(3 + 1)‐Dimensional time‐fractional modified Burgers equation for dust ion‐acoustic waves as well as its exact and numerical solutions

Abstract

In this paper, the theoretical model of dust ion‐acoustic waves in collisionless, unmagnetized dust plasma is analyzed. According to the (3 + 1)‐dimensional nonlinear dynamic propagation equations with the electron distribution in the presence of trapping particles and the kinetic equation of charge of dust particle, based on multiscale analysis and perturbation method, a new (3 + 1)‐dimensional modified Burgers equation is constructed. Because of the space property of dust plasma, (3 + 1)‐dimensional modified Burgers equation is more suitable than low‐dimensional equation to describe the dust ion‐acoustic waves. Furthermore, applying the semi‐inverse method and the fractional variational principle, time‐fractional modified Burgers equation is given, which owns potential value for comprehending the propagation feature of dust ion‐acoustic waves in three‐dimensional space. Following, the conservation laws of the time‐fractional modified Burgers equation are discussed by using the Noether method. Finally, the shock wave solutions of the new model is obtained with the help of the G′/G‐expansion method; meanwhile, the radial basis function method is also used to get the numerical solutions of the (3 + 1)‐dimensional time‐fractional modified Burgers equation.