Partial differential equations Sturm-Liouville equation

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. 

In Chapters 7 to 12 we deal with numerical solution methods, and partial differential equations (PDE) are presented. Qassical techniques, such as combination of variables and separation of variables, are covered in detail. This is followed by Chapter 11 on PDE transform methods, culminating in the generalized Sturm-Liouville transform. This allows sets of PDEs to be solved as handily as algebraic sets. Approximate and numerical methods close out the treatment of PDEs in Chapter 12. 

X here represents various variables and the equation is therefore a partial differential equation. L[ j represents a linear, homogeneous, self-adjoint differential expression of second order, ip is the desired function, p x) the density function and A the eigenvalue parameter of this Sturm-Liouville eigenvalue problem. ... 

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

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