Ritz-Galerkin method for solving a parabolic equation with non-local and time-dependent boundary conditions

• Published in: Mathematical methods in the applied sciences Vol. 39; no. 5; pp. 1241 - 1253
• Main Authors: Zhou, Jian-Rong, Li, Heng, Xu, Yongzhi
• Format: Journal Article
• Language: English
• Published: Freiburg Blackwell Publishing Ltd 01.04.2016; Wiley Subscription Services, Inc
• Subjects: Approximation; approximation solution; Bernstein polynomial basis; Boundaries; Boundary conditions; Dealing; ductal carcinoma in situ (DCIS) model; initial boundary value problem; Integrals; Mathematical analysis; Mathematical models; non-local boundary condition; parabolic equation; Partial differential equations; Ritz-Galerkin method; time-dependent boundary condition
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.3568

The paper is devoted to the investigation of a parabolic partial differential equation with non‐local and time‐dependent boundary conditions arising from ductal carcinoma in situ model. Approximation solution of the present problem is implemented by the Ritz–Galerkin method, which is a first attempt at tackling parabolic equation with such non‐classical boundary conditions. In the process of dealing with the difficulty caused by integral term in non‐local boundary condition, we use a trick of introducing the transition function G(x,t) to convert non‐local boundary to another non‐classical boundary, which can be handled with the Ritz–Galerkin method. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2015 John Wiley & Sons, Ltd.