Galerkin finite element method for one nonlinear integro-differential model

Keywords: Nonlinear integro-differential equations Finite elements Galerkin method a b s t r a c t Galerkin finite element method for the approximation of a nonlinear integro-differential equation associated with the penetration of a magnetic field into a substance is studied. First type initial-boundary value problem is investigated. The convergence of the finite element scheme is proved. The rate of convergence is given too. The decay of the numerical solution is compared with the analytical results. The goal of this paper is a study of Galerkin finite element method for approximation of a nonlinear integro-differential equation arising in mathematical modeling of the process of a magnetic field penetrating into a substance. If the coefficient of thermal heat capacity and electroconductivity of the substance highly dependent on temperature, then the Maxwell's system [1], that describe above-mentioned process, can be rewritten in the following form [2]: @W @t ¼ Àrot a Z t 0 jrotWj 2 ds rotW ! ; ð1:1Þ where W = (W 1 , W 2 , W 3) is a vector of the magnetic field and the function a = a(r) is defined for r 2 [0, 1). Let us consider magnetic field W, with the form W = (0, 0, u), where u = u(x, t) is a scalar function of time and of one spatial variables. Then rotW ¼ ð0; À @u @x ; 0Þ and Eq. (1.1) will take the form @u @t ¼ @ @x a Z t 0 @u @x 2 ds ! @u @x " # : ð1:2Þ Note that (1.2) is complex, but special cases of such type models were investigated, see [2–11]. The existence of global solutions for initial-boundary value problems of such models have been proven in [2–4,10] by using the Galerkin and compactness methods [12,13]. The asymptotic behavior of the solutions of (1.2) have been the subject of intensive research in recent years (see e.g. [10,14–20]). In [7] some generalization of equations of type (1.1) is proposed. There it was assumed that the temperature of the considered body is depending on time, but independent of the space coordinates. If the magnetic field again has the form W = (0, 0, u) and u = u(x, t), then the same process of penetration of the magnetic field into the material is modeled by the following integro-differential equation [7]: 0096-3003/$-see front matter Published by Elsevier Inc.