Closed-Form Exact Solution of the Unified Boundary Value Problem for the Telegraph Equation on Intervals

This paper proposes a methodology for solving initial-boundary value problems for the one-dimensional telegraph equation with constant coefficients. The problem is addressed in its most general form, with the boundary conditions presented in a unified manner on an arbitrary bounded interval of the real line. Under the assumption that the data pertaining to the boundary value problem admit their Laplace transforms, it is demonstrated that there exists a unique solution in the Laplace domain. This exact operational solution can be obtained in a closed-form, facilitating the recovery of the explicit expression of the solution in the time domain, provided that the inverse Laplace transform is available as in tables of transforms. Some prior estimates are established using an integral representation of the solution, and the time domain solution is proven to be stable. As demonstrated in the illustrations, the operational solution is shown to exhibit a significantly higher degree of computational efficiency in comparison to both classical and generalized series solutions. The latter are hardly obtainable for the same problem using the Fourier decomposition approach. This enhancement can be attributed largely to the efficacy of the algorithms employed for the numerical inverse Laplace transform. Moreover, the exact closed-form operational solution can be extended to unbounded domains. The telegraph equation is fundamental in various fields of applied mathematics. Its exact solution in time or Laplace domains serves as a benchmark for numerical and semi-analytical methods.