Development of Integral Transform Pairs

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). 

To demonstrate the development of the integral transform pair in a practical way, consider the Fickian diffusion or Fourier heat conduction problem in a slab object (Fig. 11.2) 

We note that the boundary conditions (11.2) are homogeneous. We define the spatial domain as (0,1) but any general domain (a, b) can be readily converted to (0,1) by a simple linear transformation 

Now multiply the LHS and RHS of Eq. 11.1 by a continuous function K (x) (called the kernel) such that the function K (.x) belongs to an infinite-dimensional space of twice differentiable functions in the domain (0,1). Eq. 11.1 then becomes 

We can see analogies with the Laplace transform, which has an unbounded domain and the kernel is 

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