In this article we aim to approximate linear time fractional advection diffusion equations (TFADE) with Atangana-Baleanu- Caputo(ABC) derivative using local meshless method and Laplace transformation(LT). The method comprises of three steps. In the first step the the time variable is eliminated using LT. In the second step the reduced problem is solved using local meshless method. In the third step the solution of TFADE with ABC derivative is retrieved from local meshless methods solution by representing it as Bromwich integral. We then approximate the integral using some suitable quadrature rule. The stability and convergence of the method are discussed. The local meshless method is utilized to overcome the ill-conditioning issue of the interpolation matrices in global meshless methods and to over come the shape parameters sensitivity. Also in comparison with time stepping methods the LT is employed and contour integration technique is utilized to deal with the ABC derivative, which circumvent the calculation of costly convolution integrals in the approximation of ABC derivative, and also avoids the effect of time step on the stability and accuracy. Some test problems are considered in one and two dimensions to validate the proposed numerical method. The two dimensional problem is solved in regular and irregular domains. The computational experiments confirms that this method is computationally efficient and highly accurate for such type of problems.

在本文中，我们旨在使用局部无网格方法和拉普拉斯变换 (LT) 用 Atangana-Baleanu-Caputo(ABC) 导数近似线性时间分数平流扩散方程 (TFADE)。该方法包括三个步骤。在第一步中，使用 LT 消除时间变量。在第二步中，使用局部无网格方法解决简化问题。在第三步中，通过将其表示为 Bromwich 积分，从局部无网格方法解中检索出具有 ABC 导数的 TFADE 解。然后我们使用一些合适的求积规则来近似积分。讨论了该方法的稳定性和收敛性。局部无网格方法用于克服全局无网格方法中插值矩阵的病态问题和克服形状参数敏感性。同样与时间步长方法相比，采用LT和轮廓积分技术处理ABC导数，避免了ABC导数逼近中昂贵的卷积积分计算，也避免了时间步长对稳定性的影响和准确性。在一维和二维中考虑一些测试问题以验证所提出的数值方法。在规则和不规则域中解决二维问题。计算实验证实，该方法对于此类问题具有计算效率和高度准确度。避免了ABC导数逼近中代价高昂的卷积积分的计算，也避免了时间步长对稳定性和精度的影响。在一维和二维中考虑一些测试问题以验证所提出的数值方法。在规则和不规则域中解决二维问题。计算实验证实，该方法对于此类问题具有计算效率和高度准确度。避免了ABC导数逼近中代价高昂的卷积积分的计算，也避免了时间步长对稳定性和精度的影响。在一维和二维中考虑一些测试问题以验证所提出的数值方法。在规则和不规则域中解决二维问题。计算实验证实，该方法对于此类问题具有计算效率和高度准确度。