Final value theorem Laplace transforms

When diffusion is not fast enough to avoid some spatial inhomogeneity, the value of the kernel must be considered. In general, eqn. (224) is too awkward to invert, so that the limiting value for long times can be determined using the final value theorem of Laplace, transforms, wherein g(s) is the Laplace transform of f(f)... 

This is the equation describing the dynamics of the longitudinal component of the magnetization obtained by the same method of truncation of the continued fraction as that employed in Section IV. This method, also used by Morita for dielectric relaxation [56], is a consequence of the final value theorem for Laplace transforms, which is... 

The final-value theorem allows us to compute the value that a function approaches as t- oo when its Laplace transform is known. 

This can be proved more rigorously by using the final-value theorem of Laplace transforms ... 

Laplace Transform Definition, Common Laplace Transforms, and Final-Value Theorem... 

The Laplace transform of the final value theorem is expressed by [18]... 

For a more elaborate treatment of Laplace transform and the initial and final value theorems, the reader is referred to the many textbooks on process control. 

The initial value of 1 corresponds to the initial condition given in Eq. 3-26. The final value of 0.5 agrees with the time-domain solution in Eq. 3-37. Both theorems are useful for checking mathematical errors that may occur in obtaining Laplace transform solutions. 

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