Numerical solution of the nonlinear parabolic problem related to inverse elastoplastic torsional problem

Abstract

In this article, we study a class of direct and inverse coefficient problems defined for a nonlinear parabolic equation. An existence of a quasi-solution of the considered inverse problem is obtained in the appropriate class of admissible coefficients. Direct problem is solved numerically by using semi-implicit finite difference scheme and then some examples are presented related to steady-state solution of the nonlinear direct problem and ill-posedness of the inverse problem. IntroductionThe determination of unknown coefficients in partial differential equations from measured output data are well-known in the literature as inverse coefficient problems (ICPs). Especially, the nonlinear ICPs have been investigated by many authors in the past years. Among these, the torsion problems in linear elasticity have been well-studied in the classical literature [1][2][3][4]. In the case of elastoplastic torsion, the problems related to qualitative properties and an existence of a weak solution of the nonlinear direct problem are also well-studied in mathematical literature [5,6]. Within the range of monotone potential operator theory, the solvability of the nonlinear direct problem has been proved in [7], in the class of admissible coefficients. Many analytical and computational methods are given in the scientific literature, especially when small amounts of twisting are reversible and the beam returns to its original shape after releasing the twisting force. This is called the elastic behaviour. However, inverse problems in torsional deformation are less well-known and have received relatively little attention in the mathematical as well as in engineering literature [8][9][10]. The quasi-static mathematical model of torsional creep within J 2 -deformation theory of plasticity is given in [11]. In this model, one seeks the solution u(x, y), (x, y) 2 & R 2 , of the following nonlinear boundary-value problem:

Analysis of direct and inverse problems for a fractional elastoplasticity model

“…Hence the torque, defined by (6), may be considered as a function of the angle ϕ > 0. The direct problem (5) and the inverse problem (7) have been well-studied both theoretically and numerically in the mathematical literature (see [11], [12], [13], [33], [34], [35]).…”