DOI link for Mathematical Introduction
Mathematical Introduction book
This chapter presents general mathematical methods used to analyze such systems, and considers one-dimensional unsteady processes. Many systems of continuum mechanics, such as gasdynamical and magnetohydrodynamical equations, elasticity and the Maxwell equations belong to the considered class of the conservation laws systems. Discontinuity surfaces can be boundaries of regions inside which the functions ui are continuous, differentiable and satisfy differential equations. Dynamic equations of elasticity are essentially hyperbolic, quasilinear differential equations expressing conservation laws. The system of hyperbolic equations expressing conservation laws which describe a continuous medium behavior has an important property, namely, one more divergent equation can obtained as a formal consequence of the correctly written equations of continuous media. Boundary conditions should be imposed on a boundary separating two regions where the solutions are described by hyperbolic systems. The number of various solutions of the Riemann wave type is determined by the number of linearly independent eigenvectors. The Riemann waves provide a natural generalization of small perturbation waves.