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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.

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.

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.

The paper shows that this new technique is easy to implement and produces accurate results.

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.

The solution is collocated by quintic B‐spline and then the integral equation is approximated by the Gauss‐Kronrod‐Legendre quadrature formula.

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.

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.

The paper provides new method to solve the linear and nonlinear Fredholm and Volterra integral equations.

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.

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.

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.

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.

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.

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.

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.

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.

By using sinc collocation method in strip, the authors try to estimate a numerical solution for this kind of integral equation.

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

The paper presents a method for solving first kind integral equations which are ill‐posed.

The contactless inductive flow tomography is a procedure that enables the reconstruction of the global three-dimensional flow structure of an electrically conducting fluid by measuring the flow-induced magnetic flux density outside the melt and by subsequently solving the associated linear inverse problem. The purpose of this study is to improve the accuracy of the computation of the forward problem, since the forward solution primarily determines the accuracy of the inversion.

The tomography procedure is described by a system of coupled integral equations where the integrals contain a singularity when a source point coincides with a field point. The integrals need to be evaluated to a high degree of precision to establish an accurate foundation for the inversion. The contribution of a singular point to the value of the surface and volume integrals in the system is determined by analysing the behaviour of the fields and integrals in the close proximity of the singularity.

A significant improvement of the accuracy is achieved by applying higher order elements and by attributing special attention to the singularities inherent in the integral equations.

The contribution of a singular point to the value of the surface integrals in the system is dependent upon the geometry of the boundary at the singular point. The computation of the integrals is described in detail and the improper surface and volume integrals are shown to exist. The treatment of the singularities represents a novelty in the contactless inductive flow tomography and is the focal point of this investigation.

This paper seeks to present an original method for transforming multiple integrals into simple integrals.

This can be done by using α‐dense curves invented by Y. Cherruault and A. Guillez at the beginning of the 1980s.

These curves allow one to approximate the space Rⁿ (or a compact of Rⁿ) with the accuracy α. They generalize fractal curves of Mandelbrobdt. They can be applied to global optimization where the multivariables functional is transformed into a functional depending on a single variable.

Applied to a multiple integral, the α‐dense curves using Chebyshev's kernels permit one to obtain a simple integral approximating the multiple integral. The accuracy depends on the choice of α.

The paper presents an original method for transforming integrals into simple integrals.

This paper presents a modification of the classical boundary integral equation method (BIEM) for two‐dimensional potential boundary‐value problem. The proposed modification consists in describing the boundary geometry by means of Hermite curves. As a result of this analytical modification of the boundary integral equation (BIE), a new parametric integral equation system (PIES) is obtained. The kernels of these equations include the geometry of the boundary. This new PIES is no longer defined on the boundary, as in the case of the BIE, but on the straight line for any given domain. The solution of the new PIES does not require boundary discretization as it can be reduced merely to an approximation of boundary functions. To solve this PIES a pseudospectral method has been proposed and the results obtained compared with exact solutions.

This paper presents a boundary element method (BEM) based on a subdomain approach for the solution of non‐Newtonian fluid flow problems which include thermal effects and viscous dissipation. The volume integral arising from non‐linear terms is converted into equivalent boundary integrals by the multi‐domain dual reciprocity method (MD‐DRM) in each subdomain. Augmented thin plate splines interpolation functions are used for the approximation of field variables. The iterative numerical formulation is achieved by viewing the material as divided into small elements and on each of them the integral representation formulae for the velocity and temperature are applied and discretised using linear boundary elements. The final system of non‐linear algebraic equations is solved by a modified Newton's method. The numerical examples include non‐Newtonian problems with viscous dissipation, temperature‐dependent viscosity and natural convection due to bouyancy forces.

The purpose of this paper is to supply a numerical analysis tool to solve eddy currents induced in nonlinear materials such as steel by boundary element method (BEM), and then apply it to design and analysis of power devices.

Utilizing integral formulas derived on the basis of rapid attenuation of the electromagnetic fields, the paper formulates eddy currents in steel. In the formulation, nonlinear terms are regarded as virtual sources, which are improved iteratively with the electromagnetic fields on the surface. The periodic electromagnetic fields are expanded in Fourier series and each harmonic is analyzed by BEM. The surface and internal electromagnetic fields are obtained numerically one after the other until convergence by the Newton‐Raphson method.

It is confirmed that this approach gives accurate solutions with meshes much larger than the skin depth and therefore is adequate to apply to a large‐scale application.

The eddy current is formulated by utilizing the impedance boundary condition in order to meet a large‐scale application, and so solutions near the edge are poor. In the case of better solutions being required, some modifications are necessary.

To lessen computer memory consumption, the parallel component of the currents to the steel surface is analyzed as a 2D problem and the normal component is obtained from the parallel component. One 2D equation for one analyzing region is discretized by dividing the region into layers adaptively and then solved. Next, another is solved sequentially. This method gives a compatible numerical analysis tool with finite element method.

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.

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.

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.

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.