Reconstruction of a space-dependent source in the inexact order time-fractional diffusion equation

• Published in: Chaos, solitons and fractals solitons and fractals, 2020-05, Vol.134, p.109724, Article 109724
• Main Authors: Trong, Dang Duc, Hai, Dinh Nguyen Duy, Minh, Nguyen Dang
• Format: Article
• Language: English
• Subjects: Ill-posed problem; Inverse source problem; Order optimal error bounds; Regularization; Time-fractional diffusion equation
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2020.109724

•A general modified quasi-boundary value regularization method is proposed to solve the inverse source problem of the time-fractional diffusion equation with inexact order.•The proposed technique is extendable to solve the continuous spectrum of the inverse and ill-posed parabolic partial differential equations.•Order optimal convergence rates for the worst case error of the method are derived under the usual source condition by using an a-priori and an a-posteriori regularization parameter choice rules, respectively.•Several numerical examples are provided to illustrate the effectiveness of the proposed method. An inverse problem to recover a space-dependent factor of a source term in the inexact order time-fractional diffusion equation from final data is considered. The problem arises in many applications, but it is in general ill-posed. The ill-posedness is since small errors in the input data cause large errors in the output solution. To overcome this instability we propose the stable approximation solution via a general modified quasi-boundary value regularization method. Order optimal convergence rates for the worst case error of the method are derived under the usual source condition by using an a-priori and an a-posteriori regularization parameter choice rules, respectively. Finally, several numerical examples are provided to illustrate the effectiveness of the proposed method.