The diffusion equation with nonlocal data

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:466

Article

Miller, P. D.¹  Smith, D. A.² 

[1] Univ Michigan, Dept Math, Ann Arbor, MI 48109 USA

[2] Yale NUS Coll, Singapore, Singapore

关键词: Evolution partial differential equation;    Nonlocal boundary condition;    Initial-boundary value problem;    Fokas method;   

DOI  :  10.1016/j.jmaa.2018.05.064

来源: Elsevier

【 摘 要 】

We study the diffusion (or heat) equation on a finite 1-dimensional spatial domain, but we replace one of the boundary conditions with a nonlocal condition, through which we specify a weighted average of the solution over the spatial interval. We provide conditions on the regularity of both the data and weight for the problem to admit a unique solution, and also provide a solution representation in terms of contour integrals. The solution and well-posedness results rely upon an extension of the Fokas (or unified) transform method to initial nonlocal value problems for linear equations; the necessary extensions are described in detail. Despite arising naturally from the Fokas transform method, the uniqueness argument appears to be novel even for initial boundary value problems. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free