The direct meshless local Petrov-Galerkin technique with its error estimate for distributed-order time fractional Cable equation

Distributed-order fractional calculus is a quickly growing concept of the more general area of fractional calculus that has significant and extensive usage for designing complex systems. This work is used the direct meshless local Petrov-Galerkin (DMLPG) technique for the numerical solution of the distributed-order time fractional Cable equation. DMLPG implements a generalized moving least square (GMLS) process to discretize the equation in space variables. By using this scheme, the test function is approximated via the values at nodes, directly. Thus, this algorithm passes integration with the MLS shape functions substituting with a more inexpensive integration than polynomials. Here, the distributed integral is discretized by the M-point Gauss-Legendre quadrature rule. Then, the finite difference scheme is applied to approximate the fractional derivative discretization. Also, the unconditionally stability and rate of convergence O(tau(2-max{alpha,beta})) of the time-discrete technique are demonstrated. Moreover, the current method converts the problem into a system of linear algebraic equations. To demonstrate the capability and flexibility of our scheme, some examples with different geometric domains are supposed in two-dimensional cases.