Nonlinear Inverse Problems for Equations with Dzhrbashyan–Nersesyan Derivatives

Fractal and Fractional

article

Vladimir E. Fedorov¹  Marina V. Plekhanova¹  Daria V. Melekhina¹ 

[1] Mathematical Analysis Department, Chelyabinsk State University

关键词: fractional differential equation;    Dzhrbashyan–Nersesyan fractional derivative;    inverse problem;    identification problem;    equation with unknown coefficients;    initial boundary value problem34G20;    35R30;    35R11;    34K29;   

DOI  :  10.3390/fractalfract7060464

学科分类：社会科学、人文和艺术（综合）

来源: mdpi

【 摘 要 】

The unique solvability in the sense of classical solutions for nonlinear inverse problems to differential equations, solved for the oldest Dzhrbashyan–Nersesyan fractional derivative, is studied. The linear part of the equation contains a bounded operator, a continuous nonlinear operator that depends on lower-order Dzhrbashyan–Nersesyan derivatives, and an unknown element. The inverse problem is given by an equation, special initial value conditions for lower Dzhrbashyan–Nersesyan derivatives, and an overdetermination condition, which is defined by a linear continuous operator. Applying the fixed-point method for contraction mapping a theorem on the existence of a local unique solution is proved under the condition of local Lipschitz continuity of the nonlinear mapping. Analogous nonlocal results were obtained for the case of the nonlocally Lipschitz continuous nonlinear operator in the equation. The obtained results for the problem in arbitrary Banach spaces were used for the research of nonlinear inverse problems with time-dependent unknown coefficients at lower-order Dzhrbashyan–Nersesyan time-fractional derivatives for integro-differential equations and for a linearized system of dynamics of fractional Kelvin–Voigt viscoelastic media.

【 授权许可】

CC BY