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<p>The first boundary-value problem for the telegraph equation on an interval which<br />one end is mobile is considered. The method of the solution of such problem is developed and its exact solution is obtained. This method is based on integrated representation of solutions of the telegraph equation and generalization of a method of re°ections with reference to areas with variable border. Variants of movement of the mobile end with subsonic, sound and supersonic speeds, and also with arbitrary speed are considered.</p>