Geodesic
Approximate solution of the generalized mixed boundary value problem
Numerical methods and programming, Tome 5 (2004) no. 1, pp. 268-280.

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Some a priori inequalities with negative norm are obtained for differential parabolic- and hyperbolic-type equations with third-kind boundary conditions in the case when the right-hand sides belong to the space of generalized functions. The existence and uniqueness of the generalized solution as well as the convergence of the approximate method proposed are proved. The work was supported by the Russian Foundation for Basic Research (04-01-00026).
Keywords: mixed boundary value problem, negative norms, parabolic- and hyperbolic-type equations, generalized functions, numerical methods. 
@article{VMP_2004_5_1_a24,
     author = {I. V. Kolos and M. V. Kolos},
     title = {Approximate solution of the generalized mixed boundary value problem},
     journal = {Numerical methods and programming},
     pages = {268--280},
     publisher = {mathdoc},
     volume = {5},
     number = {1},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2004_5_1_a24/}
}
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I. V. Kolos; M. V. Kolos. Approximate solution of the generalized mixed boundary value problem. Numerical methods and programming, Tome 5 (2004) no. 1, pp. 268-280. http://geodesic.mathdoc.fr/item/VMP_2004_5_1_a24/