分数阶微分方程的积分条件的理论与数值研究
On Theoretical and Numerical Aspect of Fractional Differential Equations with Purely Integral Conditions
摘要：
本文研究了一类分数阶导数具有积分型边界条件的问题。该问题表示一个Caputo型对流扩散方程，其中分数阶对时间的导数为$1<α<2$ .利用能量不等式的方法证明了该问题解的存在唯一性。为了找到所考虑问题的近似解，还引入了有限差分法对问题进行数值研究。给出了一些数值例子，结果令人满意。
英文摘要：
In this paper, we are interested in the study of a problem with fractional derivatives having boundary conditions of integral types. The problem represents a Caputo type advection-diffusion equation where the fractional order derivative with respect to time with $1<α<2$. The method of the energy inequalities is used to prove the existence and the uniqueness of solutions of the problem. The finite difference method is also introduced to study the problem numerically in order to find an approximate solution of the considered problem. Some numerical examples are presented to show satisfactory results.
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