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1 – 10 of 44Jalil 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.
Details
Keywords
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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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.
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Keywords
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.
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Godwin Amechi Okeke and Daniel Francis
The authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modular G-metric spaces. The authors…
Abstract
Purpose
The authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modular G-metric spaces. The authors apply the results in solving nonlinear Volterra-Fredholm-type integral equations. The results extend generalize compliment and include several known results as special cases.
Design/methodology/approach
The results of this paper are theoretical and analytical in nature.
Findings
The authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modular G-metric spaces. apply the results in solving nonlinear Volterra-Fredholm-type integral equations. The results extend, generalize, compliment and include several known results as special cases.
Research limitations/implications
The results are theoretical and analytical.
Practical implications
The results were applied to solving nonlinear integral equations.
Social implications
The results has several social applications.
Originality/value
The results of this paper are new.
Details
Keywords
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
Keywords
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 u‐Nu = f…
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
Keywords
Farshid Mirzaee and Nasrin Samadyar
The purpose of this paper is to develop a new method based on operational matrices of Bernoulli wavelet for solving linear stochastic Itô-Volterra integral equations, numerically.
Abstract
Purpose
The purpose of this paper is to develop a new method based on operational matrices of Bernoulli wavelet for solving linear stochastic Itô-Volterra integral equations, numerically.
Design/methodology/approach
For this aim, Bernoulli polynomials and Bernoulli wavelet are introduced, and their properties are expressed. Then, the operational matrix and the stochastic operational matrix of integration based on Bernoulli wavelet are calculated for the first time.
Findings
By applying these matrices, the main problem would be transformed into a linear system of algebraic equations which can be solved by using a suitable numerical method. Also, a few results related to error estimate and convergence analysis of the proposed scheme are investigated.
Originality/value
Two numerical examples are included to demonstrate the accuracy and efficiency of the proposed method. All of the numerical calculation is performed on a personal computer by running some codes written in MATLAB software.
Details