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Article

Ping He and Yangmin Li

The purpose of this paper is to investigate the analytical solution of a hyperbolic partial differential equation (PDE) and its application.

Abstract

Purpose

The purpose of this paper is to investigate the analytical solution of a hyperbolic partial differential equation (PDE) and its application.

Design/methodology/approach

The change of variables and the method of successive approximations are introduced. The Volterra transformation and boundary control scheme are adopted in the analysis of the reaction-diffusion system.

Findings

A detailed and complete calculation process of the analytical solution of hyperbolic PDE (1)-(3) is given. Based on the Volterra transformation, a reaction-diffusion system is controlled by boundary control.

Originality/value

The introduced approach is interesting for the solution of hyperbolic PDE and boundary control of the reaction-diffusion system.

Details

International Journal of Intelligent Computing and Cybernetics, vol. 10 no. 2
Type: Research Article
ISSN: 1756-378X

Keywords

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Article

Jiao Wang

This paper aims to propose an efficient and convenient numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional…

Abstract

Purpose

This paper aims to propose an efficient and convenient numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types).

Design/methodology/approach

The main idea of the presented algorithm is to combine Bernoulli polynomials approximation with Caputo fractional derivative and numerical integral transformation to reduce the studied two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations to easily solved algebraic equations.

Findings

Without considering the integral operational matrix, this algorithm will adopt straightforward discrete data integral transformation, which can do good work to less computation and high precision. Besides, combining the convenient fractional differential operator of Bernoulli basis polynomials with the least-squares method, numerical solutions of the studied equations can be obtained quickly. Illustrative examples are given to show that the proposed technique has better precision than other numerical methods.

Originality/value

The proposed algorithm is efficient for the considered two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations. As its convenience, the computation of numerical solutions is time-saving and more accurate.

Details

Engineering Computations, vol. ahead-of-print no. ahead-of-print
Type: Research Article
ISSN: 0264-4401

Keywords

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Article

Yves Cherruault and Virginie Seng

Aims to solve Fredholm and Volterra non‐linear integral equations of the first kind. Uses the Adomian method, but since these equations are not under the canonical form…

Abstract

Aims to solve Fredholm and Volterra non‐linear integral equations of the first kind. Uses the Adomian method, but since these equations are not under the canonical form u‐Nu = f, proposes some transformations for reducing the integral equations to integral equations of the second kind, much more appropriate. Uses a perturbation method for Fredholm equations. Concerning Volterra equations, uses a differentiation of the original equation, under sufficient regularity conditions, for obtaining a canonical form of Adomian.

Details

Kybernetes, vol. 26 no. 2
Type: Research Article
ISSN: 0368-492X

Keywords

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Article

M. Inc and Y. Cherruault

Based on the original methods of Adomian a decomposition method has been developed to find the analytic approximation of the linear and nonlinear Volterra‐Fredholm (V‐F…

Abstract

Purpose

Based on the original methods of Adomian a decomposition method has been developed to find the analytic approximation of the linear and nonlinear Volterra‐Fredholm (V‐F) integro‐differential equations under the initial or boundary conditions.

Design/methodology/approach

Designed around the methods of Adomian and later researchers. The methodology to obtain numerical solutions of the V‐F integro‐differential equations is one whose essential features is its rapid convergence and high degree of accuracy which it approximates. This is achieved in only a few terms of its iterative scheme which is devised to avoid linearization, perturbation and any transformation in order to find solutions to given problems.

Findings

The scheme was shown to have many advantages over the traditional methods. In particular it provided discretization and provided an efficient numerical solution with high accuracy, minimal calculations as well as an avoidance of physical unrealistic assumptions.

Research limitations/implications

A reliable method for obtaining approximate solutions of linear and nonlinear V‐F integro‐differential using the decomposition method which avoids the tedious work needed by traditional techniques has been developed. Exact solutions were easily obtained.

Practical implications

The new method had most of its symbolic and numerical computations performed using the Computer Algebra Systems‐Mathematica. Numerical results from selected examples were presented.

Originality/value

A new effective and accurate methodology has been developed and demonstrated.

Details

Kybernetes, vol. 34 no. 7/8
Type: Research Article
ISSN: 0368-492X

Keywords

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Article

Kazem Nouri

The purpose of this paper is to discuss a numerical method for solving system of Volterra integral equations.

Abstract

Purpose

The purpose of this paper is to discuss a numerical method for solving system of Volterra integral equations.

Design/methodology/approach

An expansion method known as Chebyshev collocation method is chosen to convert the system of integral equations to the linear algebraic system of equations, so by solving the linear algebraic system an approximate solution is concluded.

Findings

Some numerical results support the accuracy and efficiency of the stated method.

Originality/value

The paper presents a method for solving first and second kind system of integral equations.

Abstract

Details

Kybernetes, vol. 41 no. 7/8
Type: Research Article
ISSN: 0368-492X

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Article

M. Hadizadeh and K. Maleknejad

The Adomian decomposition method is used and applied to the mathematical model of a biosensor. This model consists of a heat equation with non‐linear and non‐local…

Abstract

The Adomian decomposition method is used and applied to the mathematical model of a biosensor. This model consists of a heat equation with non‐linear and non‐local boundary conditions. To obtain a canonical form of Adomian, an equivalent non‐linear Volterra integral equation with a weakly singular kernel is set up. In addition, the asymptotic behaviour of the solution as t → 0 and t → • by asymptotic decomposition is obtained. Finally, numerical results are given which support the theoretical results.

Details

Kybernetes, vol. 27 no. 4
Type: Research Article
ISSN: 0368-492X

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Article

Shuhua Mao, Mingyun Gao and Min Zhu

The purpose of this paper is to elevate the accuracy when predicting the gross domestic product (GDP) on research and development (R&D) and to develop the grey delay Lotka…

Abstract

Purpose

The purpose of this paper is to elevate the accuracy when predicting the gross domestic product (GDP) on research and development (R&D) and to develop the grey delay Lotka-Volterra model.

Design/methodology/approach

Considering the lag effects between input in R&D and output in GDP, this paper estimated the delay value via grey delay relation analysis. Taking the delay into original Lotka-Volterra model and combining with the thought of grey theory and grey transform, the authors proposed grey delay Lotka-Volterra model, estimated the parameter of model and gave the discrete time analytic expression.

Findings

Collecting the actual data of R&D and GDP in Wuhan China from 1995 until 2008, this paper figure out that the delay between R&D and GDP was 2.625 year and found the dealy time would would gradually be reduced with the economy increasing.

Practical implications

Constructing the grey delay Lotka-Volterra model via above data, this paper shown that the precision was satisfactory when fitting the data of R&D and GDP. Comparing the forecasts with the actual data of GDP in Wuhan from 2009 until 2012, the error was small.

Social implications

The result shows that R&D and GDP would be both growing fast in future. Wuhan will become a city full of activity.

Originality/value

Considering the lag between R&D and GDP, this work estimated the delay value via a grey delay relation analysis and constructed a novel grey delay Lotka-Volterra model.

Details

Grey Systems: Theory and Application, vol. 5 no. 1
Type: Research Article
ISSN: 2043-9377

Keywords

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Article

Guishu Liang and Yulan Yang

This paper aims to analyze soil electrical properties based on fractional calculus theory due to the fact that the frequency dependence of soil electrical parameters at…

Abstract

Purpose

This paper aims to analyze soil electrical properties based on fractional calculus theory due to the fact that the frequency dependence of soil electrical parameters at high frequencies exhibits a fractional effect. In addition, for the fractional-order formulation, this paper aims to provide a more accurate numerical algorithm for solving the fractional differential equations.

Design/methodology/approach

This paper analyzes the frequency-dependence of soil electrical properties based on fractional calculus theory. A collocation method based on the Puiseux series is proposed to solve fractional differential equations.

Findings

The algorithm proposed in this paper can be used to solve fractional differential equations of arbitrary order, especially for 0.5th-order equations, obtaining accurate numerical solutions. Calculating the impact response of the grounding electrode based on the fractional calculus theory can obtain a more accurate result.

Originality/value

This paper proposes an algorithm for solving fractional differential equations of arbitrary order, especially for 0.5th-order equations. Using fractional calculus theory to analyze the frequency-dependent effect of soil electrical properties, provides a new idea for ground-related transient calculation.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering , vol. 39 no. 2
Type: Research Article
ISSN: 0332-1649

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Article

Feyed Ben Zitoun and Yves Cherruault

The purpose of this paper is to present a method for solving nonlinear integral equations of the second and third kind.

Abstract

Purpose

The purpose of this paper is to present a method for solving nonlinear integral equations of the second and third kind.

Design/methodology/approach

The method converts the nonlinear integral equation into a system of nonlinear equations. By solving the system, the solution can be determined. Comparing the methodology with some known techniques shows that the present approach is simple, easy to use, and highly accurate.

Findings

The proposed technique allows the authors to obtain an approximate solution in a series form. Test problems are given to illustrate the pertinent features of the method. The accuracy of the numerical results indicates that the technique is efficient and well‐suited for solving nonlinear integral equations.

Originality/value

The present approach provides a reliable technique that avoids the difficulties and massive computational work if compared with the traditional techniques and does not require discretization in order to find solutions to the given problems.

Details

Kybernetes, vol. 38 no. 5
Type: Research Article
ISSN: 0368-492X

Keywords

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