An initial-boundary value problem of subdiffusion type is considered; the temporal component of the differential operator has the form $\sum_{i=1}^{\ell}q_i(t)\, D _t ^{\alpha_i} u(x,t)$, where the $q_i$ are continuous functions, each $D _t ^{\alpha_i}$ is a Caputo derivative, and the $\alpha_i$ lie in $(0,1]$. Maximum/comparison principles for this problem are proved under weak hypotheses. A new positivity result for the multinomial Mittag-Leffler function is derived. A posteriori error bounds are obtained in $L_2(\Omega)$ and $L_\infty(\Omega)$, where the spatial domain $\Omega$ lies in $\bR^d$ with $d\in\{1,2,3\}$. An adaptive algorithm based on this theory is tested extensively and shown to yield accurate numerical solutions on the meshes generated by the algorithm.
翻译：考虑子扩散类型的初始- 边界值问题; 差分操作员的时间值成分为 $\ sum ⁇ i=1 ⁇ ell}q_i( t)\, D_ t ⁇ alpha_i}u( x, t)$, 美元是连续函数, 美元为$_ t ⁇ alpha_ i} 美元为卡普托衍生物, 美元为$\ alpha_ i 美元为$( 0. 1) 。 这一问题的最大/ 比较原则在虚弱的假设下被证明。 多名Mittag- Leffler 函数的新假设结果被推导出。 事后错误的界限以$[2(\\\\ Omega] 美元) 和 $Linfty(\ $\ omega) 美元获得, 空间域 $\\ bR_ d$ 美元为$ d\\\\\\\\ $1, 2, 3 美元。 基于这个理论的适应算算法经过广泛测试, 并显示在算算算算算的中间制的中间的调算出准确的数字解决方案上的数字解决方案上的数字解决办法。
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