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We consider a system of singular integral equations whose kernels contain solutionsof transmission problems. We construct the general solution of this system by quadratures.
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In this work, we generalize the numerical method discussed in [Z. Avazzadeh, M. Heydari, G.B. Loghmani, Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem, Appl. math, modelli...
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In this work, we generalize the numerical method discussed in [Z. Avazzadeh, M. Heydari, G.B. Loghmani, Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem, Appl. math, modelling, 35 (2011) 2374-2383] for solving linear and nonlinear Fredholm integral and integro-differential equations of the second kind. The presented method can be used for solving integral equations in high dimensions. In this work, we describe the integral mean value method (IMVM) as the technical algorithm for solving high dimensional integral equations. The main idea in this method is applying the integral mean value theorem. However the mean value theorem is valid for multiple integrals, we apply one dimensional integral mean value theorem directly to fulfill required linearly independent equations. We solve some examples to investigate the applicability and simplicity of the method. The numerical results confirm that the method is efficient and simple.
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Simple and efficient numerical procedures using singularity cancellation methods are presented for evaluating singular and near-singular potential integrals. Four different transformations are compared and the advantages of the ra...
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Simple and efficient numerical procedures using singularity cancellation methods are presented for evaluating singular and near-singular potential integrals. Four different transformations are compared and the advantages of the radial-angular transform are demonstrated. A method is then described for optimizing this integration scheme.
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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 ma...
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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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We consider the linear Volterra equation x.t/ D a.t/ Z t 0 K.t; s/ x.s/ ds and suppose that the kernel K and forcing function a depend on some parameters 2 Rd. We prove that, under suitable conditions, the solutions depend on as s...
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We consider the linear Volterra equation x.t/ D a.t/ Z t 0 K.t; s/ x.s/ ds and suppose that the kernel K and forcing function a depend on some parameters 2 Rd. We prove that, under suitable conditions, the solutions depend on as smoothly the functions a and K. The proof is based on the contraction mapping principle and the variational equation. Though our conditions are not the most generally possible, they nonetheless include many important examples.
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In this paper we study the solvability of a singular integral equation arising in the theory of boundary value problems for the heat equation in an infinite angular domain. The particular case of the corresponding homogeneous inte...
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In this paper we study the solvability of a singular integral equation arising in the theory of boundary value problems for the heat equation in an infinite angular domain. The particular case of the corresponding homogeneous integral equation was investigated earlier in [1, 2] and it was shown that in a weight class of essentially bounded functions it has, along with a trivial solution, a family of non-trivial solutions up to a constant factor. In this paper we study the more general case of a nonhomogeneous integral equation for which a representation of the general solution is found with using the resolvent constructed by us. Estimates of the resolvent and of the solution of the boundary value problem are established. (C) 2021 Institute of Mathematics and Mathematical Modeling, Ministry of Education and Science of Republic of Kazakhstan. Published by Elsevier Inc.
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We demonstrate the use of the modified decomposition method for the analytic treatment of non-linear Fredholm integral equations, non-linear Volterra integral equations and systems of non-linear integral equations. The proper impl...
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We demonstrate the use of the modified decomposition method for the analytic treatment of non-linear Fredholm integral equations, non-linear Volterra integral equations and systems of non-linear integral equations. The proper implementation of the modified method can dramatically reduce the amount of work required and may provide the exact solution using only a few iterations. The analysis is accompanied by numerical illustrations that show the pertinent features of the technique.
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One of the methods for solving definite integrals is modified trapezoid method, which is obtained by using Hermit interpolation [J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Second Edition, Springer-Verlag, 1993]. In...
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One of the methods for solving definite integrals is modified trapezoid method, which is obtained by using Hermit interpolation [J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Second Edition, Springer-Verlag, 1993]. In this article we intend to make a quadrature method for solving the linear integral equations such as repeated trapezoid and repeated Simpson quadrature by using repeated modified trapezoid formula and by doing so, we solve the linear integral equations more accurately. For further information on quadrature methods see [L.M. Delves, J.L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, 1985] and [L.M. Delves, J. Walsh, Numerical Solution of Integral Equations, Oxford University Press, 1974].
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Abstract In this article, a numerical method for the solution of neutrosophic Fredholm integral equation has been investigated. In addition, the neutrosophic Fredholm integral equation has been presented in the sense of (α,β,γ)...
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Abstract In this article, a numerical method for the solution of neutrosophic Fredholm integral equation has been investigated. In addition, the neutrosophic Fredholm integral equation has been presented in the sense of (α,β,γ)-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,\beta ,\gamma )-$$\end{document}cut using Riemann integration approach. Some basic properties of neutrosophic calculus such as neutrosophic integral, neutrosophic continuity have been introduced. An iterative method has been modified in neutrosophic environment to find the numerical solution of Fredholm integral equation of second kind. The convergence of the iterative method in neutrosophic environment has been demonstrated in terms of some theorems. In the iterative method, trapezoidal rule has been used to evaluate the integral and find the approximate solution of the equation. In addition, the convergence of the trapezoidal approximations has been provided in terms of theorem. The algorithm of the proposed method has been given in the numerical method section, which briefly helps to understand the proposed method. A comparison of our method with other existing methods has been discussed to show the efficiency and reliability of our proposed method. In addition, a brief discussion about the advantages and limitations of our method has been provided. Some numerical examples have been examined to show the validation and effectiveness of the proposed method. In addition, different types of error analysis have been investigated by providing different types of tables and figures.
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In this paper, several integral equations are solved by He's variational iteration method in general case, then we consider the convergence of He's variational iteration method for solving integral equations.
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