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The corner of stone of this paper is the numerical treatment of the Robot arm control problem and its integral representation. The semi-analytic and numerical solutions are introduced using two impressive techniques. The first is the Chebyshev collocation method and the second is the differential transform method. The comparison from the error point of view between the exact solution and the numerical solution obtained by two techniques used are considered. Also, the integral representation of the Robot arm control system of equations are constructed and gives the same result obtained for the differential form. The advantage of the integral form is the non-locality and globality. The results and comparison between our techniques with other classic techniques which solve the same problem are introduced in the form of tables and figures with the help of Mathematica program to investigate and explain the efficiency and applicability of differential transform and Chebyshev collocation methods.

This study presents analytical and numerical‐analytical decomposition methods for determining complex one‐parameter generalized inverse Moore–Penrose matrices. The analytical approach is based on the third Moore–Penrose condition, offering three solution options. The first option employs complex decompositions of the given matrix and its Moore–Penrose inverse. The second option combines the first and third Moore–Penrose conditions, while the third option integrates the second and third conditions. For the first and third options, if any derived iterative procedure converges, the Moore–Penrose inverse matrix can be constructed using the corresponding matrix blocks. In contrast, the second option provides simplified relations, enabling the direct computation of the Moore–Penrose inverse matrix. Numerical‐analytical methods build on the second analytical solution, utilizing differential Pukhov transformations as the primary mathematical tool. A model example featuring a rectangular complex matrix is analyzed. A numerical‐analytical solution is derived using three matrix discretes, from which corresponding matrix blocks are reconstructed. The Moore–Penrose inverse matrix is then obtained through its complex decomposition.

A local thermal non-equilibrium analysis of heat and fluid flow in a channel fully filled with aluminum foam is performed for three cases: (a) pore density of 5 PPI (pore per inch), (b) pore density of 40 PPI, and (c) two different layers of 5 and 40 PPI. The dimensionless forms of fully developed heat and fluid flow equations for the fluid phase and heat conduction equation for the solid phase are solved analytically. The effects of interfacial heat transfer coefficient and thermal dispersion conductivity are considered. Analytical expressions for temperature profile of solid and fluid phases, and also the channel Nusselt number (NuH) are obtained. The obtained results are discussed in terms of the channel-based Reynolds number (ReH) changing from 10 to 2000, and thickness ratio between the channel height and sublayers. The Nusselt number of the channel with 40 PPI is always greater than that of the 5 PPI channel. It is also greater than the channel with two-layer aluminum foams until a specific Reynolds number then the Nusselt number of the channel with two-layer aluminum foams becomes greater than the uniform channels due to the higher velocity in the outer region and considerable increase in thermal dispersion.

The wetting pattern which occurs under surface drip irrigation is an important component for the optimum design of the system and for irrigation programming. The aim of this investigation was to devise a model which enables estimation of the 2D cross-sectional area of the wetting pattern which occurs under a surface dripper by an analytical method. The main parameters of the wetting pattern are the wetting diameter on the soil surface, the maximum wetted depth and maximum wetted width in the soil profile, and the depth of this maximum wetted width from the soil surface. In the laboratory experiments, water applications were carried out on two soil textures (clay loam and clay) with homogeneous soil profiles at two discharge rates over 125 min and 170 min periods. The sizes of the parameters of the wetting pattern were measured as consecutive series over 5-min intervals during the water application period. The general/main shape which represents the wetting patterns which occur under diferent irrigation conditions was defined and mathematically modelled. When the results were evaluated statistically, a correlation of 0.9128 was found between the momentary rates of change of the maximum depth of the wetting pattern predicted by the model and those measured in the experiment. The correlation between the momentary variations of accelerations of the same parameter was 0.9205. In addition, the size of the wetting pattern showed an increment in reducing velocities during the water application period. The results indicate that the model devised in this investigation can be used in the prediction of the cross-sectional area of the wetting pattern which occurs under a surface dripper.

This research paper targets the fractional Hirota’s analytical solutions–Satsuma (HS) equations. The conformable fractional derivative is employed to convert the fractional system into a system with an integer–order. The extended simplest equation (ESE) and modified Kudryashov (MKud) methods are used to construct novel solutions of the considered model. The solutions’ accuracy is investigated by handling the computational solutions with the Adomian decomposition method. The solutions are explained in some different sketches to demonstrate more novel properties of the considered model.

We study the impact of a spatially homogeneous yet non-stationary dielectric permittivity on the dynamical and spectral properties of light. Our choice of potential is motivated by the interest in   -symmetric systems as an extension of quantum mechanics. Because we consider a homogeneous and non-stationary medium,   symmetry reduces to time-reversal symmetry in the presence of balanced gain and loss. We construct the instantaneous amplitude and angular frequency of waves within the framework of Maxwell's equations and demonstrate the modulation of light amplification and attenuation associated with the well-defined temporal domains of gain and loss, respectively. Moreover, we predict the splitting of extrema of the angular frequency modulation and demonstrate the associated shrinkage of the modulation period. Our theory can be extended for investigating similar time-dependent effects with matter and acoustic waves in   -symmetric structures.

Darboux transformation is an efficient method for solving different nonlinear partial differential equations. In this paper, on the basis of a Lie super-algebras, a generalized super-NLS-mKdV equation is solved by the Darboux transformation. The analytic solutions are presented with the help of symbolic computation. Besides, two special cases are given to make the solution intuitive. Dynamic properties of solitons are also discussed.

In this article, the iteration transform method is used to evaluate the solution of a fractional-order dark optical soliton, bright optical soliton, and periodic solution of the nonlinear Schrödinger equations. The Caputo operator describes the fractional-order derivatives. The solutions of some illustrative examples are presented to show the validity of the proposed technique without using any polynomials. The proposed method provides the series form solutions, which converge to the exact results. Using the present methodology, the solutions of fractional-order problems as well as integral-order problems are calculated. The present method has less computational costs and a higher rate of convergence. Therefore, the suggested algorithm is constructive to solve other fractional-order linear and nonlinear partial differential equations.

A polynomial trajectory is a time-traveled distance function used to describe trajectory of the robot. Optimal high-degree polynomial trajectories considering initial and the final velocity conditions besides the acceleration constraints are desired. In this paper, a trajectory optimization problem aiming travel maximum distance for a robot that follows an arc based path is formulated. Along the path, the robot requires observing initial and final zero velocity conditions as well as certain acceleration limits. A high-degree polynomial equation along the trajectory is proposed inside of the optimization problem. The closed-form solution of the problem had been obtained analytically. The solution includes the coefficients of the any high-degree trajectory polynomial equation where the coefficients are obtained in closed-form. Simulations several experiments show that the resulting high-degree trajectories satisfy the initial and final zero velocity conditions as well as acceleration constraint.

The objective of this study was to evaluate the applicability of a flow model with different numbers of spatial dimensions in a hydraulic features solution, with parameters such a free surface profile, water depth variations, and averaged velocity evolution in a dam-break under dry and wet bed conditions with different tailwater depths. Two similar three-dimensional (3D) hydrodynamic models (Flow-3D and MIKE 3 FM) were studied in a dam-break simulation by performing a comparison with published experimental data and the one-dimensional (1D) analytical solution. The results indicate that the Flow-3D model better captures the free surface profile of wavefronts for dry and wet beds than other methods. The MIKE 3 FM model also replicated the free surface profiles well, but it underestimated them during the initial stage under wet-bed conditions. However, it provided a better approach to the measurements over time. Measured and simulated water depth variations and velocity variations demonstrate that both of the 3D models predict the dam-break flow with a reasonable estimation and a root mean square error (RMSE) lower than 0.04, while the MIKE 3 FM had a small memory footprint and the computational time of this model was 24 times faster than that of the Flow-3D. Therefore, the MIKE 3 FM model is recommended for computations involving real-life dam-break problems in large domains, leaving the Flow-3D model for fine calculations in which knowledge of the 3D flow structure is required. The 1D analytical solution was only effective for the dam-break wave propagations along the initially dry bed, and its applicability was fairly limited.