By Theodore W Gamelin
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More General Applications: Direct use of the Laplace trans- form on equations such as (48) depends on first finding the asymptotic form of the solution near the origin, and then using this information in such a way that unknown quantities such as f(O) are eliminated from the transformed equation. Since these steps are only intermediate, the final result being a particular integral representation of the desired solution, it is better to write down the solution as an integral from the outset. This approach is generally known as Laplace's method, and is the subject of Section 20.
This represents the PART I: 52 at the boundary, with the same amplitude ¢ reflection of Further reflections occur at but opposite sign. , as the disturbance travels up and down the string. Infinite Transmission Line: Coaxial cables and other elec- trical transmission lines are systems which are amenable to We consider here simple analysis via the Laplace transform. a line which has no resistive a rather simple example: The line is described by two parameters, the indue- losses. tance L C per unit length.
3. See, for example, KAPLAN (1962) and VAN DER POL & BREMMER (1955), Ch. 8. 4. Logarithmic terms appear in the second solution whenever v is an integer. 6. §4. Partial differential equations §4. 1. As an example to show how the Laplace transform may be applied to the solution of partial differential equations, we consider the diffusion of heat in an isotropic solid body. For temperature gradients which are not too large, it is a good approximation to assume that the heat flow is proportional to the temperature gradient, so that the temperature u(E,t) must satisfy the partial differential equation au at (1) K- where K is a constant given by density, c K (We have assumed that of the material.