This paper focuses on the fourth-order nonlinear potential Ito equation, which describes wave patterns in shallow waters. To reveal the integrable characteristics of the considered equation, such as ...bilinear BT and Lax pair, the Bell polynomials method is used. By employing this technique, the bilinear representation in the form of classical Hirota operators is derived. Moreover, with the help of bilinear form, the bilinear Bäcklund transformation and the Lax pair of the considered equation are obtained successfully. The three wave method in the form of test function is adopted to generate the analytical solutions of the considered equation. By applying this approach, ten analytical solutions are obtained successfully. For each of the obtained solutions, a 3D graph has been plotted by varying free parametric values. These graphs show the different kinds of wave behaviour, including kink-soliton, anti-kink soliton, periodic wave, dark-soliton, bright-soliton, and some complex kink and periodic-type wave solutions.
High-harmonic generation is a cornerstone of nonlinear optics. It has been demonstrated in dielectrics, semiconductors, semi-metals, plasmas, and gases, but, until now, not in metals. Here we report ...high harmonics of 800-nm-wavelength light irradiating metallic titanium nitride film. Titanium nitride is a refractory metal known for its high melting temperature and large laser damage threshold. We show that it can withstand few-cycle light pulses with peak intensities as high as 13 TW/cm
, enabling high-harmonics generation up to photon energies of 11 eV. We measure the emitted vacuum ultraviolet radiation as a function of the crystal orientation with respect to the laser polarization and show that it is consistent with the anisotropic conduction band structure of titanium nitride. The generation of high harmonics from metals opens a link between solid and plasma harmonics. In addition, titanium nitride is a promising material for refractory plasmonic devices and could enable compact vacuum ultraviolet frequency combs.
•Painlevé analysis and auto-Bäcklund transformation are used.•(2 + 1)-dimensional extended Sakovich equation considered.•(3 + 1)-dimensional extended Sakovich equation also considered.•New solitary ...wave soliton solutions are explored.•The soliton solutions are involving the exponential and rational functions.
This article considers time-dependent variable coefficients (2+1) and (3+1)-dimensional extended Sakovich equation. Painlevé analysis and auto-Bäcklund transformation methods are used to examine both the considered equations. Painlevé analysis is appeared to test the integrability while an auto-Bäcklund transformation method is being presented to derive new analytic soliton solution families for both the considered equations. Two new family of exact analytical solutions are being obtained successfully for each of the considered equations. The soliton solutions in the form of rational and exponential functions are being depicted. The results are also expressed graphically to illustrate the potential and physical behaviour of both equations. Both the considered equations have applications in ocean wave theory as they depict new solitary wave soliton solutions by 3D and 2D graphs.
Grid integration of solar photovoltaic (PV) systems has been escalating in recent years, with two main motivations: reducing greenhouse gas (GHG) emission and minimizing energy cost. However, the ...intermittent nature of solar PV generated power can significantly affect the grid voltage stability. Therefore, intermittent solar PV power generation and uncertainties associated with load demand are required to be accounted for to gain a holistic understanding on power grid voltage stability with the high penetration of PV energy sources. This article presents a framework for power grid voltage stability analysis considering uncertainties associated with PV power generation and load demand using Monte Carlo simulation. Commonly used voltage stability indicators such as critical eigenvalue, line loss, reactive power margin have been considered in the proposed framework. The proposed methodology has been verified by analyzing the voltage stability of the IEEE 14 bus test system, with the high penetration of PV energy sources and considering uncertainties associated with load demand. The results provide a clear insight into the voltage stability of power grid with different penetration levels of PV energy sources into the power grid.
In this paper, a new method based on the Chebyshev wavelet expansion is proposed for solving a coupled system of nonlinear ordinary differential equations to model the unsteady flow of a nanofluid ...squeezing between two parallel plates. Chebyshev wavelet method is applied to compute the numerical solution of coupled system of nonlinear ordinary differential equations in order to model squeezing unsteady nanofluid flow. The approximate solutions of nonlinear ordinary differential equations thus obtained by Chebyshev wavelet method are compared with those of obtained by Adomian decomposition method (ADM), fourth order Runge–Kutta method and homotopy analysis method (HAM). The results obtained by the above methods are illustrated graphically and are discussed in details. The present scheme is very simple, effective and appropriate for obtaining numerical solution of squeezing unsteady nanofluid flow between parallel plates.
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•Chebyshev wavelet method is applied.•Compared the solutions with ADM, fourth order RK method and homotopy analysis method (HAM)•To the best information of the authors, this model has not been solved ever before.
The main aim of this article is to demonstrate the collocation method based on the barycentric rational interpolation function to solve nonlinear stochastic differential equations driven by ...fractional Brownian motion. First of all, the corresponding integral form of the nonlinear stochastic differential equations driven by fractional Brownian motion is introduced. Then, collocation points followed by the Gauss-quadrature formula and Simpson’s quadrature method are used to reduce them into a system of algebraic equations. Finally, the approximate solution is obtained using Newton’s method. The rigorous convergence and error analysis of the presented method has been discussed in detail. The proposed method has been applied to some well-known stochastic models, such as the stock model and a few other examples, to demonstrate the applicability and plausibility of the discussed method. Also, the numerical results of the collocation method based on the barycentric rational interpolation function and barycentric Lagrange interpolation function get compared with an exact solution.
•Optimal homotopy asymptotic method has been proposed for Boussinesq–Burger equation.•In this paper soliton solutions of Boussinesq–Burger equation have been analyzed.•The obtained results have been ...compared with HPM method solutions.•The results reveal that the OHAM is very effective and simple.•The numerical solutions are presented graphically in 3D surfaces and 2D plots.
In this article, a comparative study between homotopy perturbation method (HPM) and optimal homotopy asymptotic method (OHAM) is presented. Homotopy perturbation method is applied to compute the numerical solutions of non-linear partial differential equations like Boussinesq–Burger equations. The approximate solutions of the Boussinesq–Burger equation are compared with the optimal homotopy asymptotic method as well as with the exact solutions. Comparison between our solutions and the exact solution shows that both the methods are effective and accurate in solving nonlinear problems whereas OHAM is accurate with less number of iterations in compared to HPM. In OHAM the convergence region can be easily adjusted and controlled. OHAM provides a simple and easy way to control and adjust the convergence region for strong nonlinearity and is applicable to highly nonlinear fluid problem like Boussinesq–Burger equations.
This paper presents efficient numerical techniques for solving fractional optimal control problems (FOCP) based on orthonormal wavelets. These wavelets are like Legendre wavelets, Chebyshev wavelets, ...Laguerre wavelets and Cosine And Sine (CAS) wavelets. The formulation of FOCP and properties of these wavelets are presented. The fractional derivative considered in this problem is in the Caputo sense. The performance index of FOCP has been considered as function of both state and control variables and the dynamic constraints are expressed by fractional differential equation. These wavelet methods are applied to reduce the FOCP as system of algebraic equations by applying the method of constrained extremum which consists of adjoining the constraint equations to the performance index by a set of undetermined Lagrange multipliers. These algebraic systems are solved numerically by Newton's method. Illustrative examples are discussed to demonstrate the applicability and validity of the wavelet methods.
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