A Triangular Spectral Element Method for the 2-D Viscous Burgers Equation

A triangular spectral element method is established for the two-dimensional viscous Burgers equation. In the spatial direction, a new type of mapping is applied. We splice the local basis functions on each triangle into a global basis function. The second-order Crank-Nicolson/ leap-frog (CNLF) method is used for discretization in the time direction. Due to the use of a quasi-interpolation operator, the nonlinear term can be handled conveniently. We give the fully discrete scheme of the method and the implementation process of the algorithm. Numerical examples verify the effectiveness of this method.

Impact of Mode-Area Dispersion on Nonlinear Pulse Propagation in Gas-Filled Anti-Resonant Hollow-Core Fiber

We numerically investigate the effect of mode-area dispersion in a tubular-type anti-resonant hollow-core fiber by using a modified generalized nonlinear Schrödinger equation that takes into account the wavelength-dependent mode area in its nonlinear term. The pulse evolution dynamics with and without the effect of mode-area dispersion are compared and analyzed. We show that strong dispersion of the mode area in the proximity of the cladding wall thickness-induced resonances has a significant impact on the soliton pulse propagation, resulting in considerable changes in the conversion efficiencies in nonlinear frequency mixing processes. The differences become more prominent when the pump has higher energy and is nearer to a resonance. Hence, the mode-area dispersion must be accounted for when modeling such a case.

Analysis of the Fractional-Order Kaup–Kupershmidt Equation via Novel Transforms

In this article, we develop a technique to determine the analytical result of some Kaup–Kupershmidt equations with the aid of a modified technique called the new iteration transform method. This technique is a mixture of the novel integral transformation Elzaki transformation and the new iteration technique. The nonlinear term can be handled easily by a new iteration technique. The results show that the combination of the Elzaki transformation and the new iteration technique is quite capable and basically well suited for applying in such problems and that it can be implemented to other nonlinear models. This technique is viewed as an effective alternative approach to certain existing approaches for such accurate models.

A Simplified Lindstedt-Poincaré Method for Saving Computational Cost to Determine Higher Order Nonlinear Free Vibrations

In order to improve the Lindstedt-Poincaré method to raise the accuracy and the performance for the application to strongly nonlinear oscillators, a new analytic method by engaging in advance a linearization technique in the nonlinear differential equation is developed, which is realized in terms of a weight factor to decompose the nonlinear term into two sides. We expand the constant preceding the displacement in powers of the introduced parameter so that the coefficients can be determined to avoid the appearance of secular solutions. The present linearized Lindstedt-Poincaré method is easily implemented to provide accurate higher order analytic solutions of nonlinear oscillators, such as Duffing and van Der Pol nonlinear oscillators. The accuracy of analytic solutions is evaluated by comparing to the numerical results obtained from the fourth-order Runge-Kotta method. The major novelty is that we can simplify the Lindstedt-Poincaré method to solve strongly a nonlinear oscillator with a large vibration amplitude.

On Glassey’s conjecture for semilinear wave equations in Friedmann–Lemaître–Robertson–Walker spacetime

Consider nonlinear wave equations in the spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes. We show blow-up in finite time of solutions and upper bounds of the lifespan of blow-up solutions to give the FLRW spacetime version of Glassey’s conjecture for the time derivative nonlinearity. We also show blow-up results for the space derivative nonlinear term.

Existence and regularity of time-dependent pullback attractors for the non-autonomous nonclassical diffusion equations

In this paper, we prove the existence and regularity of pullback attractors for non-autonomous nonclassical diffusion equations with nonlocal diffusion when the nonlinear term satisfies critical exponential growth and the external force term $h \in L_{l o c}^{2}(\mathbb {R} ; H^{-1}(\Omega )).$ Under some appropriate assumptions, we establish the existence and uniqueness of the weak solution in the time-dependent space $\mathcal {H}_{t}(\Omega )$ and the existence and regularity of the pullback attractors.

Passive Fault-Tolerant Control of a 2-DOF Robotic Helicopter

The presence of faults in dynamic systems causes the potential loss of some of the control objectives. For that reason, a fault-tolerant controller is required to ensure a proper operation, as well as to reduce the risk of accidents. The present work proposes a passive fault-tolerant controller that is based on robust techniques, which are utilized to adjust a proportional-derivative scheme through a linear matrix inequality. In addition, a nonlinear term is included to improve the accuracy of the control task. The proposed methodology is implemented in the control of a two degrees of a freedom robotic helicopter in a simulation environment, where abrupt faults in the actuators are considered. Finally, the proposed scheme is also tested experimentally in the Quanser® 2-DOF Helicopter, highlighting the effectiveness of the proposed controller.

The well-posedness and long-time behavior of the nonlocal diffusion porous medium equations with nonlinear term

In this paper, we mainly consider the well-posedness and long-time behavior of solutions for the nonlocal diffusion porous medium equations with nonlinear term. Firstly, we obtain the well-posedness of the solutions in L1(Ω) for the equations. Secondly, we prove the existence of a global attractor by proving there exists a compact absorbing set. Finally, we apply index theory to consider the dimension of the attractor and prove that there exists an infinite dimensional attractor of the equations under proper conditions.

Strongly nonlinear elliptic unilateral problems without sign condition and with free obstacle in Musielak-Orlicz spaces

In this paper, we prove the existence of solutions to an elliptic problem containing two lower order terms, the first nonlinear term satisfying the growth conditions and without sign conditions and the second is a continuous function on R.