this equation lead to many classical functions such as Bessel functions, Legendre polynomials, Hemite polynomials, Laguerre polynomials and Chebyshev polynomials. In addition the Schrodinger time independent wave equation is a form of the Sturm-Liouville problem. 

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