Double Laplace Transformation in Mixed Boundary-Initial Value Problems

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Abstract
A comprehensive treatment of the application of the double Laplace transform to the combined boundary‐initial value problem for systems of linear, hyperbolic, partial differential equations with constant coefficients is given. The treatment is restricted to two independent variables x and t. A discussion of the relations among the boundary and initial conditions necessary for a well‐formulated problem is given, and it is shown that the solution subject to these restrictions does satisfy the boundary and initial conditions. An extensive examination of the solution in different regions of space‐time is given, and the connection between this solution and the more common ``normal‐mode'' solution proportional to exp[ikx − iωl (k)t] or exp[ikl (ω)x − iωt] is discussed. A generalized wave equation is used throughout the paper for illustration.

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