Using an example of the Cauchy problem for the one-dimensional heat equation, we study the approximation of the solution to the initial condition in the Hausdorff metric. The simplest discontinuous function u0(x) = sgn x is taken for the initial condition. Based on the asymptotic behavior of the Lambert W function and its modification, we obtain a two-sided estimate and an asymptotics for the Hausdorff distance between the solution given by the Poisson formula and the function u0(x). Similar results are obtained for a similar model problem for the Laplace equation in the upper half-plane.
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References
Bl. Sendov, “Some questions of the theory of approximations of functions and sets in the Hausdorff metric,” Russian Math. Surveys 24, No. 5, 143–183 (1969).
E. P. Dolzhenko and E. A. Sevast’yanov, “Approximations of functions in the Hausdorff metric by piecewise monotonic (in particular, rational) functions,” Math. USSR, Sb. 30, No. 4, 449–477 (1976).
Bl. Sendov, Hausdorff Approximations, Kluwer, Dordrecht etc. (1990).
M. Abramowitz (Ed.) and I. A. Stegun (Ed.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, John Wiley and Sons, New York etc. (1972).
R. M. Corless et al., “On the Lambert W function,” Adv. Comput. Math. 5, No. 4, 329–360 (1996).
A. Hoorfar and M. Hassani, “Inequalities on the Lambert W function and hyperpower function,” JIPAM J. Inequal. Pure Appl. Math. 9, No. 2, Paper No. 51 (2008).
S. T. Rachev, “Hausdorff metric structure in the space of probability measures,” J. Math. Sci. 17, No. 6, 2275–2288 (1981).
B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge (1954).
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Kostin, A.B., Sherstyukov, V.B. Application of the Hausdorff Metric in Model Problems with Discontinuous Functions in Boundary Conditions. J Math Sci 274, 511–522 (2023). https://doi.org/10.1007/s10958-023-06616-6
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DOI: https://doi.org/10.1007/s10958-023-06616-6