2010/08/23 by Guang-Qing Bi, Bi, Guang-Qing, Yuekai Bi +2 · 3 citations
Computer Science · Mathematics · #35A10 #35G16 #44A10 #47F05 #Algebraic and Geometric Analysis #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Matrix Theory and Algorithms #math.AP #math.FA #msc:35A10 #msc:35G16 #msc:44A10 #msc:47F05
paper · pdf · doi:10.48550/arxiv.1008.3808
28 pages
openalex publication_date 2010/08/23 · arxiv created 2016/06/23 · arxiv updated 2016/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We first strictly expressed the basic notions and research methods of abstract operators, which systematically expounded the main results of abstract operator theory. By combining abstract operators with the Laplace transform, we can easily apply this Laplace transform to n+1 dimensional partial differential equations. Further, all the analytic solutions to an initial value problem of an arbitrary order linear partial differential equation are expressed in these abstract operators. By writing abstract operators in this class into integral forms, the solutions in operator form are represented into integral forms. We thus solved the important problem of representing the solutions of linear higher-order partial differential equations into the integrations of some given functions. By introduction of abstract operators on Hilbert space, we further discuss the solvability of initial-boundary value problem for the linear higher-order partial differential equations.