Generalized Sturm-Liouville Integral Transform

Integrate the result of Step 3 and use the boundary conditions of the equation as well as the eigenproblem an equation for the image of y, that is, y,K ) should be obtained. 

Obtain the inverse y from the image by using either Eq. 11.24 or 11.39. 

We have presented the method of Sturm-Liouville integral transforms, and applied it to a number of problems in chemical engineering. Additional reading on this material can be found in Sneddon (1972) or Tranter (1958). 

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In this section, we will apply the finite integral transform to a general Sturm-Liouville system, and the integral transform is therefore called the Sturm-Liouville integral transform. Thus, all finite integral transforms are covered at once Fourier, Hankel, and so forth. 

The application of the Sturm-Liouville integral transform using the general linear differential operator (11.45) has now been demonstrated. One of the important new components of this analysis is the self-adjoint property defined in Eq. 11.50. The linear differential operator is then called a self-adjoint dijferential operator. 

Before we apply the Sturm-Liouville integral transform to practical problems, we should inspect the self-adjoint property more carefully. Even when the linear differential operator (Eq. 11.45) possesses self-adjointness, the self-adjoint property is not complete since it actually depends on the type of boundary conditions applied. The homogeneous boundary condition operators, defined in Eq. 11.46, are fairly general and they lead naturally to the self-adjoint property. This self-adjoint property is only correct when the boundary conditions are unmixed as defined in Eq. 11.46, that is, conditions at one end do not involve the conditions at the other end. If the boundary conditions are mixed, then the self-adjoint property may not be applicable. 

As we have observed with the Sturm-Liouville integral transform and we will observe later for this generalized integral transform, there will arise an infinite set of eigenvalues and an infinite set of corresponding eigenfunctions. We then rewrite Eqs. 11.191 to 11.193 as follows to represent the nth values... 

In the previous section, we developed the finite integral transform for a general Sturm-Liouville system. Homogeneous boundary conditions were used in the analysis up to this point. Here, we would like to discuss cases where the boundary conditions are not homogeneous, and determine if complications arise which impede the inversion process. 

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