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 $1<\alpha <2$. 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.

中文翻译：

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具有纯积分条件的分数阶微分方程的理论和数值方面

在本文中，我们对具有整数类型边界条件的分数导数的问题感兴趣。该问题表示Caputo型对流扩散方程，其中相对于时间的分数阶导数为$ 1 <\ alpha <2 $。用能量不等式的方法证明问题解的存在性和唯一性。还引入了有限差分法来对问题进行数值研究，以便找到所考虑问题的近似解。给出了一些数值示例以显示令人满意的结果。