Abstract

The inverse problem of determining a source in the one-dimensional heat equation in the case of a Dirichlet boundary value problem is investigated. The trace of the solution of the direct problem on straight-line segments inside the domain at the final time is specified as overdetermination (i.e., additional information on the solution of the direct problem). A Fredholm alternative theorem for this problem is proved, and sufficient conditions for its unique solvability are obtained. The inverse problem is considered in classes of smooth functions with derivatives satisfying the Hölder condition.

中文翻译：

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一维热方程的紧支撑源的确定

摘要

研究了在Dirichlet边值问题的情况下在一维热方程中确定源的反问题。最终时间在域内的直线段上的直接问题的解的轨迹被指定为超确定（即，有关直接问题的解的附加信息）。证明了该问题的Fredholm替代定理，并为其唯一的可解性获得了充分的条件。在具有满足Hölder条件的导数的光滑函数类中考虑了反问题。