TY - CHAP

T1 - Quasilinear integrodifferential Riccati-type equations

AU - Vaskevich, V. L.

AU - Shcherbakov, A. I.

N1 - Publisher Copyright: © Springer Nature Switzerland AG 2020.

PY - 2020/4/3

Y1 - 2020/4/3

N2 - Equations under study have the form in which the time derivative of an unknown function u(t, k) is expressed by the linear combination of u(t, k) and a double integral over the integration domain P(k) with respect to the spatial variables (k1, k2). The double integral is from the weighted quadratic expression of u(t, k1) and u(t, k2). The coefficient a(t) of the linear part of the equation is a continuous function; the integration domain P(k) is unbounded and does not depend on time, but depends on the spatial variable k. The properties of solutions to the equation are determined by the kernel W(k, k1, k2) of the integral operator in the right-hand side of the equation, as well as the behavior of the unknown solution as k tends to zero and as k tends to infinity. The kernel W(k, k1, k2) of the double integral is a continuous function in the first octant of the three-dimensional real space and satisfies some additional conditions. We introduce special functional classes associated with the equation under study and consider the Cauchy problem with initial data on the half-axis k > 0. In application to the Cauchy problem, we consider the method of successive approximations and estimate the successive approximations quality in dependence on the number of the iterated solution.

AB - Equations under study have the form in which the time derivative of an unknown function u(t, k) is expressed by the linear combination of u(t, k) and a double integral over the integration domain P(k) with respect to the spatial variables (k1, k2). The double integral is from the weighted quadratic expression of u(t, k1) and u(t, k2). The coefficient a(t) of the linear part of the equation is a continuous function; the integration domain P(k) is unbounded and does not depend on time, but depends on the spatial variable k. The properties of solutions to the equation are determined by the kernel W(k, k1, k2) of the integral operator in the right-hand side of the equation, as well as the behavior of the unknown solution as k tends to zero and as k tends to infinity. The kernel W(k, k1, k2) of the double integral is a continuous function in the first octant of the three-dimensional real space and satisfies some additional conditions. We introduce special functional classes associated with the equation under study and consider the Cauchy problem with initial data on the half-axis k > 0. In application to the Cauchy problem, we consider the method of successive approximations and estimate the successive approximations quality in dependence on the number of the iterated solution.

UR - http://www.scopus.com/inward/record.url?scp=85114658141&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-38870-6_49

DO - 10.1007/978-3-030-38870-6_49

M3 - Chapter

AN - SCOPUS:85114658141

SN - 9783030388690

SP - 375

EP - 380

BT - Continuum Mechanics, Applied Mathematics and Scientific Computing

PB - Springer International Publishing AG

ER -