An Efficient Local Formulation for Time–Dependent PDEs

Imtiaz Ahmad,Muhammad Ahsan,Zaheer-Ud Din,Ahmad Masood,Poom Kumam

doi:10.3390/math7030216

Abstract

In this paper, a local meshless method (LMM) based on radial basis functions (RBFs) is utilized for the numerical solution of various types of PDEs. This local approach has flexibility with respect to geometry along with high order of convergence rate. In case of global meshless methods, the two major deficiencies are the computational cost and the optimum value of shape parameter. Therefore, research is currently focused towards localized RBFs approximations, as proposed here. The proposed local meshless procedure is used for spatial discretization, whereas for temporal discretization, different time integrators are employed. The proposed local meshless method is testified in terms of efficiency, accuracy and ease of implementation on regular and irregular domains.

Highlights

Partial differential equation (PDE) of reaction-convection-diffusion type are physically very rich
To check the accuracy and efficiency of the local meshless method (LMM) various test problems in one and two dimensional cases are considered and the results are compared with the existence methods reported in literature
Accuracy of the LMM is measured though different error norms given as follows

Summary

Introduction

Partial differential equation (PDE) of reaction-convection-diffusion type are physically very rich. In general it is difficult and sometimes impossible to get exact solution of such type of nonlinear PDEs. Researchers employed different numerical techniques which include discrete Adomian decomposition method [12], Haar wavelet. Hirota-Satsuma introduced the nonlinear coupled Kortewege-de Vries equations [28] This model has numerous applications in physical sciences. Researchers have used various numerical techniques for the solution of this model equations These include RBFs collocation (Kansa) method [29], meshless RBFs method of lines [30], variational iteration method [31] and spectral collocation method [32].

Partial Differential Equation Models

Local Meshless Numerical Scheme

Results and Discussion

Conclusions