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ISSN 0012-2661, Differential Equations, 2006, Vol. 42, No. 8, pp. 1134–1139. c Pleiades Publishing, Inc., 2006. Original Russian Text c N.L. Laˇ zeti´ c, 2006, published in Differentsial’nye Uravneniya, 2006, Vol. 42, No. 8, pp. 1072–1077. PARTIAL DIFFERENTIAL EQUATIONS On the Classical Solvability of the Mixed Problem for a Second-Order One-Dimensional Hyperbolic Equation N. L. Laˇ zeti´ c Belgrade University, Belgrade, Serbia Received December 6, 2004 DOI: 10.1134/S0012266106080088 1. INTRODUCTION In the present paper, we analyze the existence and uniqueness of the classical solution of mixed problems for a one-dimensional inhomogeneous hyperbolic second-order equation in a closed rect- angle with arbitrary homogeneous self-adjoint boundary conditions. Let G =(a, b) be a finite interval of the real line R,and let T be an arbitrary positive number. We consider the problem on the existence and uniqueness of a real function u = u(x, t) defined on the closed rectangle Ω=[a, b] × [0,T ] and satisfying the equation 2 2 ∂ u ∂ u (x, t) − (x, t)+ q(x)u(x, t)= f (x, t), (x, t) ∈ Ω, (1) 2 2 ∂t ∂x the initial conditions u(x, 0) = ϕ(x),u (x, 0) = ψ(x),x ∈ G, (2) and the
Differential Equations – Springer Journals
Published: Oct 7, 2006
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