论文标题
在其定义领域之外的最初有限价值问题的解决方案的分析扩展
The analytic extension of solutions to initial-boundary value problems outside their domain of definition
论文作者
论文摘要
我们检查了在定义的空间域之外，线性，恒定的初始价值问题的解决方案的分析扩展。我们使用FOKA的统一变换方法或方法，该方法为半行和有限间隔初始边界值问题提供了一种表示，作为具有显式空间和时间依赖性的内核的积分。这些解决方案表示形式是在问题的空间域中定义的。我们通过在这些空间域之外的泰勒串联获得这些表示公式的扩展，并找到引起扩展解决方案解决的全线初始值问题的初始条件的扩展。通常，除非边界和初始条件满足兼容性条件，否则扩展的初始条件不是可区分或连续的。我们分析耗散和分散问题，以及连续和离散空间变量的问题。
We examine the analytic extension of solutions of linear, constant-coefficient initial-boundary value problems outside their spatial domain of definition. We use the Unified Transform Method or Method of Fokas, which gives a representation for solutions to half-line and finite-interval initial-boundary value problems as integrals of kernels with explicit spatial and temporal dependence. These solution representations are defined within the spatial domain of the problem. We obtain the extension of these representation formulae via Taylor series outside these spatial domains and find the extension of the initial condition that gives rise to a whole-line initial-value problem solved by the extended solution. In general, the extended initial condition is not differentiable or continuous unless the boundary and initial conditions satisfy compatibility conditions. We analyze dissipative and dispersive problems, and problems with continuous and discrete spatial variables.
