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Article
Publication date: 2 April 2020

A numerical method for solving optimal control problems via Legendre polynomials

Yajing Gu, Hongyan Yan and Yuanguo Zhu

The purpose of this paper is to propose an iterative Legendre technique to deal with a continuous optimal control problem (OCP).

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Abstract

Purpose

The purpose of this paper is to propose an iterative Legendre technique to deal with a continuous optimal control problem (OCP).

Design/methodology/approach

For the system in the considered problem, the control variable is a function of the state variables and their derivatives. State variables in the problem are approximated by Legendre expansions as functions of time t. A constant matrix is given to express the derivatives of state variables. Therefore, control variables can be described as functions of time t. After that, the OCP is converted to an unconstrained optimization problem whose decision variables are the unknown coefficients in the Legendre expansions.

Findings

The convergence of the proposed algorithm is proved. Experimental results, which contain the controlled Duffing oscillator problem demonstrate that the proposed technique is faster than existing methods.

Originality/value

Experimental results, which contained the controlled Duffing oscillator problem demonstrate that the proposed technique can be faster while securing exactness.

Details

Engineering Computations, vol. 37 no. 8
Type: Research Article
DOI: https://doi.org/10.1108/EC-07-2019-0326
ISSN: 0264-4401

Keywords

  • Optimal control
  • Legendre polynomial
  • State parametrisation
  • 49M30
  • 74H15

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Article
Publication date: 11 November 2019

Polynomials for numerical solutions of space-time fractional differential equations (of the Fokker–Planck type)

Jiao Wang

Fokker–Planck equation appears in various areas in natural science, it is used to describe solute transport and Brownian motion of particles. This paper aims to present an…

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Abstract

Purpose

Fokker–Planck equation appears in various areas in natural science, it is used to describe solute transport and Brownian motion of particles. This paper aims to present an efficient and convenient numerical algorithm for space-time fractional differential equations of the Fokker–Planck type.

Design/methodology/approach

The main idea of the presented algorithm is to combine polynomials function approximation and fractional differential operator matrices to reduce the studied complex equations to easily solved algebraic equations.

Findings

Based on Taylor basis, simple and useful fractional differential operator matrices of alternative Legendre polynomials can be quickly obtained, by which the studied space-time fractional partial differential equations can be transformed into easily solved algebraic equations. Numerical examples and error date are presented to illustrate the accuracy and efficiency of this technique.

Originality/value

Various numerical methods are proposed in complex steps and are computationally expensive. However, the advantage of this paper is its convenient technique, i.e. using the simple fractional differential operator matrices of polynomials, numerical solutions can be quickly obtained in high precision. Presented numerical examples can also indicate that the technique is feasible for this kind of fractional partial differential equations.

Details

Engineering Computations, vol. 36 no. 9
Type: Research Article
DOI: https://doi.org/10.1108/EC-02-2019-0061
ISSN: 0264-4401

Keywords

  • Numerical solutions
  • Alternative Legendre polynomials
  • Differential operator matrices
  • Space-time fractional differential equations of Fokker–Planck type

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Article
Publication date: 12 August 2019

A modified method for solving non-linear time and space fractional partial differential equations

Umer Saeed and Muhammad Umair

The purpose of the paper is to extend the differential quadrature method (DQM) for solving time and space fractional non-linear partial differential equations on a…

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Abstract

Purpose

The purpose of the paper is to extend the differential quadrature method (DQM) for solving time and space fractional non-linear partial differential equations on a semi-infinite domain.

Design/methodology/approach

The proposed method is the combination of the Legendre polynomials and differential quadrature method. The authors derived and constructed the new operational matrices for the fractional derivatives, which are used for the solutions of non-linear time and space fractional partial differential equations.

Findings

The fractional derivative of Lagrange polynomial is a big hurdle in classical DQM. To overcome this problem, the authors represent the Lagrange polynomial in terms of shifted Legendre polynomial. They construct a transformation matrix which transforms the Lagrange polynomial into shifted Legendre polynomial of arbitrary order. Then, they obtain the new weighting coefficients matrices for space fractional derivatives by shifted Legendre polynomials and use these in conversion of a non-linear fractional partial differential equation into a system of fractional ordinary differential equations. Convergence analysis for the proposed method is also discussed.

Originality/value

Many engineers can use the presented method for solving their time and space fractional non-linear partial differential equation models. To the best of the authors’ knowledge, the differential quadrature method has never been extended or implemented for non-linear time and space fractional partial differential equations.

Details

Engineering Computations, vol. 36 no. 7
Type: Research Article
DOI: https://doi.org/10.1108/EC-01-2019-0011
ISSN: 0264-4401

Keywords

  • Legendre polynomials
  • Differential quadrature method
  • Caputo derivative
  • Fractional differential equations
  • Adam Bashforth method
  • Runge–Kutta method

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Article
Publication date: 8 July 2019

A novel technique based on Bernoulli wavelets for numerical solutions of two-dimensional Fredholm integral equation of second kind

S. Saha Ray and S. Behera

A novel technique based on Bernoulli wavelets has been proposed to solve two-dimensional Fredholm integral equation of second kind. Bernoulli wavelets have been created by…

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Abstract

Purpose

A novel technique based on Bernoulli wavelets has been proposed to solve two-dimensional Fredholm integral equation of second kind. Bernoulli wavelets have been created by dilation and translation of Bernoulli polynomials. This paper aims to introduce properties of Bernoulli wavelets and Bernoulli polynomials.

Design/methodology/approach

To solve the two-dimensional Fredholm integral equation of second kind, the proposed method has been used to transform the integral equation into a system of algebraic equations.

Findings

Numerical experiments shows that the proposed two-dimensional wavelets technique can give high-accurate solutions and good convergence rate.

Originality/value

The efficiency of newly developed two-dimensional wavelets technique has been validated by different illustrative numerical examples to solve two-dimensional Fredholm integral equations.

Details

Engineering Computations, vol. 36 no. 6
Type: Research Article
DOI: https://doi.org/10.1108/EC-11-2018-0500
ISSN: 0264-4401

Keywords

  • Bernoulli wavelets
  • Two-dimensional Fredholm integral equation
  • Bernoulli polynomial
  • Bernoulli numbers
  • Legendre wavelets
  • Karhunen–Loéve expansion

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Article
Publication date: 1 April 2003

Analytic solution of partial differential equations with Adomian's decomposition

Antonio García‐Olivares

Adomian's method is completed to obtain the analytic solution of any partial differential equation with boundary conditions defined on the four sides of a rectangle…

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Abstract

Adomian's method is completed to obtain the analytic solution of any partial differential equation with boundary conditions defined on the four sides of a rectangle. Adomian's decomposition method is first used to obtain the N‐order approximant to the one direction partial solution that satisfies the boundary conditions on that direction. Then the functions obtained are variationally modified on the four sides to make them compatible with a given experimental error. The product of these transformations is an analytic approximation to the solution which is compatible with both the weak norm imposed on the boundaries and the accuracy imposed in the whole domain.

Details

Kybernetes, vol. 32 no. 3
Type: Research Article
DOI: https://doi.org/10.1108/03684920310458584
ISSN: 0368-492X

Keywords

  • Cybernetics
  • Partial differential equations
  • Boundary conditions
  • Decomposition method

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Article
Publication date: 7 August 2017

Solutions of the Blasius and MHD Falkner-Skan boundary-layer equations by modified rational Bernoulli functions

Velinda Calvert and Mohsen Razzaghi

This paper aims to propose a new numerical method for the solution of the Blasius and magnetohydrodynamic (MHD) Falkner-Skan boundary-layer equations. The Blasius and MHD…

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Abstract

Purpose

This paper aims to propose a new numerical method for the solution of the Blasius and magnetohydrodynamic (MHD) Falkner-Skan boundary-layer equations. The Blasius and MHD Falkner-Skan equations are third-order nonlinear boundary value problems on the semi-infinite domain.

Design/methodology/approach

The approach is based upon modified rational Bernoulli functions. The operational matrices of derivative and product of modified rational Bernoulli functions are presented. These matrices together with the collocation method are then utilized to reduce the solution of the Blasius and MHD Falkner-Skan boundary-layer equations to the solution of a system of algebraic equations.

Findings

The method is computationally very attractive and gives very accurate results.

Originality/value

Many problems in science and engineering are set in unbounded domains. One approach to solve these problems is based on rational functions. In this work, a new rational function is used to find solutions of the Blasius and MHD Falkner-Skan boundary-layer equations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 27 no. 8
Type: Research Article
DOI: https://doi.org/10.1108/HFF-05-2016-0190
ISSN: 0961-5539

Keywords

  • Operational matrices
  • Blasius equation
  • MHD Falkner-Skan equation
  • Modified rational Bernoulli functions
  • Semi-infinite domain

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Article
Publication date: 1 December 2003

Parametric error modeling and software error compensation for rapid prototyping

Kun Tong, E. Amine Lehtihet and Sanjay Joshi

This paper is motivated by the need for a generic approach to evaluate the volumetric accuracy of rapid prototyping (RP) machines. The approach presented in this paper is…

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Abstract

This paper is motivated by the need for a generic approach to evaluate the volumetric accuracy of rapid prototyping (RP) machines. The approach presented in this paper is inspired in large part by the techniques developed over the years for the parametric evaluation of coordinate measuring machine (CMM) errors. In CMM metrology, the parametric error functions for the machine are determined by actual measurement of a master reference artifact with known characteristics. In our approach, the RP machine is used to produce a generic artifact, which is then measured by a master CMM, and measurement results are used to infer the RP machine's parametric error functions. The results presented demonstrate the feasibility of such an approach on a two‐dimensional model.

Details

Rapid Prototyping Journal, vol. 9 no. 5
Type: Research Article
DOI: https://doi.org/10.1108/13552540310502202
ISSN: 1355-2546

Keywords

  • Parametric measures
  • Computer software
  • Rapid prototypes

Content available
Article
Publication date: 3 August 2012

A bibliography of the theory and applications of the Adomian decomposition method, 1961-2011

Randolph Rach

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Abstract

Details

Kybernetes, vol. 41 no. 7/8
Type: Research Article
DOI: https://doi.org/10.1108/k.2012.06741gaa.007
ISSN: 0368-492X

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Article
Publication date: 1 April 2014

Modelling of a Phase Change Material melting process heated from below using spectral collocation methods

Jean Batina, Serge Blancher and Tarik Kouskou

Mathematical and numerical models are developed to study the melting of a Phase Change Material (PCM) inside a 2D cavity. The bottom of the cell is heated at constant and…

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Abstract

Purpose

Mathematical and numerical models are developed to study the melting of a Phase Change Material (PCM) inside a 2D cavity. The bottom of the cell is heated at constant and uniform temperature or heat flux, assuming that the rest of the cavity is completely adiabatic. The paper used suitable numerical methods to follow the interface temporal evolution with a good accuracy. The purpose of this paper is to show how the evolution of the latent energy absorbed to melt the PCM depends on the temperature imposed on the lower wall of the cavity.

Design/methodology/approach

The problem is written with non-homogeneous boundary conditions. Momentum and energy equations are numerically solved in space by a spectral collocation method especially oriented to this situation. A Crank-Nicolson scheme permits the resolution in time.

Findings

The results clearly show the evolution of multicellular regime during the process of fusion and the kinetics of phase change depends on the boundary condition imposed on the bottom cell wall. Thus the charge and discharge processes in energy storage cells can be controlled by varying the temperature in the cell PCM. Substantial modifications of the thermal convective heat and mass transfer are highlighted during the transient regime. This model is particularly suitable to follow with a good accuracy the evolution of the solid/liquid interface in the process of storage/release energy.

Research limitations/implications

The time-dependent physical properties that induce non-linear coupled unsteady terms in Navier-Stokes and energy equations are not taken into account in the present model. The present model is actually extended to these coupled situations. This problem requires smoother geometries. One can try to palliate this disadvantage by constructing smoother approximations of non-smooth geometries. The augmentation of polynomials developments orders increases strongly the computing time. When the external heat flux or temperature imposed at the PCM is much greater than the temperature of the PCM fusion, one must choose carefully some data to assume the algorithms convergence.

Practical implications

Among the areas where this work can be used, are: buildings where the PCM are used in insulation and passive cooling; thermal energy storage, the PCM stores energy by changing phase, solid to liquid (fusion); cooling and transport of foodstuffs or pharmaceutical or medical sensitive products, the PCM is used in the food industry, pharmaceutical and medical, to minimize temperature variations of food, drug or sensitive materials; and the textile industry, PCM materials in the textile industry are used in microcapsules placed inside textile fibres. The PCM intervene to regulate heat transfer between the body and the outside.

Originality/value

The paper's originality is reflected in the precision of its results, due to the use of a high-accuracy numerical approximation based on collocation spectral methods, and the choice of Chebyshev polynomials basis in both axial and radial directions.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 24 no. 3
Type: Research Article
DOI: https://doi.org/10.1108/HFF-03-2012-0062
ISSN: 0961-5539

Keywords

  • Phase Change Material
  • Charge and discharge processes
  • Energy storage cells
  • Convective heat transfer
  • Spectral collocation methods
  • High-accuracy numerical approximation
  • Crank-Nicolson resolution in time

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Article
Publication date: 1 February 1992

A GENERALIZED FORMULATION OF THE DISCRETE ELEMENT METHOD

GRAHAM G.W. MUSTOE

A generalized weighted residual method is used to formulate the discrete element method (DEM) for rigid or deformable bodies. It is shown that this approach provides a…

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Abstract

A generalized weighted residual method is used to formulate the discrete element method (DEM) for rigid or deformable bodies. It is shown that this approach provides a unified methodology for deriving many of the different discrete element techniques in current use today. This procedure is used to develop a number of different element formulations for use in problems in which the distinct bodies exhibit complex deformation behaviour such as beam or plate flexure, membrane action, and additional reinforcement of a jointed discontinuum. A covergence proof for the two‐dimensional beam element is given for mathematical validation. A number of examples are also presented which illustrate the usefulness of different discrete element types in engineering analyses of discontinuum problems.

Details

Engineering Computations, vol. 9 no. 2
Type: Research Article
DOI: https://doi.org/10.1108/eb023857
ISSN: 0264-4401

Keywords

  • Discrete fracture
  • Discontinuous materials

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