On the inverse initial-value problem for a quasilinear differential equation with a high-degree multidimensional Whitham operator

G. A. Dyikanov ; K. Kh. Shabadikov ; T. K. Yuldashev (Iuldashev)

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential equations, geometry, and topology, Tome 201 (2021), pp. 44-52.

Résumé

In this paper, we examine the solvability of the inverse initial-value problem for a quasilinear partial differential equation with a high-degree multidimensional Whitham operator. The expression of high-order partial differential equations as the superposition of first-order partial differential operators allowed us to represent the higher-order equation as an ordinary differential equation for an unknown function along characteristics. The unique solvability of the direct initial-value problem is proved by the method of successive approximations. An estimate for the convergence of the Picard iterative process is obtained. The problem of the search for the unknown coefficient is reduced to a Volterra integral equation of the first kind.

Détail

Keywords: inverse problem, multidimensional Whitham operator, method of successive approximations, Volterra integral equation of the first kind, unique solvability.

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