# Forthcoming articles

International Journal of Applied Nonlinear Science

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 International Journal of Applied Nonlinear Science (8 papers in press)  Regular Issues  Influence of Mass and Length of Biped Robot on its Passive Walking   by Nita Shah, Mahesh Yeolekar Abstract: Abstract: The motion of a biped robot can be explained by a set of nonlinear ordinary differential equations. In this paper, we investigate the linearized form of a system of nonlinear ordinary differential equations with impulse effect which model a simple planer biped robot without knee. It demonstrated the periodic motion in a sagittal plane in absence of external forces except gravity. This paper explains the bifurcation study of the system of biped robot with respect to the bifurcation parameters, mass and length. The results exhibit that the continuous change in the value of any one of its parameters the stable symmetric gait leads to chaotic gait. We observed that the symmetric gaits of robot are more sensitive for the values of length than the values of mass. Keywords: KeywordsrnBiped robot; limit cycle walking; passive dynamic walking; Poincare map; orbital stability; bifurcation diagramrn. Design of a non-linear controller using Feedback Linearization based on back-stepping technique for Magnetic Levitation system   by Shiv Narayan, Bhawna Tandon, Jagdish Kumar Abstract: This study proposes a feedback linearization based on the back-stepping method to design a nonlinear controller with a goal of improving both steady state and transient stability of a Magnetic Levitation System. The feedback linearization based on back-stepping technique is the combination of the two techniques mentioned i.e. Feedback Linearization and backstepping. It uses backstepping design process, designs a sequence of virtual systems of relative degree one, reduces the relative degree by one by choosing a virtual input, achieves passivity with respect to a virtual output, and the last virtual output is used to close feedback loop.Unlike the Direct Feedback Linearization method, the proposed method does not require a linear controller. Moreover, there is no need to know the exact nonlinear model of the system. Back stepping process of the controller guarantees its robustness against disturbances and uncertainties. Keywords: Backstepping technique; Feedback Linearization; nonlinear control; Magnetic Levitation system. A fourth order regula-falsi Newton-like method for enclosing simple zeros of nonlinear equations   by Shwet Nisha, P.K. Parida Abstract: In this paper, we have developed an improved regula falsi method of order four for finding simple roots of nonlinear equations $f(x)=0$, where $f: [a, b]\subset \mathbb{R}\rightarrow \mathbb{R}$ is a given continuously differentiable function. This is done by combining a Newton-like method of order four to solve $f(x)=0$ and the usual regula-falsi method. Convergence analysis for the method has been given in this paper. Finally some numerical examples are presented and comparison has been made with existing results. Keywords: Nonlinear equations; Order of convergence; Regula-falsi method; Newton-like methods. Neural Network Augmented Backstepping Control For Uncertain Nonlinear Systems - Application To Laboratory Antilock Braking System   by Abdelhamid Rabhi Abstract: A new control approach is proposed to address the tracking problem of a class of uncertain nonlinear systems. In this approach, one relies first on a partially known model of the system to be controlled using a Backstepping control strategy. The obtained controller is then augmented by an online Artificial Neural Network (ANN) that serves as an approximator for the neglected dynamics and modelling errors. Thus, the developed method combines Bakstepping approach and ANN to address the tracking problem for uncertain systems. The proposed approach is systematic, and exploits the known nonlinear dynamics to derive the stepwise virtual stabilizing control laws. At the final step, an augmented Lyapunov function is introduced to derive the adaptation laws of the network weights. The suggested control algorithm is tested experimentally on a Laboratory ABS system showing satisfactory results although the system is highly nonlinear and with unknown physical parameters. Keywords: Neural Network; ABS; backstepping control; Lyapunov; uncertain nonlinear systems. An optimally scaled polynomial-Fourier-series method for the numerical solution of the Duffing oscillator   by Najeeb Khan, Chein-Shan Liu, Fatima Riaz Abstract: In this paper, a novel multiple-scale polynomial-Fourier-series method (PFSM) is developed to be used in the data interpolation, in which the multiple-scale R_k can be determined exactly and optimally in terms of the data nodes. For solving the nonlinear duffing equation, an optimally-scaled harmonic balance method (OSHB)is derived, which is better than the classic harmonic balance method (HB). In terms of the OSHB, the periodic solutions of the Duffing oscillator, and reconstruction of the frequency response curves, which exhibit a hysteresis within which the multiple solutions can happen in an interval of frequency near to the resonant frequency can be precisely solved. The PFSM is further adapted to solve the initial value problem of the Duffing equation, and the periodic solution can be obtained more accurately than the HB. The proposed method has also been tested to solve the boundary value problem and the initial value problem of some nonlinear ordinary differential equations (ODEs). The conclusion can be drawn that the present OSHB and PFSM are effective to solve nonlinear ODEs, including the nonlinear Duffing equation as a demonstrative example. Keywords: Duffing equation; Multiple-scale polynomial Fourier-series interpolation; Harmonic balance method (HB); Optimally scaled harmonic balance method (OSHB); Optimally scaled polynomial-Fourier-series method (PFSM). A companion of the generalized trapezoid inequality for functions of two variables with bounded variation and applications   by Hüseyin BUDAK, Mehmet Zeki SARIKAYA Abstract: In this paper, a companion of trapezoid inequality for functions of two independent variables with bounded variation is established and some applications for general quadrature formula are given. Keywords: Function of bounded variation; Ostrowski type inequalities; Riemann-Stieltjes integrals; Quadrature formula. A hybrid numerical treatment of nonlinear reaction-diffusion equations with memory: A prototypical Fisher-Kolmogorov-Petrovskii-Piskunov equation   by Okey Oseloka Onyejekwe Abstract: In reaction-diffusion systems with non-standard diffusion, the memory of the transport process causes a coupling of reaction and diffusion. A generalization of the Ficks law has been suggested to account for this coupling. Furthermore, the resultant effects of the interplay of transport , memory and reaction lend themselves to some interesting physics which is still not well understood because the governing equation as well as the accompanying memory integral and nonlinear reaction terms are not always amenable to tidy analytic or numerical expressions. Hence the derivation of a suitable governing integro-differential equation as well as the approximate solution demand a non-standard numerical procedure. The main focus of this work can be seen as a contribution towards this objective. In this report, we develop and apply an hybrid boundary integral-finite element finite difference numerical procedure to investigate an integro-differential-FKPP(Fisher-Kolmogorov-Petrovskii-Piskunov) type kinetics. We also focus on scalar evolution for cases where the reaction coefficient takes on relatively large values. Although we are still far from a rigorous mathematical analysis, it has been found that the numerical results obtained compared favorably with existing benchmark solutions. This not only validates the current numerical formulation but also justifies the physics of the resulting front propagation. Keywords: Hybrid numerical formulation; boundary integral; finite element; finite differerence; integro-differential partial differential equation; Fisher-Klomogorov-Petrovskii- Piskunov equation; memory term; nonlinear; reaction-diffusion. Pressure curves for compressible flows with slip through asymmetric local constrictions   by Salahaldeen Rabba, Katrin Rohlf Abstract: A second-order non-linear differential equation is derived for the pressure of a compressible flow with slip at the wall through a constricted cylinder. The ideal gas equation of state is used, and the Karman-Pohlhausen method is utilized to derive the pressure differential equation from the Navier-Stokes equations of motion for a Newtonian viscous fluid. The solution for pressure is determined numerically and assessed in various flow geometries. This work is an extension of existing assessments in that non-linear terms are kept in the differential equation for pressure, as well as second-order derivative terms. Additionally, wall slip and compressibility are incorporated in the equations, as well as geometries that are asymmetric with respect to the location of maximum constriction. Keywords: Pressure; Gradient; Compressible; Stenosis; Navier-Stokes; Slip; Karman-Pohlhausen; Asymmetric.