Geodesic
Projection difference scheme for a parabolic functional differential equation with two-dimensional transformation of arguments
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 10, pp. 1848-1859

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A nonlinear parabolic functional differential equation with the functional part containing a generalized superposition of the unknown solution and a transformation of the two-dimensional spatial argument is considered. A projection difference scheme for the approximation of the initial Dirichlet boundary value problem in a rectangle is proposed for a wide class of measurable, including noninvertible, transformations. An estimate of the rate of convergence to the generalized solutions of the initial problem of order $O(\tau^{1/4-\gamma}+h^{1/2-2\gamma})$ in the norm $L_2(Q)$ without a priori assumptions on the invertibility of the transformation and without any mesh size matching is obtained. 
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     author = {A. V. Razgulin},
     title = {Projection difference scheme for a parabolic functional differential equation with two-dimensional transformation of arguments},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1848--1859},
     publisher = {mathdoc},
     volume = {45},
     number = {10},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_10_a9/}
}
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A. V. Razgulin. Projection difference scheme for a parabolic functional differential equation with two-dimensional transformation of arguments. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 10, pp. 1848-1859. http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_10_a9/