Pseudospectral Methods Research Papers - Academia.edu (original) (raw)

Calculation of eigenvalue problems for the second-order ordinary differential equations is relevant for the physics problems. The second-order ordinary differential equation with homogeneous Dirichlet boundary condition was considered. The Chebyshev pseudospectral method (CPM) was used for the problem of eigenvalues basing on the Chebyshev-Gauss-Lobatto points to create the differential matrices. The Mathematica version 10.4 to write computing programs was used. In the applications, the Chebyshev pseudospectral method was used to find eigenvalues that were approximated gradually to the exact eigenvalues of the problem.

Autonomous nonlinear differential equations constituted a system of ordinary differential equations, which often applied in different areas of mechanics, quantum physics, chemical engineering science, physical science, and applied mathematics. It is assumed that the second-order autonomous nonlinear differential equations have the types u ′′ (x) − u ′ (x) = f [u(x)] and u ′′ (x) + f [u(x)]u ′ (x) + u(x) = 0 on the range [−1, 1] with the boundary values u[−1] and u[1] provided. We use the pseudospectral method based on the Chebyshev differentiation matrix with Chebyshev-Gauss-Lobatto points to solve these problems. Moreover, we build two new iterative procedures to find the approximate solutions. In this paper, we use the programming language Mathematica version 10.4 to represent the algorithms, numerical results and figures. In the numerical results, we apply the well-known Van der Pol oscillator equation and gave good results. Therefore, they will be able to be applied to other nonlinear systems such as the Rayleigh equations, the Lienard equations, and the Emden-Fowler equations.

— The k-wave toolbox enables simple and efficient simulation of time domain wave propagation. It is widely used for simulating the behaviour of ultrasonic waves in biological media. This paper proposes its use as teaching aid in subjects related to physical acoustics. By adding a small number of instructions it is possible to produce a MATLAB/Octave function that simulates acoustic field propagation for complex two-dimensional conditions, based on a reference image (BMP file) in which points of different colours represent specific elements within the simulation. This enables experimentation of different situation using the k-wave toolbox with minimal MATLAB know-how. Examples are shown in which red dots are used for sound sources, green dots for microphones, and black dots for perfectly reflecting materials. Any alteration on the images by simple cut, paste, move or rotate operation results in new simulation conditions. Examples of wave behaviour in tubes are showcased for studying a variety of situations. In addition to that, simulations are carried out in COMSOL Multiphysics with the purpose of comparing results. Resumen— El toolbox k-wave permite realizar simulaciones en el dominio del tiempo de la propagación de ondas de un modo eficiente y sencillo. Se utiliza principalmente para simular el comportamiento de ondas ultrasónicas en medios biológicos. Este trabajo propone utilizarlo como apoyo a la enseñanza de temas de Acústica Física. Incorporando un pequeño número de instrucciones es posible generar una función en MATLAB/Octave para poder simular situaciones complejas de propagación de campos acústicos en dos dimensiones tomando como referencia una imagen (archivo BMP) en el cual los puntos de diferentes colores representan elementos específicos de la simulación. Esto permite experimentar distintas situaciones utilizando el toolbox k-wave con mínimos conocimientos de uso del MATLAB. Se muestran ejemplos que utilizan puntos rojos para fuentes, verdes para micrófonos, y negros para elementos perfectamente reflectantes. Cualquier alteración en las imágenes mediante operaciones simples de cortar, pegar, trasladar y rotar genera nuevas condiciones de simulación. Se presentan ejemplos de uso para estudiar el comportamiento de ondas en tubos en variedad de situaciones. Se presentan además simulaciones realizadas en COMSOL Multiphysics con el fin de poder comparar resultados. Palabras clave: enseñanza de acústica; método pseudoespectral del espacio k; simulación numérica.

- by Jorge Petrosino
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- Acoustics, Pseudospectral Methods

This research focuses on proposing an optimal trajectory planning and control method of two link rigid-flexible manipulators (TLRFM) for Dynamic Object Manipulation (DOM) missions. For the first time, achievement of DOM task using a rotating one flexible link robot was taken into account in [20]. The authors do not aim to contribute on either trajectory tracking or vibration control of the End-Effector (EE) of the manipulator; On the contrary, utilizing the powerful tool optimal control accomplishing a point-to-point task for TLRFM is the purpose of the current research. Towards this goal, the pseudospectral method will be developed to meet the optimality conditions subject to system dynamics and boundary conditions. The complicated optimal trajectory planning is formulated as a nonlinear programming problem and solved by SNOPT nonlinear solver. To make robust the response of optimal control against external disturbances as well as model parameter uncertainties, the control partitioning concept is employed. The controlled input is composed of an optimal control-based feedforward part and a PID-based feedback component. The obtained simulation results reveal the usefulness and robustness of the developed composite scheme, in DOM missions.

This paper presents a different numerical solution to compute eigenvalues of the Schrödinger equation with the potentials in graphene structures [1]. The research subjects include the Schrödinger equation and the exchange-correlation energy of the graphene structures in Grachev's article. Specifically, we used the pseudospectral method basing on the Chebyshev-Gauss-Lobatto grid to determine the approximate numerical results of the problem. The results are the discrete energy spectra and the corresponding eigenfunctions of the nonlinear spin waves in the graphene structure. Additionally, these results can be applied to create the nonlinear spin waves in the graphene structures.

- by PhD. Anh Nhat Le
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- Spine, Graphene, Pseudospectral Methods, Nonlinear Models

— Auralization is a term introduced to describe the recreation of the experience of acoustic phenomena a listener would perceive in a specific soundfield. Sound propagation in a soundfield can be simulated with geometric based models or wave based models. Each one offers particular advantages and disadvantages. For wave based models, the finite element method, the boundary element method or the finite difference method are widely mentioned. They are characterized for achieving very precise results for individual frequencies applied to small and moderately sized rooms. Geometric methods lead to the ray tracing method or the image source method. These methods achieve good results for high frequencies and are efficient in large rooms and complex structures, but are not able to represent in a simple manner specific wave phenomena such as diffraction. Commercial software used to produce auralizations is usually based on a hybrid model combining ray tracing and image sources. This paper proposes an exploration on possible advantages and difficulties on the use of the k-space pseudospectral method for wave based auralizations. Resumen— El término auralización se refiere a la recreación en los oídos de un oyente de la sensación que percibiría en un espacio acústico determinado. La propagación del sonido en un espacio acústico puede simularse mediante procesos basados en el modelo geométrico o en el modelo ondulatorio. Cada uno ofrece ventajas y dificultades particulares. Entre los modelos basados en ondas pueden mencionarse principalmente el método de elementos finitos, el de contornos finitos o el de diferencias finitas. Se caracterizan por lograr resultados muy precisos para frecuencias únicas aplicadas a recintos de tamaño pequeño o medio. Por otro lado, los modelos geométricos dan lugar al método del trazado de rayos o al método de las fuentes imagen. Estos métodos logran buenos resultados en frecuencias altas y resultan eficientes en salas de gran tamaño con estructuras complejas, pero no pueden dar cuenta en forma sencilla de fenómenos específicamente ondulatorios como la difracción. Los programas comerciales utilizados para obtener auralizaciones suelen utilizar un modelo híbrido combinando el trazado de rayos y las fuentes imagen. El presente trabajo tiene la intención de iniciar una exploración sobre las posibles ventajas y dificultades del uso del método pseudoespectral del espacio k para obtener auralizaciones basadas en el modelo ondulatorio. Palabras clave: auralización; método pseudoespectral del espacio k; métodos numéricos.

- by Jorge Petrosino and +1
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- Pseudospectral Methods, Auralization

A new algorithm used the Chebyshev pseudospectral method to solve the nonlinear second-order Lienard differential equations Abstract. This article presents a numerical method to determine the approximate solutions of the Lienard equations. It is assumed that the second-order nonlinear Linard differential equations on the range [-1, 1] with the given boundary values. We have to build a new algorithm to find approximate solutions to this problem. This algorithm based on the pseudospectral method using the Chebyshev differentiation matrix (CPM). In this paper, we used the Mathematica version 10.4 to represent the algorithm, numerical results and graphics. In the numerical results, we made a comparison between the CPMs numerical results and the Mathematica's numerical results. The biggest odds were very small. Therefore, they will be able to be applied to other nonlinear systems such as the Rayleigh equations and Emden-fowler equations.

Internal solitary waves have been documented in several parts of the world. This paper intends to look at the effects of the variable topography and rotation on the evolution of the internal waves of depression. Here, the wave is considered to be propagating in a two-layer fluid system with the background topography is assumed to be rapidly and slowly varying. Therefore, the appropriate mathematical model to describe this situation is the variable-coefficient Ostrovsky equation. In particular, the study is interested in the transition of the internal solitary wave of depression when there is a polarity change under the influence of background rotation. The numerical results using the Pseudospectral method show that, over time, the internal solitary wave of elevation transforms into the internal solitary wave of depression as it propagates down a decreasing slope and changes its polarity. However, if the background rotation is considered, the internal solitary waves decompose and form a wave packet and its envelope amplitude decreases slowly due to the decreasing bottom surface. The numerical solutions show that the combination effect of variable topography and rotation when passing through the critical point affected the features and speed of the travelling solitary waves.

- by sudheer gopinathan
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- Pseudospectral Methods

- by PhD. Anh Nhat Le
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- Pseudospectral Methods, Odes, Chebyshev, Mathieu

The article searched on mathematics and numerical solutions for the nonlinear pendulum (Chaotic pendulum). The numerical solution that was used for our research suitably the pseudospectral methods. With these equations, we studied and calculated on the interval [−1; 1], with boundary conditions already known. We used the software Mathematica 10.4 to calculate the results of the problems.

This article presents an approximate numerical solution for nonlinear Duffing Oscillators by pseudospectral (PS) method to compare boundary conditions on the interval [-1, 1]. In the PS method, we have been used differentiation matrix for Chebyshev points to calculate numerical results for nonlinear Duffing Oscillators. The results of the comparison show that this solution had the high degree of accuracy and very small errors. The software used for the calculations in this study was Mathematica V.10.4.