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1 – 10 of 133Imtiyaz 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.
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This paper aims to propose an efficient and convenient numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential…
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.
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Elçin Yusufoğlu and Barış Erbaş
This paper sets out to introduce a numerical method to obtain solutions of Fredholm‐Volterra type linear integral equations.
Abstract
Purpose
This paper sets out to introduce a numerical method to obtain solutions of Fredholm‐Volterra type linear integral equations.
Design/methodology/approach
The flow of the paper uses well‐known formulations, which are referenced at the end, and tries to construct a new approach for the numerical solutions of Fredholm‐Volterra type linear equations.
Findings
The approach and obtained method exhibit consummate efficiency in the numerical approximation to the solution. This fact is illustrated by means of examples and results are provided in tabular formats.
Research limitations/implications
Although the method is suitable for linear equations, it may be possible to extend the approach to nonlinear, even to singular, equations which are the future objectives.
Practical implications
In many areas of mathematics, mathematical physics and engineering, integral equations arise and most of these equations are only solvable in terms of numerical methods. It is believed that the method is applicable to many problems in these areas such as loads in elastic plates, contact problems of two surfaces, and similar.
Originality/value
The paper is original in its contents, extends the available work on numerical methods in the solution of certain problems, and will prove useful in real‐life problems.
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Jalil Rashidinia and Zahra Mahmoodi
The purpose of this paper is to develop a numerical method based on quintic B‐spline to solve the linear and nonlinear Fredholm and Volterra integral equations.
Abstract
Purpose
The purpose of this paper is to develop a numerical method based on quintic B‐spline to solve the linear and nonlinear Fredholm and Volterra integral equations.
Design/methodology/approach
The solution is collocated by quintic B‐spline and then the integral equation is approximated by the Gauss‐Kronrod‐Legendre quadrature formula.
Findings
The arising system of linear or nonlinear algebraic equations can solve the linear combination coefficients appearing in the representation of the solution in spline basic functions.
Practical implications
The error analysis of proposed numerical method is studied theoretically. Numerical results are given to illustrate the efficiency of the proposed method. The results are compared with the results obtained by other methods to verify that this method is accurate and efficient.
Originality/value
The paper provides new method to solve the linear and nonlinear Fredholm and Volterra integral equations.
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Lazhar Bougoffa, Manal Al‐Haqbani and Randolph C. Rach
In this paper, Fredholm integral equations of the first kind, the Schlomilch integral equation, and a class of related integral equations of the first kind with constant limits of…
Abstract
Purpose
In this paper, Fredholm integral equations of the first kind, the Schlomilch integral equation, and a class of related integral equations of the first kind with constant limits of integration are transformed in such a manner that the Adomian decomposition method (ADM) can be applied. Some examples with closed‐form solutions are studied in detail to further illustrate the proposed technique, and the results obtained indicate this approach is indeed practical and efficient. The purpose of this paper is to develop a new iterative procedure where the integral equations of the first kind are recast into a canonical form suitable for the ADM. Hence it examines how this new procedure provides the exact solution.
Design/methodology/approach
The new technique, as presented in this paper in extending the applicability of the ADM, has been shown to be very efficient for solving Fredholm integral equations of the first kind, the Schlomilch integral equation and a related class of nonlinear integral equations with constant limits of integration.
Findings
By using the new proposed technique, the ADM can be easily used to solve the integral equations of the first kind, the Schlomilch integral equation, and a class of related integral equations of the first kind with constant limits of integration.
Originality/value
The paper shows that this new technique is easy to implement and produces accurate results.
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Rahman Farnoosh and Ebrahimi Morteza
The purpose of this paper is to provide a Monte Carlo variance reduction method based on Control variates to solve Fredholm integral equations of the second kind.
Abstract
Purpose
The purpose of this paper is to provide a Monte Carlo variance reduction method based on Control variates to solve Fredholm integral equations of the second kind.
Design/methodology/approach
A numerical algorithm consisted of the combined use of the successive substitution method and Monte Carlo simulation is established for the solution of Fredholm integral equations of the second kind.
Findings
Owing to the application of the present method, the variance of the solution is reduced. Therefore, this method achieves several orders of magnitude improvement in accuracy over the conventional Monte Carlo method.
Practical implications
Numerical tests are performed in order to show the efficiency and accuracy of the present paper. Numerical experiments show that an excellent estimation on the solution can be obtained within a couple of minutes CPU time at Pentium IV‐2.4 GHz PC.
Originality/value
This paper provides a new efficient method to solve Fredholm integral equations of the second kind and discusses basic advantages of the present method.
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Keywords
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…
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.
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K. Maleknejad, M. Alizadeh and R. Mollapourasl
The purpose of this paper is to discuss a numerical method for solving Fredholm integral equations of the first kind with degenerate kernels and convergence of this numerical…
Abstract
Purpose
The purpose of this paper is to discuss a numerical method for solving Fredholm integral equations of the first kind with degenerate kernels and convergence of this numerical method.
Design/methodology/approach
By using sinc collocation method in strip, the authors try to estimate a numerical solution for this kind of integral equation.
Findings
Some numerical results support the accuracy and efficiency of the stated method.
Originality/value
The paper presents a method for solving first kind integral equations which are ill‐posed.
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Majeed Ahmed AL-Jawary, Ghassan Hasan Radhi and Jure Ravnik
In this paper, the exact solutions of the Schlömilch’s integral equation and its linear and non-linear generalized formulas with application are solved by using two efficient…
Abstract
Purpose
In this paper, the exact solutions of the Schlömilch’s integral equation and its linear and non-linear generalized formulas with application are solved by using two efficient iterative methods. The Schlömilch’s integral equations have many applications in atmospheric, terrestrial physics and ionospheric problems. They describe the density profile of electrons from the ionospheric for awry occurrence of the quasi-transverse approximations. The paper aims to discuss these issues.
Design/methodology/approach
First, the authors apply a regularization method combined with the standard homotopy analysis method to find the exact solutions for all forms of the Schlömilch’s integral equation. Second, the authors implement the regularization method with the variational iteration method for the same purpose. The effectiveness of the regularization-Homotopy method and the regularization-variational method is shown by using them for several illustrative examples, which have been solved by other authors using the so-called regularization-Adomian method.
Findings
The implementation of the two methods demonstrates the usefulness in finding exact solutions.
Practical implications
The authors have applied the developed methodology to the solution of the Rayleigh equation, which is an important equation in fluid dynamics and has a variety of applications in different fields of science and engineering. These include the analysis of batch distillation in chemistry, scattering of electromagnetic waves in physics, isotopic data in contaminant hydrogeology and others.
Originality/value
In this paper, two reliable methods have been implemented to solve several examples, where those examples represent the main types of the Schlömilch’s integral models. Each method has been accompanied with the use of the regularization method. This process constructs an efficient dealing to get the exact solutions of the linear and non-linear Schlömilch’s integral equation which is easy to implement. In addition to that, the accompanied regularization method with each of the two used methods proved its efficiency in handling many problems especially ill-posed problems, such as the Fredholm integral equation of the first kind.
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Mohsen Rabbani and Khosrow Maleknejad
The purpose of this paper is to use Alpert wavelet basis and modify the integrand function approximation coefficients to solve Fredholm‐Hammerstein integral equations.
Abstract
Purpose
The purpose of this paper is to use Alpert wavelet basis and modify the integrand function approximation coefficients to solve Fredholm‐Hammerstein integral equations.
Design/methodology/approach
L2[0, 1] was considered as solution space and the solution was projected to the subspaces of L2[0, 1] with finite dimension so that basis elements of these subspaces were orthonormal.
Findings
In this process, solution of Fredholm‐Hammerstein integral equation is found by solving the generated system of nonlinear equations.
Originality/value
Comparing the method with others shows that this system has less computation. In fact, decreasing of computations result from the modification.
Details