论文标题
弱单数卷积积分的复合正交方法
Composite Quadrature Methods for Weakly Singular Convolution Integrals
论文作者
论文摘要
众所周知的Caputo分数衍生物和相应的Caputo分数积分自然出现在许多方程式中，这些方程式在不均匀培养基下进行了物理现象。通过将拉普拉斯变换应用于给定方程式，可以很容易地获得两个分数项之间的关系。我们寻求使用泰勒串联膨胀积分来在数值上近似Caputo分数积分。自然而然地，能够为更广泛的卷积积分内核$ k（T-s）$近似卷积积分。这种方法下的主要优点之一是能够在数值上近似弱的奇异核，而这些核无法在传统的正交方法下融合。我们为这些复合二次化提供了稳定性和收敛分析，它们为$ c^γ[0，t] $中近似功能提供了最佳的收敛，其中$α\ leqleqγ\ leq 5 $和$ 0 <α<1 $。对于$γ= 1,2,3,4,5 $方案的顺序，结果近似为$ o（τ^γ）$精确度，其中$τ$是时域分区的大小。通过使用分数泰勒系列扩展，我们可以以$γ\ in（0,5） - \ {1,2,3,4 \} $订单方案获得，从而产生了$ O（τ^γ）$的近似值，并持续依赖于可改善转化顺序的内核功能。这使得可以近似较宽的函数，并且通过加强规律性假设，我们能够获得更准确的结果。通用卷积积分表现出这些结果高达$γ= 2 $，而不会假设$ k $减少。最后，提出了一些数值示例，这些示例验证了我们的发现。
The well-known Caputo fractional derivative and the corresponding Caputo fractional integral occur naturally in many equations that model physical phenomena under inhomogeneous media. The relationship between the two fractional terms can be readily obtained by applying the Laplace transform to a given equation. We seek to numerically approximate Caputo fractional integrals using a Taylor series expansion for convolution integrals. This naturally extends into being able to approximate convolution integrals for a wider class of convolution integral kernels $K(t-s)$. One of the main advantages under this approach is the ability to numerically approximate weakly singular kernels, which fail to converge under traditional quadrature methods. We provide stability and convergence analysis for these composite quadratures, which offer optimal convergence for approximating functions in $C^γ[0,T]$, where $α\leq γ\leq 5$ and $0<α< 1$. For the order $γ= 1,2,3,4,5$ scheme, the resulting approximation is $O(τ^γ)$ accurate, where $τ$ is the size of the partition of the time domain. By instead utilizing a fractional Taylor series expansion, we are able to obtain for $γ\in (0,5)-\{1,2,3,4\}$ order scheme, which yields an approximation of $O(τ^γ)$ with a constant dependent on the kernel function which improves the order of convergence. This allows for a far wider class of functions to be approximated, and by strengthening the regularity assumption, we are able to obtain more accurate results. General convolution integrals exhibit these results up to $γ= 2$ without the assumption of $K$ being decreasing. Finally, some numerical examples are presented, which validate our findings.
