Abstract: Due to many applications and problems in the fields of plasma physics, geophysics, and other many topics, the interaction between the strain field and the magnetic field has to be considered. Adomian introduced the decomposition method for solving linear and nonlinear functional equations. This method leads to accurate, computable, approximately convergent solutions of linear and nonlinear partial and ordinary differential equations even the equations with variable coefficients. This paper is dealing with a mathematical model of generalized thermoelasticity of a half-space conducting medium. A magnetic field with constant intensity acts normal to the bounding plane has been assumed. Adomian’s decomposition method has been used to solve the model when the bounding plane is taken to be traction free and thermally loaded by harmonic heating. The numerical results for the temperature increment, the stress, the strain, the displacement, the induced magnetic, and the electric fields have been represented in figures. The magnetic field, the relaxation time, and the angular thermal load have significant effects on all the studied fields.
Abstract: In this work, functionally graded materials (FGMs), subjected to loading, which varies with time has been studied. The material properties of FGM are changing through the thickness of material as power law distribution. The conical shells have been chosen for this study so in the first step capability equations for FGM have been obtained. With Galerkin method, these equations have been replaced with time dependant differential equations with variable coefficient. These equations have solved for different initial conditions with variation methods. Important parameters in loading conditions are semi-vertex angle, external pressure and material properties. Results validation has been done by comparison between with those in previous studies of other researchers.
Abstract: We develop a method based on polynomial quintic
spline for numerical solution of fourth-order non-homogeneous
parabolic partial differential equation with variable coefficient. By
using polynomial quintic spline in off-step points in space and
finite difference in time directions, we obtained two three level
implicit methods. Stability analysis of the presented method has been
carried out. We solve four test problems numerically to validate the
derived method. Numerical comparison with other methods shows
the superiority of presented scheme.
Abstract: The analytical bright two soliton solution of the 3-
coupled nonlinear Schrödinger equations with variable coefficients in
birefringent optical fiber is obtained by Darboux transformation
method. To the design of ultra-speed optical devices, Soliton
interaction and control in birefringence fiber is investigated. Lax pair
is constructed for N coupled NLS system through AKNS method.
Using two-soliton solution, we demonstrate different interaction
behaviors of solitons in birefringent fiber depending on the choice of
control parameters. Our results shows that interactions of optical
solitons have some specific applications such as construction of logic
gates, optical computing, soliton switching, and soliton amplification
in wavelength division multiplexing (WDM) system.
Abstract: As p-Laplacian equations have been widely applied in field of the fluid mechanics and nonlinear elastic mechanics, it is necessary to investigate the periodic solutions of functional differential equations involving the scalar p-Laplacian. By using Mawhin’s continuation theorem, we study the existence of periodic solutions for p-Laplacian neutral Rayleigh equation (ϕp(x(t)−c(t)x(t − r))) + f(x(t)) + g1(x(t − τ1(t, |x|∞))) + β(t)g2(x(t − τ2(t, |x|∞))) = e(t), It is meaningful that the functions c(t) and β(t) are allowed to change
signs in this paper, which are different from the corresponding ones of known literature.
Abstract: A method based on the power series solution is proposed to solve the natural frequency of flapping vibration for the rotating inclined Euler beam with constant angular velocity. The vibration of the rotating beam is measured from the position of the corresponding steady state axial deformation. In this paper the governing equations for linear vibration of a rotating Euler beam are derived by the d'Alembert principle, the virtual work principle and the consistent linearization of the fully geometrically nonlinear beam theory in a rotating coordinate system. The governing equation for flapping vibration of the rotating inclined Euler beam is linear ordinary differential equation with variable coefficients and is solved by a power series with four independent coefficients. Substituting the power series solution into the corresponding boundary conditions at two end nodes of the rotating beam, a set of homogeneous equations can be obtained. The natural frequencies may be determined by solving the homogeneous equations using the bisection method. Numerical examples are studied to investigate the effect of inclination angle on the natural frequency of flapping vibration for rotating inclined Euler beams with different angular velocity and slenderness ratio.
Abstract: We have solved the Burgers-Fisher (BF) type equations,
with time-dependent coefficients of convection and reaction terms,
by using the auxiliary equation method. A class of solitary wave
solutions are obtained, and some of which are derived for the first
time. We have studied the effect of variable coefficients on physical
parameters (amplitude and velocity) of solitary wave solutions. In
some cases, the BF equations could be solved for arbitrary timedependent
coefficient of convection term.
Abstract: We seek exact solutions of the coupled Klein-Gordon-Schrödinger equation with variable coefficients with the aid of Lie classical approach. By using the Lie classical method, we are able to derive symmetries that are used for reducing the coupled system of partial differential equations into ordinary differential equations. From reduced differential equations we have derived some new exact solutions of coupled Klein-Gordon-Schrödinger equations involving some special functions such as Airy wave functions, Bessel functions, Mathieu functions etc.
Abstract: Water pollution assessment problems arise frequently
in environmental science. In this research, a finite difference method
for solving the one-dimensional steady convection-diffusion equation
with variable coefficients is proposed; it is then used to optimize
water treatment costs.