Sultana Farheen, Pandey Rajesh K, Singh Deeksha, Agrawal Om P
Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Varanasi, 221005, Uttar Pradesh, India.
Mechanical Engineering and Energy Processes, Southern Illinois University, Carbondale, IL-62901, USA.
MethodsX. 2022 Nov 4;9:101905. doi: 10.1016/j.mex.2022.101905. eCollection 2022.
This paper presents a high order approximation scheme to solve the generalized fractional telegraph equation (GFTE) involving the generalized fractional derivative (GFD). The GFD is characterized by a scale function and a weight function . Thus, we study the solution behavior of the GFTE for different and . The scale function either stretches or contracts the solution while the weight function dramatically shifts the numerical solution of the GFTE. The time fractional GFTE is approximated using quadratic scheme in the temporal direction and the compact finite difference scheme in the spatial direction. To improve the numerical scheme's accuracy, we use the non-uniform mesh. The convergence order of the whole discretized scheme is, where and are the temporal and spatial step sizes respectively. The outcomes of the work are as follows: •The error estimate for approximation of the GFD on non-uniform meshes is established.•The numerical scheme's stability and convergence are examined.•Numerical results for four examples are compared with those obtained using other method. The study shows that the developed scheme achieves higher accuracy than the scheme discussed in literature.
本文提出了一种高阶近似格式来求解包含广义分数阶导数（GFD）的广义分数阶电报方程（GFTE）。GFD由一个尺度函数和一个权函数表征。因此，我们研究了不同的尺度函数和权函数下GFTE的解的行为。尺度函数会拉伸或压缩解，而权函数会显著地移动GFTE的数值解。时间分数阶GFTE在时间方向上采用二次格式近似，在空间方向上采用紧致有限差分格式近似。为了提高数值格式的精度，我们使用非均匀网格。整个离散格式的收敛阶为 ，其中 和 分别是时间步长和空间步长。工作成果如下：•建立了非均匀网格上GFD近似的误差估计。•研究了数值格式的稳定性和收敛性。•将四个例子的数值结果与使用其他方法得到的结果进行了比较。研究表明，所提出的格式比文献中讨论的格式具有更高的精度。
