Generalized Collocation Method Research Articles

A Special Version of the Collocation Method for One Class of Integro-Differential Equations

A linear integro-differential equation with a singular differential operator in the main part is studied. To find its approximate solution in the space of generalized functions, a special version of the generalized collocation method is proposed and justified

A generalized collocation method in reproducing kernel space for solving a weakly singular Fredholm integro-differential equations

A generalized collocation method for solving a weakly singular Fredholm integro-differential equation with Kalman kernel is proposed in reproducing kernel space. To obtain the generalized collocation method, the multiwaves basis in reproducing kernel space Wn+1[0,b] is constructed based on Legendre multiwaves in L2[0,1]. Using the multiwaves basis, we propose ε-approximate solutions and use the method of searching the minimum to obtain the best approximate solution of the equation. Meanwhile, convergence order and stability of the generalized collocation method are studied. It is worth to show that the generalized collocation method proposed in the paper is stable and could be applied to solve other integral equations or differential equations.

MODELING AND SIMULATION OF THE NANOFLUID TRANSPORT VIA ELASTIC SHEETS

The field of nanofluidics research has spanned over the past decade with a variety of promising applications. We investigate the “laminar boundary layer flow” of a Newtonian nanofluid past a moving extendable/contractable horizontal plate with surface velocity and thermal slip effects. The passively controlled nanofluid model (PCM) is considered. Such models are physically more realistic as compared to the “actively controlled models” (ACM). Using Lie symmetry group method, the governing equations are reduced by a set of highly coupled nonlinear ODE’s with thermo-solutal coupled boundary conditions. The reduced equations are solved numerically by a generalized collocation method. The influences of the emerging parameters on the local skin friction factor and the local Nusselt number are depicted numerically. The skin friction is decreased as the thermophoresis and buoyancy ratio parameters are decreased. The heat transfer rates reduce with thermophoresis and buoyancy ratio parameters. Velocity slip also leads to a rise in wall temperature gradient. This study is relevant to near-wall flows in nanofluid fuel cells, nano-materials processing, etc.

Modeling the Effect of Imperfections in Glassblown Micro-Wineglass Fused Quartz Resonators

In this paper, we developed an analytical model, supported by experimental results, on the effect of imperfections in glassblown micro-wineglass fused quartz resonators. The analytical model predicting the frequency mismatch due to imperfections was derived based on a combination of the Rayleigh's energy method and the generalized collocation method. The analytically predicted frequency of the n = 2 wineglass mode shape was within 10% of the finite element modeling results and within 20% of the experimental results for thin shells, showing the fidelity of the predictive model. The postprocessing methods for improvement of the resonator surface quality were also studied. We concluded that the thermal reflow of fused quartz achieves the best result, followed in effectiveness by the RCA-1 surface treatment. All the analytical models developed in this paper are to guide the manufacturing methods to reduce the frequency and damping mismatch, and to increase the mechanical quality factor of the device.

Hall Effect on Falkner—Skan Boundary Layer Flow of FENE-P Fluid over a Stretching Sheet

The Falkner—Skan boundary layer steady flow over a flat stretching sheet is investigated in this paper. The mathematical model consists of continuity and the momentum equations, while a new model is proposed for MHD Finitely Extensible Nonlinear Elastic Peterlin (FENE-P) fluid. The effects of Hall current with the variation of intensity of non-zero pressure gradient are taken into account. The governing partial differential equations are first transformed to ordinary differential equations using appropriate similarity transformation and then solved by Adomian decomposition method (ADM). The obtained results are validated by generalized collocation method (GCM) and found to be in good agreement. Effects of pertinent parameters are discussed through graphs and tables. Comparison with the existing studies is made as a limiting case of the considered problem at the end.

A multipoint Birkhoff type boundary value problem

AbstractA multipoint boundary value problem is considered. The existence and uniqueness of solution is proved. Then, for the numerical solution, a general collocation method is proposed.Numerical experiments confirm theoretical results.

Collocation for high order differential equations with two-points Hermite boundary conditions

For the numerical solution of high even order differential equations with two-points Hermite boundary conditions a general collocation method is derived and studied. Computation of the integrals which appear in the coefficients are generated by a recurrence formula and no integrals are involved in the calculation. An application to the solution of the beam problem is given. Numerical experiments provide favorable comparisons with other existing methods.

A method for high-order multipoint boundary value problems with Birkhoff-type conditions

In this paper an efficient numerical method for solving a class of multipoint boundary value problems with special boundary conditions of Birkhoff-type is presented. After a quick reference to Birkhoff-type interpolation polynomial which satisfies the particular conditions, and a result on the existence and uniqueness of solution of the given problem, an algorithm is introduced to find a polynomial that approximates the solution. It is a general collocation method. Then an a priori estimation of the error of this approximation is given. Finally, to show the efficiency and the applicability of the method, numerical results are presented. These numerical experiments provide favourable comparisons with the NDSolve command of Mathematica.

Robustness of Operational Matrices of Differentiation for Solving State-Space Analysis and Optimal Control Problems

The idea of approximation by monomials together with the collocation technique over a uniform mesh for solvingstate-space analysisandoptimal controlproblems (OCPs) has been proposed in this paper. After imposing the Pontryagins maximum principle to the main OCPs, the problems reduce to a linear or nonlinear boundary value problem. In the linear case we propose a monomial collocation matrix approach, while in the nonlinear case, the general collocation method has been applied. We also show the efficiency of the operational matrices of differentiation with respect to the operational matrices of integration in our numerical examples. These matrices of integration are related to the Bessel, Walsh, Triangular, Laguerre, and Hermite functions.

Elastoplastic Large Deformation Using Meshless Integral Method

In this paper, the meshless integral method based on the regularized boundary integral equation [1] has been extended to analyze the large deformation of elastoplastic materials. The updated Lagrangian governing integral equation is obtained from the weak form of elastoplasticity based on Green-Naghdi’s theory over a local sub-domain, and the moving least-squares approximation is used for meshless function approximation. Green-Naghdi’s theory starts with the additive decomposition of the Green-Lagrange strain into elastic and plastic parts and considers aJ2elastoplastic constitutive law that relates the Green-Lagrange strain to the second Piola-Kirchhoff stress. A simple, generalized collocation method is proposed to enforce essential boundary conditions straightforwardly and accurately, while natural boundary conditions are incorporated in the system governing equations and require no special handling. The solution algorithm for large deformation analysis is discussed in detail. Numerical examples show that meshless integral method with large deformation is accurate and robust.

Generalized collocation method for linear and nonlinear convection-diffusion models

The generalized collocation method has proved to be a simple and useful numerical approach for the simulation of a number of one- and two-dimensional nonlinear differential problems. Its remarkable efficiency to solve convection-diffusion models is shown in this note. Some typical test cases are dealt with and the good results that have been obtained suggest that the proposed generalized collocation method could be useful to simulate a class of equations usually adopted to describe several important processes in water engineering.

An Approximation Algorithm for the solution of astrophysics equations using rational scaled generalized Laguerre function collocation method based on transformed Hermite-Gauss nodes

In this paper we propose a collocation method for solving Lane-Emden type equation which is nonlinear or-dinary differential equation on the semi-infinite domain. This equation is categorized as singular initial value problems. We solve this equation by the generalized Laguerre polynomial collocation method based on Her-mite-Gauss nodes. This method solves the problem on the semi-infinite domain without truncating it to a fi-nite domain and transforming domain of the problem to a finite domain. In addition, this method reduces so-lution of the problem to solution of a system of algebraic equations.

Transport–diffusion models with nonlinear boundary conditions and solution by generalized collocation methods

This paper deals with the derivation of a class of nonlinear transport and diffusion models implemented with nonlinear boundary conditions. Mathematical tools to treat the initial-boundary value problems are developed, based on generalized collocation methods, focused on the treatment of nonlinear boundary conditions in one space dimension. Applications refer to a problem of interest in applied sciences.

Generalized collocation method for two-dimensional reaction-diffusion problems with homogeneous Neumann boundary conditions

Nonlinear reaction-diffusion problems are ubiquitous in science, ranging from physics to engineering, from economy to biology. In particular, in this paper two-dimensional problems with generic Dirichlet boundary conditions or homogeneous Neumann boundary conditions are focused on. To simulate this class of mathematical problems, a quite simple generalized collocation method is proposed and developed. Moreover, a test case and an application to the Schnakenberg model are described, in order to show the efficiency of the simulation method and the goodness of the numerical results.

Elastoplastic meshless integral method

In this paper, the meshless integral method based on regularized boundary integral equation [A. Bodin, J. Ma, X.J. Xin, P. Krishnaswami, A meshless integral method based on regularized boundary integral equation, Comput. Methods Appl. Mech. Engrg. 195 (2006) 6258–6286] has been extended to elastoplastic materials. In the formulation, the domain of interest is populated with a set of nodes using an automatic node generation algorithm. The sub-domain and support domain associated with each node are also generated automatically using algorithms that have been developed for this purpose. The governing integral equation is obtained from the weak form of elastoplasticity over a local sub-domain, and the moving least-squares approximation is used for meshless function approximation. The constitutive law is the small deformation, rate-independent flow theory based on von Mises yielding criterion with isotropic hardening. The generalized collocation method is employed to enforce the essential boundary conditions exactly, which is simple and computationally efficient. Natural boundary conditions are incorporated in the system governing equation and require no special handling. The solution algorithm for elastoplastic analysis is discussed in detail. The proposed method can handle any prescribed loading profile, including unloading and reversed loading. Numerical examples show that the elastoplastic integral meshless method is accurate and robust.

Implicit extension of taylor series method for initial value problems

The Taylor series method is one of the earliest analytic-numeric algorithms for approximate solution of initial value problems for ordinary differential equations. The main idea of the rehabilitation of these algorithms is based on the approximate calculation of higher derivatives using well-known technique for the partial differential equations. The approximate solution is given as a piecewise polynomial function defined on the subintervals of the whole interval. This property offers different facility for adaptive error control. This paper describes several explicit Taylor series with implicit extension algorithms and examines its consistency and stability properties. The implicit extension based on a collocation term added to the explicit truncated Taylor series. This idea is different from the general collocation method construction, which conduces to the implicit R-K algorithms [12]. It demonstrates some numerical test results for stiff systems herewith we attempt to prove the efficiency of these new-old algorithms.

Nonlinear convection-dispersion models with a localized pollutant source, II—A class of inverse problems

This paper deals with the modelling and solution of a class of nonlinear inverse problems related to a transport diffusion model with a source term. Specifically, the first part of the paper deals with the description of the physical system and of the related mathematical problems. The second part deals with the development of the solution method which decomposes the space domain into a suitable set of subdomains. In each of them, the initial boundary value problem is direct and is solved by generalized collocation method. Combining the solutions in all domains gives the general solution to the inverse problem. Some simulations visualize the application of the method.

Free Vibration Analysis of Spherical Caps Using a G.D.Q. Numerical Solution

In this paper we present the frequency evaluation of spherical shells by means of the generalized differential quadrature method (G.D.Q.M.), an effective numerical procedure which pertains to the class of generalized collocation methods. The shell theory used in this study is a first-order shear deformation theory with transverse shearing deformations and rotatory inertia included. The shell governing equations in terms of mid-surface displacements are obtained and, after expansion in partial Fourier series of the circumferential coordinate, solved with the G.D.Q.M. Several comparisons are made with available results, showing the reliability and modeling capability of the numerical scheme in argument.

Nonlinear convection-dispersion models with a distributed pollutant source I: Direct initial boundary value problems

This paper deals with the modeling and solution of a class of nonlinear direct problems related to a transport diffusion model with a source term. Specifically, the first part of the paper deals with the derivation of a class of transport and diffusion models (with a distributed source term) in one space dimensions with variable properties along the channel and nonlinear decay term. The second part with simulations, that is the approximation to the solution of nonlinear initial boundary value problems by generalized collocation methods. The third part develops a critical analysis mainly addressed to research perspectives on the solution of inverse problems related to the identification of the source term.

Implicit Extension of Taylor Series Method with Numerical Derivatives

AbstractThe Taylor series method is one of the earliest analytic‐numeric algorithms for approximate solution of initial value problems for ordinary differential equations. The main idea of the rehabilitation of these algorithms is based on the approximate calculation of higher order derivatives using well‐known technique for the partial differential equations. The implicit extension based on a collocation term added to the explicit truncated Taylor series. This idea is different from the general collocation method construction, which led to the implicit R‐K algorithms [1].