[01] Rufina Abdullina, Department of Physics and Mathematics, Sterlitamak Branch of the Bashkir State University, Sterlitamak, Russia. [02] Alena Agafonova, Department of Physics and Mathematics, Sterlitamak Branch of the Bashkir State University, Sterlitamak, Russia.

The first general solution of the problem of Сaushy for an extensive class of partial differential equations was given by Riemann almost a century ago in his well-known paper on the propagation of sound waves of finite amplitude. Although stated only for certain special equations, it is applicable to any linear equation of hyperbolic type of the second order in two independent variables; it depends ultimately on finding a certain subsidiary function, often called the Riemann function, which is the solution of a characteristic boundary value problem for the adjoint equation. This paper is of a synthetic nature, being a result of combining Riemann’s method for integrating second-order linear hyperbolic equations with Lie’s classification of such equations. In paper was found the solution of the Cauchy problem by the Riemann method for a hyperbolic equation. It was also shown the invariance of the Riemann function relatively to the symmetry of the fundamental solutions.

Problem Cauchy, Riemann’s Function, Hyperbolic Equation, Group Analysis

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