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1 – 10 of 177
Article
Publication date: 1 August 2005

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

Article
Publication date: 2 March 2012

Feyed Ben Zitoun and Yves Cherruault

The purpose of this paper is to present a method for solving nonlinear integro‐differential equations with constant/variable coefficients and with initial/boundary conditions.

Abstract

Purpose

The purpose of this paper is to present a method for solving nonlinear integro‐differential equations with constant/variable coefficients and with initial/boundary conditions.

Design/methodology/approach

The method converts the given problem into a system of nonlinear algebraic equations. By solving this system, the solution is 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 an approximate solution in a series form to be obtained. Test problems are solved 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 integro‐differential equations.

Originality/value

The present approach provides a reliable technique which avoids the tedious work needed by the classical techniques.

Article
Publication date: 1 April 1986

S.D. MILUSHEVA and D.D. BAINOV

The paper justifies the averaging method for nonlinear integro‐differential equations of the type (x ∈ Rn, ε > 0 is a small parameter, ω is a constant), experiencing impulse…

Abstract

The paper justifies the averaging method for nonlinear integro‐differential equations of the type (x ∈ Rn, ε > 0 is a small parameter, ω is a constant), experiencing impulse effect.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 5 no. 4
Type: Research Article
ISSN: 0332-1649

Content available

Abstract

Details

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

Article
Publication date: 3 August 2020

Yaser Rostami

This paper aims to present a new method for the approximate solution of two-dimensional nonlinear Volterra–Fredholm partial integro-differential equations with boundary conditions…

Abstract

Purpose

This paper aims to present a new method for the approximate solution of two-dimensional nonlinear Volterra–Fredholm partial integro-differential equations with boundary conditions using two-dimensional Chebyshev wavelets.

Design/methodology/approach

For this purpose, an operational matrix of product and integration of the cross-product and differentiation are introduced that essentially of Chebyshev wavelets. The use of these operational matrices simplifies considerably the structure of the computation used for a set of the algebraic system has been obtained by using the collocation points and solved.

Findings

Theorem for convergence analysis and some illustrative examples of using the presented method to show the validity, efficiency, high accuracy and applicability of the proposed technique. Some figures are plotted to demonstrate the error analysis of the proposed scheme.

Originality/value

This paper uses operational matrices of two-dimensional Chebyshev wavelets and helps to obtain high accuracy of the method.

Details

Engineering Computations, vol. 38 no. 2
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 11 April 2022

Bakhtiyar Khudayarov and Fozilzhon Turaev

The purpose of this study is to create a mathematical model, a numerical algorithm and a computer program for studying the vibration of composite pipelines based on the theory of…

Abstract

Purpose

The purpose of this study is to create a mathematical model, a numerical algorithm and a computer program for studying the vibration of composite pipelines based on the theory of beams used in the oil and gas industry, agriculture and water management, housing and communal services and other areas.

Design/methodology/approach

A mathematical model of vibration of a viscoelastic pipeline based on the theory of beams with a pulsating fluid flowing through it was developed. Using the Bubnov-Galerkin method, based on the polynomial approximation of deflections, the problem is reduced to the study of systems of ordinary integro-differential equations, the solution of which is found by a numerical method. A computational algorithm was developed for solving problems of vibrations of composite pipelines conveying pulsating liquid.

Findings

The stability and amplitude-time characteristics of vibration of composite pipelines with a pulsating fluid flowing in it are studied for wide range of changes in the parameters of deformable systems and fluid flow. The critical velocities of fluid flow at which the viscoelastic pipe loses its rectilinear equilibrium shape are found. The effect of singularity in the kernels of heredity on the vibrations of structures with viscoelastic properties was numerically studied. It is shown that with an increase in the viscosity parameter of the pipeline material, the critical flow velocity decreases. It was determined that an increase in the value of the fluid pulsation frequency and the excitation coefficient leads to a decrease in the critical velocity of the fluid flow. It was established that an increase in the parameters of the Winkler foundation and the rigidity parameter of the continuous layer leads to an increase in the critical flow velocity.

Originality/value

The study of the vibration of pipelines made of composite materials is of great theoretical and applied interest. The solution to this problem is an effective application of the theory of viscoelasticity to real processes. Therefore, the methods and problems of pipeline vibrations attract much attention from researchers. This study is devoted to solving the above problems and therefore its subject is relevant. The paper considers the results of numerical simulation of the processes of vibration of a composite pipeline based on the theory of shells during the flow of a pulsating liquid through it. A mathematical model of vibration of a composite pipeline was developed. A computational algorithm was developed for solving problems of vibrations of composite pipelines conveying pulsating liquid.

Details

Multidiscipline Modeling in Materials and Structures, vol. 18 no. 2
Type: Research Article
ISSN: 1573-6105

Keywords

Article
Publication date: 5 March 2018

Hadi Dehbovid, Habib Adarang and Mohammad Bagher Tavakoli

Charge pump phase locked loops (CPPLLs) are nonlinear systems as a result of the nonlinear behavior of voltage-controlled oscillators (VCO). This paper aims to specify jitter…

Abstract

Purpose

Charge pump phase locked loops (CPPLLs) are nonlinear systems as a result of the nonlinear behavior of voltage-controlled oscillators (VCO). This paper aims to specify jitter generation of voltage controlled oscillator phase noise in CPPLLs, by considering approximated practical model for VCO.

Design/methodology/approach

CPPLL, in practice, shows nonlinear behavior, and usually in LC-VCOs, it follows second-degree polynomial function behavior. Therefore, the nonlinear differential equation of the system is obtained which shows the CPPLLs are a nonlinear system with memory, and that Volterra series expansion is useful for such systems.

Findings

In this paper, by considering approximated practical model for VCO, jitter generation of voltage controlled oscillator phase noise in CPPLLs is specified. Behavioral simulation is used to validate the analytical results. The results show a suitable agreement between analytical equations and simulation results.

Originality/value

The proposed method in this paper has two advantages over the conventional design and analysis methods. First, in contrast to an ideal CPPLL, in which the characteristic of the VCO’s output frequency based on the control voltage is linear, in the present paper, a nonlinear behavior was considered for this characteristic in accordance with the real situations. Besides, regarding the simulations in this paper, a behavior similar to the second-degree polynomial was considered, which caused the dependence of the produced jitter’s characteristic corner frequency on the jitter’s amplitude. Second, some new nonlinear differential equations were proposed for the system, which ensured the calculation of the produced jitter of the VCO phase noise in CPPLLs. The presented method is general enough to be used for designing the CPPLL.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 37 no. 2
Type: Research Article
ISSN: 0332-1649

Keywords

Article
Publication date: 12 January 2024

Imtiyaz Ahmad Bhat, Lakshmi Narayan Mishra, Vishnu Narayan Mishra, Cemil Tunç and Osman Tunç

This study aims to discuss the numerical solutions of weakly singular Volterra and Fredholm integral equations, which are used to model the problems like heat conduction in…

Abstract

Purpose

This study aims to discuss the numerical solutions of weakly singular Volterra and Fredholm integral equations, which are used to model the problems like heat conduction in engineering and the electrostatic potential theory, using the modified Lagrange polynomial interpolation technique combined with the biconjugate gradient stabilized method (BiCGSTAB). The framework for the existence of the unique solutions of the integral equations is provided in the context of the Banach contraction principle and Bielecki norm.

Design/methodology/approach

The authors have applied the modified Lagrange polynomial method to approximate the numerical solutions of the second kind of weakly singular Volterra and Fredholm integral equations.

Findings

Approaching the interpolation of the unknown function using the aforementioned method generates an algebraic system of equations that is solved by an appropriate classical technique. Furthermore, some theorems concerning the convergence of the method and error estimation are proved. Some numerical examples are provided which attest to the application, effectiveness and reliability of the method. Compared to the Fredholm integral equations of weakly singular type, the current technique works better for the Volterra integral equations of weakly singular type. Furthermore, illustrative examples and comparisons are provided to show the approach’s validity and practicality, which demonstrates that the present method works well in contrast to the referenced method. The computations were performed by MATLAB software.

Research limitations/implications

The convergence of these methods is dependent on the smoothness of the solution, it is challenging to find the solution and approximate it computationally in various applications modelled by integral equations of non-smooth kernels. Traditional analytical techniques, such as projection methods, do not work well in these cases since the produced linear system is unconditioned and hard to address. Also, proving the convergence and estimating error might be difficult. They are frequently also expensive to implement.

Practical implications

There is a great need for fast, user-friendly numerical techniques for these types of equations. In addition, polynomials are the most frequently used mathematical tools because of their ease of expression, quick computation on modern computers and simple to define. As a result, they made substantial contributions for many years to the theories and analysis like approximation and numerical, respectively.

Social implications

This work presents a useful method for handling weakly singular integral equations without involving any process of change of variables to eliminate the singularity of the solution.

Originality/value

To the best of the authors’ knowledge, the authors claim the originality and effectiveness of their work, highlighting its successful application in addressing weakly singular Volterra and Fredholm integral equations for the first time. Importantly, the approach acknowledges and preserves the possible singularity of the solution, a novel aspect yet to be explored by researchers in the field.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 34 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 5 October 2018

Nathaniel Mahwash Kamoh and Terhemen Aboiyar

The purpose of this paper is to develop a block method of order five for the general solution of the first-order initial value problems for Volterra integro-differential equations

Abstract

Purpose

The purpose of this paper is to develop a block method of order five for the general solution of the first-order initial value problems for Volterra integro-differential equations (VIDEs).

Design/methodology/approach

A collocation approximation method is adopted using the shifted Legendre polynomial as the basis function, and the developed method is applied as simultaneous integrators on the first-order VIDEs.

Findings

The new block method possessed the desirable feature of the Runge–Kutta method of being self-starting, hence eliminating the use of predictors.

Originality/value

In this paper, some information about solving VIDEs is provided. The authors have presented and illustrated the collocation approximation method using the shifted Legendre polynomial as the basis function to investigate solving an initial value problem in the class of VIDEs, which are very difficult, if not impossible, to solve analytically. With the block approach, the non-self-starting nature associated with the predictor corrector method has been eliminated. Unlike the approach in the predictor corrector method where additional equations are supplied from a different formulation, all the additional equations are from the same continuous formulation which shows the beauty of the method. However, the absolute stability region showed that the method is A-stable, and the application of this method to practical problems revealed that the method is more accurate than earlier methods.

Details

Multidiscipline Modeling in Materials and Structures, vol. 14 no. 5
Type: Research Article
ISSN: 1573-6105

Keywords

Article
Publication date: 4 May 2010

Feyed Ben Zitoun and Yves Cherruault

The purpose of this paper is to present a method for solving nonlinear differential equations with constant and/or variable coefficients and with initial and/or boundary…

Abstract

Purpose

The purpose of this paper is to present a method for solving nonlinear differential equations with constant and/or variable coefficients and with initial and/or boundary conditions.

Design/methodology/approach

The method converts the nonlinear boundary value problem into a system of nonlinear algebraic equations. By solving this system, the solution is 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 us 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 differential equations.

Originality/value

The present approach provides a reliable technique, which avoids the tedious work needed by classical techniques and existing numerical methods. The nonlinear problem is solved without linearizing or discretizing the nonlinear terms of the equation. The method does not require physically unrealistic assumptions, linearization, discretization, perturbation, or any transformation in order to find the solutions of the given problems.

Details

Kybernetes, vol. 39 no. 4
Type: Research Article
ISSN: 0368-492X

Keywords

1 – 10 of 177