The Cauchy problem for higher-order linear partial differential equation

2010/10/05 by Guang-Qing Bi, Guangqing Bi, Bi, Guangqing +2
Mathematics · #35G05 #35G10 #Algebraic and Geometric Analysis #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods for differential equations #math.AP #math.FA #msc:35G05 #msc:35G10

paper · pdf · doi:10.48550/arxiv.1010.0761

openalex publication_date 2010/10/05 · arxiv created 2011/02/03 · arxiv updated 2011/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For the linear partial differential equation P(∂_x,∂_t)u=f(x,t), where x∈ℝⁿ, t∈ℝ¹, with P(∂_x,∂_t) is ∏^m_i=1(\frac∂∂t-a_iP(∂_x)) or ∏^m_i=1(\frac∂²∂t²-a_i²P(∂_x)), the authors give the analytic solution of the cauchy problem using the abstract operators e^tP(∂_x) and \frac\sinh(tP(∂_x)^1/2)P(∂_x)^1/2. By representing the operators with integrals, explicit solutions are obtained with an integral form of a given function.

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