In this paper, we study the Caputo–Hadamard fractional partial differential equation where the time derivative is the Caputo–Hadamard fractional derivative and the space derivative is the integer-order one. We first introduce a modified Laplace transform. Then using the newly defined Laplace transform and the well-known finite Fourier sine transform, we obtain the analytical solution to this kind of linear equation. Furthermore, we study the regularity and logarithmic decay of its solution. Since the equation has a time fractional derivative, its solution behaves a certain weak regularity at the initial time. We use the finite difference scheme on non-uniform meshes to approximate the time fractional derivative in order to guarantee the accuracy and use the local discontinuous Galerkin method (LDG) to approximate the spacial derivative. The fully discrete scheme is established and analyzed. A numerical example is displayed which support the theoretical analysis.

在本文中，我们研究了Caputo-Hadamard分数阶偏微分方程，其中时间导数是Caputo-Hadamard分数阶导数，空间导数是整数阶。我们首先介绍一个经过修改的Laplace变换。然后使用新定义的拉普拉斯变换和著名的有限傅里叶正弦变换，我们得到了这种线性方程的解析解。此外，我们研究了其解的正则性和对数衰减。由于方程具有时间分数导数，因此其解在初始时间表现出一定的弱规律性。为了保证精度，我们在非均匀网格上使用有限差分方案来近似时间分数导数，并使用局部不连续Galerkin方法（LDG）来近似空间导数。建立并分析了完全离散的方案。将显示一个数值示例，以支持理论分析。