At the present work, we propose a new numerical scheme for the solution of two and three dimensional time fractional nonlinear damped Klein–Gordon equation (DKGE). To this end, we use the Legendre spectral element method to discretize the equation in the spatial directions and for the time stepping, an alternating direction implicit (ADI) method based on a scheme of order $\mathcal{O}\left({\tau }^2\right)$ is considered. We prove that the presented method is unconditionally stable and obtain an error estimate for the fully discrete scheme. The presented numerical experiments demonstrate the good accuracy and efficiency of the proposed ADI spectral element method.

中文翻译：

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用ADI谱元法求解二维和三维时间分数阶阻尼非线性Klein-Gordon方程的数值解。

在目前的工作中，我们为二维和三维时间分数非线性阻尼Klein-Gordon方程（DKGE）的求解提出了一种新的数值方案。为此，我们使用勒让德（Legendre）谱元方法在空间方向上离散方程，对于时间步长，使用基于有序方案的交替方向隐式（ADI）方法$\mathcal{Ø}（{\tau }^{\mathrm{2个}}）$被认为。我们证明了所提出的方法是无条件稳定的，并且获得了完全离散方案的误差估计。提出的数值实验证明了所提出的ADI光谱元素方法的良好准确性和效率。