Pseudo-spectral Galerkin Research Articles

Pseudo-Spectral Galerkin Method Using Shifted Vieta-Fibonacci Polynomials for Stochastic Models: Existence, Stability, and Numerical Validation

Pseudo-Spectral Galerkin Method Using Shifted Vieta-Fibonacci Polynomials for Stochastic Models: Existence, Stability, and Numerical Validation

An innovative pseudo-spectral Galerkin algorithm for the time-fractional Tricomi-type equation

Herein, we offer semi−analytic numerical procedures for the 1−D Tricomi−type time−fractional equation (T−FTTE). We consider the Jacobi−shifted polynomials as basis functions (BFs). A novel spectral approach is implemented based on the Galerkin procedure to tackle the Tricomi−type equation. The main strength of this approach is, it reduces the diverseial problem into solving an algebraic system of equations. The constructed methodology is successfully extended to solve the 2−D T−FTTE. Some numerical test experiments are exhibited to verify the proficiency and high accuracy of the proposed method.

A high accurate and convergent numerical framework for solving high-order nonlinear Volterra integro-differential equations

This article discuss about the application of Jacobi spectral and pseudo-spectral Galerkin techniques for solving high-order nonlinear Volterra integro-differential equations. We supply strict error analysis for Jacobi spectral and pseudo-spectral Galerkin methods(JSGM,JPSGM) under reasonable assumptions, which indicate that the approximate solution errors decay exponentially in the sense of L∞ and Lω2 norm. Some numerical experiments are presented to verify the theoretical results. Superior numerical results with respect to some recent methods are provided in tables and figures.

High accurate convergent spectral Galerkin methods for nonlinear weakly singular Volterra integro-differential equations

This paper contributes to investigate the Jacobi spectral and pseudo-spectral Galerkin techniques to solve a general form of nonlinear weakly singular Volterra integro-differential equations of the first order. By applying some suitable change of variables, we have made the solution of the mentioned equations to be smooth. Then, by applying the spectral and pseudo-spectral Jacobi Galerkin schemes, accurate solutions are computed efficiently. Rigorous convergence analysis associated with both the spectral and pseudo-spectral Jacobi Galerkin approaches are discussed in detail. Some numerical test problems are given to depict the accuracy of the presented numerical schemes with respect to some recent approximate methods in the literature.

High accurate pseudo-spectral Galerkin scheme for pantograph type Volterra integro-differential equations with singular kernels

Spectral and pseudo-spectral Galerkin techniques, by using the standard Jacobi polynomials, are implemented to calculate numerically the solutions of pantograph type Volterra delay integro-differential equations that have kernels with the property of weak singularity. Because of the complex structure of the considered problems, pseudo-spectral Galerkin approaches are more desirable with respect to the spectral Galerkin approaches, since they have the property of integral approximator by using high order convergent Gauss quadrature formulas. A deep and detailed analysis of convergence of the numerical solutions to the exact solutions are given under some mild conditions such as smoothness of the solutions. Some test problems are illustrated and efficiency of the suggested numerical approach is investigated with respect to a recently proposed Jacobi pseudo-spectral collocation technique via some figures and tables experimentally.

Rigorous convergence analysis of Jacobi spectral Galerkin methods for Volterra integral equations with noncompact kernels

This article deals with the implementation of Jacobi spectral and pseudospectral Galerkin approaches for solving weakly singular Volterra integral equations with noncompact kernels. The computational domain of the main problem and intervals of the involved integrals are transformed into [-1,1] by some appropriate change of variables. Therefore, one can apply the Jacobi Galerkin method and impose the inner product conditions on the residual of the considered equation. Also, Jacobi Gauss quadrature rules are implemented to approximate the integral operator and the inner products associated to the Jacobi weight function in the pseudospectral Galerkin method. For both cases of spectral and pseudospectral Galerkin schemes, rigorous convergence analysis is provided under some mild conditions on the smoothness of kernel and source functions in L∞ and weighted L2 norms. Several numerical examples are also given to confirm the theoretical prediction.

Modelling the dynamics of nonlinear partial differential equations using neural networks

The dynamics of two nonlinear partial differential equations (PDEs) known as the Kuramoto–Sivashinsky (K–S) equation and the two-dimensional Navier–Stokes (N–S) equations are analyzed using Karhunen–Loéve (K–L) decomposition and artificial neural networks (ANN). For the K–S equation, numerical simulations using a pseudospectral Galerkin method is presented at a bifurcation parameter α=17.75, where a dynamical behavior represented by a heteroclinic connection is obtained. We apply K–L decomposition on the numerical simulation data with the task of reducing the data into a set of data coefficients. Then we use ANN to model, and predict the data coefficients at a future time. It is found that training the neural networks with only the first data coefficient is enough to capture the underlying dynamics, and to predict for the other remaining data coefficients. As for the two-dimensional N–S equation, a quasiperiodic behavior represented in phase space by a torus is analyzed at Re=14.0. Applying the symmetry observed in the two-dimensional N–S equations on the quasiperiodic behavior, eight different tori were obtained. We show that by exploiting the symmetries of the equation and using K–L decomposition in conjunction with neural networks, a smart neural model is obtained.

A Nonlinear Galerkin Method for the Shallow-Water Equations on Periodic Domains

A nonlinear Galerkin method for the shallow-water equations is developed, based on spectral transforms. The scheme is compared to a pseudo-spectral Galerkin method. Our numerical results indicate that the nonlinear scheme has the potential advantage of providing similar accuracy at a lower cost than the Galerkin method. The nonlinear method has also less restrictive stability conditions.

Designing Reduced‐Order Models for Rapid Thermal Processing Systems

A technique for developing fast process models for rapid thermal processing (RTP) systems is described. The modeling strategy uses a priori knowledge of the process behavior to construct a model with relatively few unknowns. The resulting nonlinear model can then be used to extrapolate beyond the original knowledge base. Using the proper orthogonal decomposition method, the a priori information is extracted in the form of empirical eigenfunctions from transient simulation results of a detailed physically based finite element model (FEM). Low‐order nonlinear models are then constructed using the empirical eigenfunctions as a basis set in a pseudospectral Galerkin approximation to the physical model, i.e., the governing partial differential equations. The predictions from models developed in this fashion show good agreement with steady‐state and transient responses of the physically based FEM. In particular, the low‐order models are shown to accurately replicate an actual RTF processing cycle in a typical process chamber with an order‐of‐magnitude reduction in computation time compared to the detailed FEM.

Convergence of the Kleiser Schumann method for the Navier-Stokes equations

We analyze the Kleiser-Schumann metho for the numerical approximation of Navier-Stokes' equations with two directions of periodicity. In these directions a pseudo-spectral Galerkin method is used, in the third one a pseudo-spectral Tau method is applied. We prove optimal rate of convergence for the velocity and for the pressure.

Numerical computation of certain three-dimensional stellar structures using a semidiscrete pseudospectral method

Numerical method for solving three-dimensional stellar structure problems, formulated as free boundary problems for a mildly nonlinear elliptic partial differential equation, is developed by a combination of Newton-Raphson procedure, trial free boundary method, a pseudospectral Galerkin type method (for angular dependence) and finite differences (for radial dependence). Implementation of the method on a computer is discussed. The method is applied to the computation of structure of a critical, polytropic primary in a synchronous binary system. Forn=3.0, the results are found to be in reasonable agreement with those obtained previously by an expansion method. This expansion method is not capable of computing the structure of the other interesting polytrope withn=1.5 which presented no difficulties to the present method.