Prove the Existence and Uniqueness of the Solution to an Equation for a Partial Differential Equation by Using a Continuous Function and Some Optional Conditions

The purpose of this paper is to develop the wave equation v_tt-v_xx=0 through the use of an optional function B(x) and some initial and boundary conditions, where the existence and uniqueness of the solution have been demonstrated by using the maximum and minimum principle, we give examples which show that the possibility of drawing.