NUMERICAL SOLUTION METHOD FOR A CLASS OF ILLPOSED PROBLEM OF HYPERBOLIC EQUATION ABOUT STRUCTURE IMAGE(Ⅱ): PROPERLY POSEDNESS OF THE APPROXIMATE PROBLEM AND CONVERGENCE OF ITS SOLUTION
FAN SHANGWU (Institute of Computational Mathematics and Scientific/Engineering Computing,Academia Sinica,Beijing 100080)
A class of structure image problem in geophysics may be described by illposed problem of hyperbolic partial differential equation.In this case, the solution of this illposed problem represents the reflected wave field which characterizes geological structure.In the paper[1],the formulation of this illposed problem and its approximate problem were proposed. In this paper, the properlyposedness of the approximate problem is discussed on the basis of paper [1].It means that the solution of the approximate problem is existential,unique and continuous about the initial data.Furthermore,the convergence of the solution of the approximate problem Is also proved.It means that the solution of the approximate problem with the approximate initial data converges to the solution of the original basic problem with the exactinitial data, when the approximate initial data converges to the exact initial data.Conse quently,the numerical solution problem of illposed basic problem is reduced to the numerical solution problem of the properlyposed approximate problem.The latter can be implemented by use of finite difference method or finite element method.


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