scholarly journals Numerical solution to the Falkner-Skan equation: a novel numerical approach through the new rational a-polynomials

2021 ◽  
Vol 42 (10) ◽  
pp. 1449-1460
Author(s):  
S. Abbasbandy ◽  
J. Hajishafieiha
Keyword(s):  
Numerical Solution ◽  
Collocation Method ◽  
Numerical Approach ◽  
Auxiliary Parameter ◽  
Numerical Examples ◽  
Gauss Points

AbstractThe new rational a-polynomials are used to solve the Falkner-Skan equation. These polynomials are equipped with an auxiliary parameter. The approximated solution to the Falkner-Skan equation is obtained by the new rational a-polynomials with unknown coefficients. To find the unknown coefficients and the auxiliary parameter contained in the polynomials, the collocation method with Chebyshev-Gauss points is used. The numerical examples show the efficiency of this method.

scholarly journals Solving Some Differential Equations Arising in Electric Engineering Using Hermite Polynomials

2020 ◽  
Vol 12 (4) ◽  
pp. 517-523
Author(s):  
G. Singh ◽  
I. Singh
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Collocation Method ◽  
Hermite Polynomials ◽  
Electric Circuit ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Electric Engineering ◽  
Collocation Points ◽  
Linear Algebraic Equations

In this paper, a collocation method based on Hermite polynomials is presented for the numerical solution of the electric circuit equations arising in many branches of sciences and engineering. By using collocation points and Hermite polynomials, electric circuit equations are transformed into a system of linear algebraic equations with unknown Hermite coefficients. These unknown Hermite coefficients have been computed by solving such algebraic equations. To illustrate the accuracy of the proposed method some numerical examples are presented.

scholarly journals New Algorithm for the Numerical Solutions of Nonlinear Third-Order Differential Equations Using Jacobi-Gauss Collocation Method

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
A. H. Bhrawy ◽  
W. M. Abd-Elhameed
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Spectral Method ◽  
Collocation Method ◽  
Numerical Solutions ◽  
Order Differential Equation ◽  
Third Order ◽  
Numerical Examples ◽  
Collocation Spectral Method ◽  
Gauss Points

A new algorithm for solving the general nonlinear third-order differential equation is developed by means of a shifted Jacobi-Gauss collocation spectral method. The shifted Jacobi-Gauss points are used as collocation nodes. Numerical examples are included to demonstrate the validity and applicability of the proposed algorithm, and some comparisons are made with the existing results. The method is easy to implement and yields very accurate results.

A Collocation Method for Global Approximation of General Second Order BVPs

2007 ◽  
Vol 3 (1) ◽  
pp. 23-34 ◽  
Author(s):  
F. Costabile ◽  
A. Napoli
Keyword(s):  
Numerical Solution ◽  
Boundary Value Problems ◽  
Collocation Method ◽  
Boundary Value ◽  
Second Order ◽  
Point Boundary ◽  
Global Approximation ◽  
Numerical Examples

For the numerical solution of the second order nonlinear two-point boundary value problems a family of polynomial global methods is derived.Numerical examples provide favorable comparisons with other existing methods.

Legendre Collocation Method for Volterra Integro-Differential Algebraic Equation

2019 ◽  
Vol 19 (4) ◽  
pp. 833-847
Author(s):  
Yunxia Wei ◽  
Yanping Chen ◽  
Yunqing Huang
Keyword(s):  
Approximate Solution ◽  
Numerical Solution ◽  
Collocation Method ◽  
Algebraic Equation ◽  
Numerical Experiments ◽  
Differential Algebraic Equation ◽  
Numerical Examples ◽  
Legendre Collocation Method ◽  
Approximate Derivative ◽  
Collocation Scheme

AbstractIn this work we study the numerical solution to the Volterra integro-differential algebraic equation. Two numerical examples based on the Legendre collocation scheme are designed. It follows from the convergence proof and numerical experiments that the errors of the approximate solution and the errors of the approximate derivative of the solution decay exponentially.

scholarly journals COLLOCATION METHOD FOR FUZZY VOLTERRA INTEGRAL EQUATIONS OF THE SECOND KIND

2020 ◽  
Vol 25 (1) ◽  
pp. 146-166 ◽  
Author(s):  
Zahra Alijani ◽  
Urve Kangro
Keyword(s):  
Integral Equation ◽  
Numerical Solution ◽  
Integral Equations ◽  
Collocation Method ◽  
Volterra Integral Equation ◽  
Basis Function ◽  
Convergence Rates ◽  
Numerical Examples ◽  
Convergence Estimates ◽  
Upper And Lower Functions

In this paper we consider fuzzy Volterra integral equation of the second kind whose kernel may change sign. We give conditions for smoothness of the upper and lower functions of the solution. For numerical solution we propose the collocation method with two different basis function sets: triangular and rectangular basis. The smoothness results allow us to obtain the convergence rates of the methods. The proposed methods are illustrated by numerical examples, which confirm the theoretical convergence estimates.

A Numerical Approach to Determine Some Properties of Cylindrical Pieces of Bananas During Drying

2015 ◽  
Vol 11 (3) ◽  
pp. 335-347 ◽  
Author(s):  
Wilton Pereira da Silva ◽  
Cleide M. D. P. S. e Silva ◽  
Aluizio Freire da Silva Junior ◽  
Alexandre José de Melo Queiroz
Keyword(s):  
Numerical Solution ◽  
Numerical Approach ◽  
Moisture Diffusivity ◽  
Convective Drying ◽  
Chi Square ◽  
Experimental Conditions ◽  
Liquid Diffusion ◽  
Air Temperatures ◽  
C 50 ◽  
Volume Method

Abstract This article uses several liquid diffusion models to describe convective drying of bananas cut into cylindrical pieces. A two-dimensional numerical solution of the diffusion equation with boundary condition of the third kind, obtained through the finite volume method, was used to describe the process. The cylindrical pieces were cut into the following dimensions: length of about 21 mm and average radius of 15 mm. Drying air temperatures were 40°C, 50°C, 60°C and 70°C. In order to determine the process parameters, an optimizer was coupled with the numerical solution. A model that considers the shrinkage and variable effective moisture diffusivity well describes drying for all the experimental conditions, and enables to predict the moisture distributions at any given time. For this model, the determination coefficient has varied from 0.99937 (70°C) to 0.99995 (40°C), while the chi-square ranged from 3.41 × 10−4 (40°C) to 4.15 × 10−3 (70°C).

The Equilibrium Cell-Based Smooth Finite Element Method for Shakedown Analysis of Structures

2019 ◽  
Vol 16 (05) ◽  
pp. 1840013 ◽  
Author(s):  
P. L. H. Ho ◽  
C. V. Le ◽  
T. Q. Chu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Optimization Problem ◽  
Conic Programming ◽  
Equilibrium Equations ◽  
Shakedown Analysis ◽  
Numerical Examples ◽  
Elastic Stresses ◽  
Gauss Points ◽  
Element Method

This paper presents a novel equilibrium formulation, that uses the cell-based smoothed method and conic programming, for limit and shakedown analysis of structures. The virtual strains are computed using straining cell-based smoothing technique based on elements of discretized mesh. Fictitious elastic stresses are also determined within the framework of finite element method (CS-FEM)-based Galerkin procedure, and equilibrium equations for residual stresses are satisfied in an average sense at every cell-based smoothing cell. All constrains are imposed at only one point in the smoothing domains, instead of Gauss points as in a standard FEM-based procedure. The resulting optimization problem is then handled using the highly efficient solvers. Various numerical examples are investigated, and obtained solutions are compared with available results in the literature.

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