ABOUT RUPTURE ALONG THE CHARACTERISTICS OF THE FIRST AND SECOND PARTIAL DERIVATIVES OF SOLUTIONS OF THE GENERAL FACTORIZED ONE-DIMENSIONAL WAVE EQUATION IN A QUARTER OF THE PLANE
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In this paper we study the smoothness of generalized solutions of one-dimensional wave equation (∂t - a2∂x + b2) (∂t + a1∂x + b1) u(x, t) = ƒ (x,t) and its right-hand side ƒ in the first quadrant. The purpose of the article – finding breaks the lines of the first and second partial derivatives of the generalized solutions of the wave equation in the case of the existence of these derivatives and the identification of necessary smoothness requirements on the right-hand side for the existence of classical solutions of this equation. The purpose of the article is achieved by propagating waves from the course of the equations of mathematical physics and methods of the theory of generalized functions. It is proved that the first partial derivatives of the continuous solutions of this equation can have a break only on pieces of its characteristics: x - a1t = C1, x + a2t = C2. The second partial derivatives of its continuously differentiable solutions can have a break only on the pieces of these characteristics and to pieces of the direct: x - √ a2a1 t = C3 , x + √ a2a1 t = C4 , Ci ∈ ℝ, i = 1, 4 . With these results it is established that any classical solution of the total factored linear inhomogeneous wave equation string contains a term of its unique (up to addition of classical solutions of the corresponding factored homogeneous equation) generalized solution which is twice continuously differentiable and is its classical solution in the first quadrant. This allowed us to bring the need for continuity of ƒ and the corresponding integral requirements for the smoothness on ƒ to the existence of classical solutions of this equation.
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References
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