Rasajit Kumar Bera

Analytical Solution of a Dynamic System Containing Fractional Derivative of Order One-Half by Adomian Decomposition Method

The fractional derivative has been occurring in many physical problems, such as frequency-dependent damping behavior of materials, motion of a large thin plate in a Newtonian fluid, creep and relaxation functions for viscoelastic materials, the PI λ D μ controller for the control of dynamical systems, etc. Phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry, and materials science are also described by differential equations of fractional order. The solution of the differential equation containing a fractional derivative is much involved. Instead of an application of the existing methods, an attempt has been made in the present analysis to obtain the solution of an equation in a dynamic system whose damping behavior is described by a fractional derivative of order 1/2 by the relatively new Adomian decomposition method. The results obtained by this method are then graphically represented and compared with those available in the work of Suarez and Shokooh [Suarez, L. E., and Shokooh, A., 1997, An Eigenvector Expansion Method for the Solution of Motion Containing Fraction Derivatives, ASME J. Appl. Mech., 64, pp. 629-635]. A good agreement of the results is observed.

Solution of an extraordinary differential equation by Adomian decomposition method

The aim of the present analysis is to apply the Adomian decomposition method for the solution of a fractional differential equation as an alternative method of Laplace transform.

The analytical approximate solution of the multi-term fractionally damped Van der Pol equation

The present paper deals with the analytical approximate solution of the fractionally damped Van der Pol equation by the homotopy perturbation method and a numerical method. By using initial conditions, the explicit solution of the equation is presented and then the numerical solutions are represented graphically; then the results obtained by two methods are compared.

The solution of coupled fractional neutron diffusion equations with delayed neutrons

The distribution of the neutron population in a nuclear reactor is described by using transport equations. One possible solution of the fractional neutron transport equation is given by the fractional neutron diffusion equation. This paper presents the application of an analytical approximation method for the solution of one group of fractional neutron diffusion equations with one group of delayed neutrons.

Relaxation Effects on Plane Wave Propagation in a Rotating Magneto-Thermo-Viscoelastic Medium

The present work aims at investing the propagation of plane magneto-thermo-viscoelastic waves in a homogeneous viscoelastic medium, which is rotating with a uniform angular velocity. Both the coupled dynamical thermoelasticity theory and the generalized thermoelasticity theory are applied to obtain the analytical solution for the phase velocities and the decay coefficients of the plane wave. The numerical values of the phase velocities and the decay coefficients are also computed for a suitable material and the results are presented in graphs. The effects of rotation and the thermal relaxation parameters on phase velocities and decay coefficients are discussed in details for both high and low frequency wave modes.

Large Amplitude Free Vibration of a Rotating Nonhomogeneous Beam With Nonlinear Spring and Mass System

This paper investigates the free out of plane vibration of a rotating nonhomogeneous beam with nonlinear spring and mass system. The effect of nonhomogeneity of the beam appears both in the governing equations and in the boundary conditions, but the nonlinear spring-mass effect appears in the boundary conditions only. The solution is obtained by applying the method of multiple time scales directly to the nonlinear partial differential equations and the boundary conditions. The results of the linear frequencies match well with those obtained in open literature. The effect of the nonhomogeneity of the stiffer beam (β=0.01) reduces the frequencies of vibration of the beam. A possible physical explanation of this reduced frequency of the nonhomogeneous beam is discussed. A subsequent nonlinear study of the nonhomogeneous beam indicates that the mass of the spring and its location also have a pronounced effect on the vibration of the beam. The effect of the nonhomogeneity of the beam on the relative stability of the nonlinear vibration of the beam with spring-mass system is also studied.

On generalized thermoelastic disturbances in an elastic solid with a spherical cavity

In this paper, a generalized dynamical theory of thermoelasticity is employed to study disturbances in an infinite elastic solid containing a spherical cavity which is subjected to step rise in temperature in its inner boundary and an impulsive dynamic pressure on its surface. The problem is solved by the use of the Laplace transform on time. The short time approximations for the stress, displacement and temperature are obtained to examine their discontinuities at the respective wavefronts. It is shown that the instantaneous change in pressure and temperature at the cavity wall gives rise to elastic and thermal disturbances which travel with finite velocities v1 and v2(gv1) respectively. The stress, displacement and temperature are found to experience discontinuities at the respective wavefronts. One of the significant findings of the present analysis is that there is no diffusive nature of the waves as found in classical theory.

Eigenvalue approach to study the effect of rotation and relaxation time in generalized magneto-thermo-viscoelastic medium in one dimension

The fundamental equations of the problems of generalized thermoelasticity with one relaxation parameter including heat sources in infinite rotating magneto-thermo-viscoelastic media have been derived in the form of a vector matrix differential equation in the Laplace transform domain for a one dimensional problem. These equations have been solved by the eigenvalue approach to determine deformations, stress, and temperature. The results have been compared to those available in the existing literature. The graphs have been drawn to show the effect of rotation in the medium.

Eigenfunction expansion method for the solution of magneto-thermoelastic problems with thermal relaxation and heat source in two dimensions

The paper deals with the study of the disturbances in an electrically conducting infinite elastic solid permeated by a primary magnetic field containing instantaneous point heat source. The electromagnetic equations of Maxwell and the coupled equations of thermoelasticity have been used. It is assumed that the elastic field under consideration is a homogeneous, orthotropic, electrically as well as thermally conducting one. The fundamental equations of the general three-dimensional problem of magneto-thermoelasticity have been written in the form of an inhomogeneous vector matrix differential equation and solved in the Laplace-Fourier transform-domain by eigenfunction method. Finally, the solution for space-time domain has been made by numerical methods and the graphs for stresses, etc., are drawn.

The formation of dynamic variable order fractional differential equation

In this paper, the formation of variable order (VO) model is established for continuous order fractional model. We review the definitions and properties of VO operators given by many researchers. We use the VO operator to define the new transfer function and analyze the model of a dynamic viscoelastic oscillator.