分数阶微分方程中的“相变”
"Phase Transition" in fractional differential equations
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中文总结 AI 辅助
该研究指出分数阶微分方程的线性无关基本解组维数可超出方程阶数，当系数跨越阈值时发生相变，产生超维解组，修正初边值问题表述。
中文摘要 AI 辅助
线性常微分方程理论的基本结论是，线性无关基本解组的维数等于方程的阶数，这一观点通常从整数阶推广到分数阶微分方程。本研究旨在证明该预期不正确并解释原因，我们表明线性分数阶微分方程可能存在超出其阶数的额外线性无关解，当方程系数跨越某些阈值时会发生这种现象，该相变产生超维基本解组，为分数阶微分方程提供新见解，并修正初值和边值问题的表述。
英文摘要
The basic result in the theory of linear ODEs is that the dimension of the fundamental set of linearly independent solutions is equal to the order of equation. This view is typically extended from integer-order to fractional differential equations. The purpose of this research is to demonstrate that this expectation is incorrect and explain why. We show that linear fractional differential equations may admit additional linearly independent solutions beyond their order. This phenomenon occurs when the coefficients of the equation cross certain threshold values. This phase transition produces a hyper-dimensional fundamental set of solutions and offers new insight into fractional differential equations and leads to the revisions of the statements of initial and boundary value problems.
