Comp. Appl. Math. DOI 10.1007/s40314-017-0509-y An asymmetric backward problem for the inhomogeneous parabolic equation with time-dependent diffusivity Triet Le Minh 1 · Phong Luu Hong 2 · Quan Pham Hoang 1 Received: 5 February 2017 / Revised: 1 August 2017 / Accepted: 24 August 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017 Abstract In this paper, we deal with an asymmetric case of the non-homogeneous backward parabolic problem associated with time-dependent diffusivity in polar coordinates which arises in describing the heat transfer in cylinder. In general, this problem is severely ill-posed by the Hadamard instability. To subdue the instability of this problem, we apply the modified quasi-boundary value method. According to some a priori assumptions on the exact solution, we get an explicit error estimate of Hölder type for all t ∈ (0, T ]. In addition, a numerical experiment is given to illustrate the efficiency and flexibility of our method. Keywords Backward parabolic problem · Modified quasi-boundary value method · Polar coordinates · Bessel expansion Mathematics Subject Classification 35R25 · 35R30 · 65M30 1 Introduction Over the past decades, the study of the backward parabolic problem (BPP) plays an impor- tant role in dealing with inverse problems for partial differential equations. Indeed, the BPP appears frequently in the following areas: heat conduction, pollutant detection, reaction diffu- sion, hydrologic inversion, thermo-elasticity and so on. The goal of the BPP is to reconstruct Communicated by Colin Cotter. This paper is dedicated to Professor Pham Hoang Quan on the occasion of his 51th birthday. B Triet Le Minh lmtriet@sgu.edu.vn 1 Faculty of Mathematics and Applications, Saigon University, 273 An Duong Vuong, Dist. 5, Ho Chi Minh City, Vietnam 2 Faculty of Mathematics, University of Science, Vietnam National University, 227 Nguyen Van Cu, Dist. 5, Ho Chi Minh City, Vietnam 123