(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. KEYWORDS(3 + 1)-dimensional time-fractional modified Burgers equation, conservation laws, dust ion-acoustic waves, radial basis function method, shock wave solution MSC CLASSIFICATION 35Q51; 35Q55After solving the equation Ac k =B k , we get the algebraic system at each time step, and we can obtain the solution by using Equation (50).

Analysis of Memreactance with Fractional Kinetics

“…Despite employing the fractional-order kinetics, the constitutive relations of the meminductor and memcapacitor remain as given by (1) and (4). As a result, the emulation of memreactance by using the memristor and conventional memristor to meminductor/memcapacitor emulators [33,46] remains possible. In this section, such possibility will be illustrated.…”

“…For the meminductor emulation, k M � k L must be satisfied. At this point, it can be seen that the meminductor can be emulated by using the memristor and the conventional memristor to the meminductor emulator [33] despite taking the fractional-order kinetics into account. In addition, the kinetics of the memristor must be of fractional order in order to obtain the meminductor employing such fractional-order kinetics.…”

“…Motivated by the generalization of memristor kinetics, we generalize the kinetics of the meminductor and memcapacitor which are commonly referred to as memreactance [33][34][35] and have been adopted in many applications, e.g., electronic oscillator [36] and synaptic circuit [37,38], in the fractional-order domain by applying the fractional calculus concept to the state equation of the memreactance and perform the modelling of such memreactance. By using the obtained results, the meminductances, memcapacitances, and related parameters due to both DC and periodic input waveforms have been derived.…”

Fractional Theoretical Model for Gravity Waves and Squall Line in Complex Atmospheric Motion

“…First of all, the situation of domain [0, h 0 ] is studied. Introducing the multiscale slow time and space transformations [46,47],…”

Fractional KdV and Boussenisq‐Burger's equations, reduction to PDE and stability approaches

“…While in the first case of self similar solutions can be derived. [34][35][36][37] Very recently, in studying the fractional equations that arise in shallow water waves, authors are trying to find approximate solutions either analytically or numerically. [16][17][18][19][20] The present work discusses the details about the results found in these works.…”