A source identification problem related to mathematical model of laser surface heating

作者：Otelbaev Muhtarbay; Hasanov Alemdar; Akpayev Bakytzhan

来源：Applied Mathematics Letters, 2012, 25(10): 1480-1485.

DOI：10.1016/j.aml.2011.12.028

• Source identification
• Laser surface hardening
• Optimal trajectory
• Heat equation

A mathematical model of two-dimensional laser surface heating for the hardening of metallic materials is proposed. The model is governed by the heat equation u(t) - Delta u = m(t)delta(gamma)(x - w(t)), (x, t) epsilon Omega, with the pointwise source term delta(gamma) (y), satisfying the initial u(x, 0) = g(x) and boundary u(x, t) = 0, x epsilon partial derivative Omega, conditions. The pair of source terms (m(t), w(t)) is assumed to be unknown. The two-valued (m(t) = 0 or m(t) = m(0) > 0) function m(t) is treated as the intensity of the laser beam, and the function omega(t) describes the laser beam trajectory. The identification problem consists of determining the pair of source terms < m(t), omega(t)> such that the corresponding heat function v(x, t) satisfies the condition parallel to u - v parallel to(L2(Omega)) <= epsilon, where the smooth function v(x, t) is assumed to be known (experimentally), and epsilon > 0 is a given-in-advance parameter. Besides the existence result, the structure of the optimal trajectory is also described.

• 出版日期2012-10