Volume 20, Issue 5 -

In this work, the Kudryashov method is handled to find exact solutions of nonlinear fractional partial differential equations in the sense of the modified Riemann–Liouville derivative as given by Guy Jumarie. Firstly, these fractional equations can be turned into another nonlinear ordinary differential equations by fractional complex transformation. Then, the method is applied to solve the space-time fractional Symmetric Regularized Long Wave equation and the space-time fractional generalized Hirota–Satsuma coupled KdV equation. The obtained solutions include generalized hyperbolic functions solutions.

A boundary-layer analysis is presented for steady, two-dimensional, Marangoni convection along an irregular surface. Thick radiation limit is considered to express the radiative heat flux. A coordinate transformation is adopted to convert the physical domain into computational domain. Implicit finite difference method is then used to obtain the solutions of the problem. The main purpose of this study is to demonstrate the radiation effects on the dissipative layers. Numerical solutions are presented in the form of skin friction coefficient, heat transfer coefficient, velocity and temperature profiles, streamlines and isotherms. It is observed that thermal radiation has a pronounced effect on the flow field and amplitude of the harmonic oscillations also decay with Rd$R_d$. The momentum and thermal boundary-layer thickness increases as Rd$R_d$ gets augmented.

Two numerical procedures are developed to approximate the solution of one-dimensional parabolic equations using extrapolated collocation method. By defining two different end conditions and forcing cubic spline to satisfy the interpolation conditions along with one of the end conditions, we obtain fourth- (CBS4) and sixth-order (CBS6) approximations to the solution in spatial direction. Also in time direction, a weighted finite difference discretization is used to approximate the solution at each time level. The convergence analysis is discussed in detail and some error bounds are obtained theoretically. Finally, some different examples of Burgers’ equation with applications in fluid mechanics as well as convection–diffusion problems with applications in transport are solved to show the applicability and good performance of the procedures.

A highly accurate collocation method based on barycentric interpolation (BICM) is proposed for solving linear and nonlinear vibration problems for multi-degree-of-freedom systems in this article. The mathematical model of the linear and nonlinear vibrations of multi-degree-of freedom systems is the initial value problem of the linear and nonlinear differential equations. The numerical solution of the linear differential equations can be directly solved by BICM. The numerical solution of nonlinear differential equations can be solved as following: Firstly, the nonlinear governing equation is converted to linear differential equation by assuming the initial function. Secondly, the linear differential equations are discretized into algebraic equations by using barycentric interpolation differential matrices. Thirdly, the numerical solution can be calculated by iteration method under given control precision. Finally, the numerical solution of calculation examples by using barycentric Lagrange interpolation iteration collocation method (BLIICM) and barycentric rational interpolation iteration collocation method (BRIICM) are compared with the analytical solution. Numerical results illustrate the advantages of proposed methodology are efficient, fast, simple formulations, and high precision. Comparing with BLIICM, BRIICM can still maintain its computational stability when dealing with a large number of nodes, especially the equidistant nodes.

This paper studies the general decay synchronization (GDS) of a class of fuzzy cellular neural networks (FCNNs) with general activation functions and time-varying delays. By introducing suitable Lyapunov-Krasovskii functionals and employing useful inequality techniques, some novel criteria ensuring the GDS of considered FCNNs are established via a type of nonlinear control. In addition, two examples with numerical simulations are presented to illustrate the obtained theoretical results.

A new optimization technique is developed to generate a step-input acceleration function for an input shaping harmonic system. This approach is integrated into an overhead crane model for a rest-to-rest maneuver with standard and nonstandard maneuver settings. The proposed method guarantees the satisfaction of the system constraints and desired final conditions, while it minimizes the system sensitivity to crane cable-length variations. The minimal system sensitivity is achieved through an optimization algorithm that provides zero vibration and a minimum integral of system sensitivity over a continuous range of crane cable length. Numerical simulations are conducted to demonstrate the feasibility of the proposed shaper in eliminating the residual vibration at the end of a programmed maneuver. Sensitivity analyses are also performed to verify the robustness of the new shaper. In comparison to the previous shapers, the new methodology is significantly less sensitive and can effectively handle different arbitrary maneuver times.

The global (absolute) stability of nonlinear systems with negative feedbacks and positive not necessary asymptotically stable linear parts is addressed. The characteristics u=f(e)u = f(e) of the nonlinear parts satisfy the condition k1e≤f(e)≤k2e{k_1}e \le f(e) \le {k_2}e for some positive k1{k_1}, k2{k_2}. It is shown that the nonlinear systems are globally asymptotically stable if the Nyquist plots of the positive linear parts are located in the right-hand side of the circles −1k1,−1k2\left( { - {1 \over {{k_1}}}, - {1 \over {{k_2}}}} \right).

An interesting phenomenon is observed while conducting numerical simulation of non-linear dynamic response of FGM (functionally graded material) beam having large amplitude motion under harmonic excitation. Instead of providing a frequency sweep (forward or backward), if amplitude is incremented and response frequency is searched for a particular amplitude of vibration, solution domain can be enhanced and stable as well as unstable solution can be obtained. In the present work, first non-linear differential equations of motion for large amplitude vibration of a beam, which are obtained using Timoshenko beam theory, are converted into a set of non-linear algebraic equations using harmonic balance method. Subsequently an amplitude incremental iterative technique is imposed in order to obtain steady-state solution in frequency amplitude plane. It is observed that the method not only shows very good agreement with the available research but the domain of applicability of the method is enhanced up to a considerable extent as the stable and unstable solution can be captured. Subsequently forced vibration response of FGM beams are analysed.

In this paper, we are concerned with the periodic boundary value problem of fractional differential equations on ordered Banach spaces. By introducing a concept of upper and lower solutions, we construct a new monotone iterative technique for the periodic boundary value problems of fractional differential equation, and obtain the existence of solutions between lower and upper solutions.

The effects of surface energy on the non-linear frequency response and stability analysis of piezoelectric cylindrical nano-shell as piezoelectric nanoresonator are investigated in the current paper using Gurtin–Murdoch surface elasticity and von Karman–Donnell’s theory. The nanoresonator is embedded in visco-Pasternak medium and electrostatic excitation. The governing equations and boundary conditions are derived using Hamilton’s principle and also the assumed mode method is used for changing the partial differential equations into ordinary differential equations. Complex averaging method combined with arc-length continuation is used to achieve an approximate solution for the steady-state vibrations of the system. The validation of the mentioned system is achieved with excellent agreements by comparison with numerical results. The parametric studies such as the effects of geometrical and material properties, different boundary conditions, the ratio of length to radius L/R$L/R$ for different values of the voltages VDC${V_{{\rm{DC}}}}$ and VAC${V_{{\rm{AC}}}}$, the gap width of the nanoresonator b/L$b/L$, the effect of the voltages VDC${V_{{\rm{DC}}}}$ and VAC${V_{{\rm{AC}}}}$ and also the effect of piezoelectric voltage Vp${V_p}$ are conducted on the non-linear frequency response and stability analysis of the piezoelectric nanoresonator.