Geodesic
Integro-differential equation with a higher-order two-dimensional Whitham operator
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Mathematical Analysis, Tome 156 (2018), pp. 117-125

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We examine the unique solvability of an initial-value problem for a certain higher-order quasilinear partial integro-differential equation with degenerate kernel. Expressing the higher-order partial integro-differential operators as the superposition of first-order partial differential operators, we represent the integro-differential equation considered as an ordinary integro-differential equations that describes the change of the unknown function along characteristics. Using the method of successive approximations, we prove the unique solvability of the initial-value problem and obtain an estimate for the convergence rate of the Picard iterative process.
Keywords: initial-value problem, characteristics, directional derivative, method of successive approximations, unique solvability. 
@article{INTO_2018_156_a10,
     author = {T. K. Yuldashev},
     title = {Integro-differential equation with a higher-order two-dimensional {Whitham} operator},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {117--125},
     publisher = {mathdoc},
     volume = {156},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2018_156_a10/}
}
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T. K. Yuldashev. Integro-differential equation with a higher-order two-dimensional Whitham operator. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Mathematical Analysis, Tome 156 (2018), pp. 117-125. http://geodesic.mathdoc.fr/item/INTO_2018_156_a10/