Sturm-Liouville boundary value

The problem imposed by Eq. (11.112) with boundary conditions of Eq. (11.113) is noted as the Sturm-Liouville boundary-value problem [e.g., Derrick and Grossman, 1987], if... 

Among the many problems encountered in chemical engineering science, those that appear very frequently involve the Sturm-Liouville boundary value problems. Following is an illustration of what is meant by a Sturm-Liouville problem. 

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. 

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] ... 

So since the boundary values for all eigenfunctions are homogeneous and of the Sturm-Liouville type, we can write the orthogonality condition immediately... 

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. 

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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