Numerical Simulation of PDEs by Local Meshless Differential Quadrature Collocation Method

Abstract

In this paper, a local meshless differential quadrature collocation method based on radial basis functions is proposed for the numerical simulation of one-dimensional Klein–Gordon, two-dimensional coupled Burgers’, and regularized long wave equations. Both local and global meshless collocation procedures are used for spatial discretization, which convert the mentioned partial differential equations into a system of ordinary differential equations. The obtained system has been solved by the forward Euler difference formula. An upwind technique is utilized in the case of the convection-dominated coupled Burgers’ model equation. Having no need for the mesh in the problem domain and being less sensitive to the variation of the shape parameter as compared to global meshless methods are the salient features of the local meshless method. Both rectangular and non-rectangular domains with uniform and scattered nodal points are considered. Accuracy, efficacy, and the ease of implementation of the proposed method are shown via test problems.

Solution of Multi-Term Time-Fractional PDE Models Arising in Mathematical Biology and Physics by Local Meshless Method

“…The LMM [26,31] is utilized for the solution of time-fractional convection-diffusion models. The derivatives of U(y, t) at the centers y h are approximated by the neighborhood of y h , {y h1 , y h2 , y h3 , ..., y hn h } ⊂ {y 1 , y 2 , .…”

“…Tragically, the computational cost of execution and ill-conditioning of the procedure will increment drastically by increasing the number of collocation points. To avoid these limitations, the researchers recommended local meshless techniques [25,26]. The beauty of the local meshless technique is utilizing just neighbouring collocation points which results in a sparse matrix system and ward off the main deficiency of ill-conditioning.…”

Application of local meshless method for the solution of two term time fractional-order multi-dimensional PDE arising in heat and mass transfer

“…The LMM [26] is extended to the time fractional convection-diffusion models. The derivatives of W(z ¯, t) are approximated at the centers z ¯h by the neighborhood of…”

Numerical Solution of the Multiterm Time‐Fractional Model for Heat Conductivity by Local Meshless Technique

“…e RBF-based meshless methods have also some deficiencies like other numerical methods, in which the most important one is the dense ill-conditioned matrices and the selection of the optimal value of the shape parameter. To avoid these drawbacks, local meshless methods are the best alternatives, suggested by the researchers which are considered to be accurate and stable for the solution of diverse integer and fractional-order PDE models [18,19]. e local meshless methods are less sensitive to the change in shape parameters than the global version, and it produces wellconditioned sparse matrices.…”