Approximate solutions of the Helmholtz equation on the plane
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Abstract
The study delves into the persistent analysis and reliability assessment of the solution to the Cauchy problem for the Helmholtz equation, within a defined area, using the known values on a smooth portion of the area's boundary as a reference. This analyzed situation belongs to those found within mathematical physics where the detected solutions fail to show consistent reliance on the initial data. Emphasizing real world applications, it's not only finding an approximate solution that matters, but also identifying its derivative. Presuming a solution is available and constantly differentiable within a close range, precise Cauchy data is considered. An explicit formula to expand both the solution and its derivative has been established, along with a regularization formula for instances where, under given conditions, ongoing approximations of initial Cauchy data with a defined error limit in the uniform metric are offered rather than the original data. Evaluations that assure the stability of the classical Cauchy problem solution have been provided.
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