Ritz–Galerkin method for solving a parabolic equation with non‐local and time‐dependent boundary conditions |
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Authors: | Jian‐Rong Zhou Heng Li Yongzhi Xu |
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Institution: | 1. Department of Mathematics, Foshan University, Foshan 528000, Guangdong, China;2. Department of Mathematics, University of Louisville, Louisville, KY 40292, USA |
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Abstract: | 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. |
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Keywords: | Ritz– Galerkin method Bernstein polynomial basis parabolic equation non‐local boundary condition time‐dependent boundary condition initial boundary value problem approximation solution ductal carcinoma in situ (DCIS) model |
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