In this paper, the problem of optimal path following for a high speed planing boat is addressed. First, a nonlinear mathematical model of the boat’s dynamics is derived and then the Serret-Frenet frame is presented to facilitate the path following control design. To satisfy the constraints on the states and the input controls of the boat's nonlinear dynamics and minimize both the cross tracking and heading error, a nonlinear optimal controller is formed. To solve the resulted nonlinear constrained optimal control problem, the Gauss pseudospectral METHOD (GPM) is used to transcribe the optimal control problem into a nonlinear programming problem (NLP) by discretization of states and controls. The resulted NLP is then solved by a well-developed algorithm known as SNOPT. The results illustrate the effectiveness of the proposed approach to tackle the boat path following problem.

Introduction: Convective air drying is one of the oldest and most popular drying METHODs. Designing and controlling the convective air drying needs the mathematical description of the moisture transfer during the drying process, known s d ying kinetics. Fick’ s second l w of diffusion c n be used fo modelling the moistu e dist ibution inside the moist object during drying process. Mathematical modeling of drying process is a very important tool, as it contributes to understand better moisture distributions inside the product which helps designing, improving and controlling drying operation in the food industry. Implementation of the partial differential equations subject to the correspondent initial and boundary conditions is one of the main METHODs of mathematical modeling to describe the physical phenomena such as moisture transfer during drying. In the recent decades, considerable number of research works have been devoted to numerical solution of mass transfer phenomena during convective drying of food products by using the common numerical solution such as FDMs, FEMs and FVMs. The spectral collocation (pseudospectral) METHODs is a powerful tool for the numerical solutions of smooth PDEs like mass transfer equations. Pseudospectral METHODs are able to achieve the high precision with using a small number of discretization points compared to FDMs and FEMs and with low computational time and computer memory. The objective of present research is to simulate the mass transfer phenomena in one dimension during convective drying of apple slices. The validation of the presented numerical model was done by comparing experimental drying data taken from Kaya et al. (2007) and Zarein et al. (2013). For more confirming the numerical approach, a numerical example with the exact solution is provided and the related errors were evaluated...

We propose a new approach for solving nonlinear Klein–Gordon and sine-Gordon equations based on radial basis function-pseudospectralMETHOD (RBF-PS). The proposed numerical METHOD is based on quasiinterpolation of radial basis function differentiation matrices for thediscretization of spatial derivatives combined with Runge–Kutta time stepping METHOD in order to deal with the temporal part of the problem.The METHOD does not require any linearization technique; in addition, a new technique is introduced to force approximations to satisfy exactlythe boundary conditions. The introduced scheme is tested for a number of one- and two-dimensional nonlinear problems. Numerical results andcomparisons with reported results in the literature are given to validate the presented METHOD, and the reported results show the applicabilityand versatility of the proposed METHOD.

In this paper, a pseudospectral METHOD is proposed for solving the nonlinear time-fractional Klein-Gordon and sine-Gordon equations. The METHOD is based on the Sinc operational matrices. A finite difference scheme is used to discretize the Caputo time-fractional derivative, while the spatial derivatives are approximated by the Sinc METHOD. The convergence of the full discretization of the problem is studied. Some numerical examples are presented to confirm the accuracy and efficiency of the proposed METHOD. The numerical results are compared with the analytical solution and the reported results in the literature.

In bang-bang optimal control problems, the control function is inherently piecewise constant. This feature creates substantial difficulties for the standard Legendre-Gauss-Radau pseudospectral METHOD, which relies on polynomial approximation for the control function. This study introduces a simplified approach that seamlessly integrates sigmoid-based control parameterization with the traditional Legendre-Gauss-Radau pseudospectral METHOD. This integration enables precise approximation of discontinuous control profiles while maintaining the polynomial approximation for state variables. The proposed METHOD significantly minimizes the number of decision variables in the optimization problem while precisely determining both the number and locations of switching points. This leads to notable enhancements in computational efficiency and solution accuracy. Numerical experiments conducted on two benchmark problems, a bridge crane system and a robotic arm control problem, demonstrate the exceptional precision and efficiency of the proposed METHOD. Despite its simplicity, the METHOD delivers results that are on par with those produced by more advanced and intricate techniques.