In this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

在本文中，我们提出了一种基于局部无网格方法和拉普拉斯变换的混合方法，用于逼近 Caputo-Fabrizio 分数阶导数意义上的线性一维偏微分方程的解。在我们的数值方案中，拉普拉斯变换用于避免时间步长过程，局部无网格方法用于产生稀疏微分矩阵并避免导致全局无网格方法的病态问题。我们的数值方法包括三个步骤。在第一步中，我们将给定的方程转换为等效的时间无关方程。其次，通过局部无网格方法求解简化方程。最后，原始方程的解是通过逆拉普拉斯变换获得的，将其表示为复左半平面中的轮廓积分。然后使用梯形规则近似轮廓积分。讨论了该方法的稳定性和收敛性。使用四个不同的问题来评估所提出方法的效率、功效和准确性。获得这些问题的数值近似值，并针对精确解进行验证。得到的结果表明，所提出的方法可以有效地解决此类问题。获得这些问题的数值近似值，并针对精确解进行验证。得到的结果表明，所提出的方法可以有效地解决此类问题。获得这些问题的数值近似值，并针对精确解进行验证。得到的结果表明，所提出的方法可以有效地解决此类问题。