GENERAL INTEGRAL OF THE MODEL WAVE EQUATION WITH VARIABLE RATES a1(x,t) AND a2 ( x,t) IN THE UPPER HALF-PLANE
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Abstract
A new one-dimensional two-rate linear model wave equation utt(t,x)+(a1-a2)utx(t,x) - a1a2uxx(x,t)-a2-1(a2)tut(x,t)-a1(a2)xux(x,t) = ƒ(x,t) (1) a3-i(x,t)≥a3-i (0) >0, (x,t)∈G=]-∞,+∞ [x[0,+∞[, a3-i ∈ C2(G), i=1,2. A particular classical solution F of this two-rate model wave equation in the upper half-plane G is calculated. A double verification of this solution is made by substituting F into equation (1) and into the corresponding canonical form of equation (1), from which the function F was calculated. A smoothness criterion for the right-hand side ƒ of Eq. (1) for the classical solution F in the upper half-plane G is found. A smoothness criterion on ƒ for twice continuous differentiability F in the first quarter of the plane is discussed. With the help of the classical solution F, the general integral of equation (1) is derived from the set of all its classical solutions u ∈ C2(G), which is needed in solving the Cauchy problem and initial-boundary problems for equation (1). These results are obtained by applying the new "implicit characteristic method" of the equation developed earlier by the author.
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References
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