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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.

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.

Numerical experiments shows that the proposed two-dimensional wavelets technique can give high-accurate solutions and good convergence rate.

The efficiency of newly developed two-dimensional wavelets technique has been validated by different illustrative numerical examples to solve two-dimensional Fredholm integral equations.

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.

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.

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.

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

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).

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.

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.

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.

The purpose of this study is to use the method of lines to solve the two-dimensional nonlinear advection–diffusion–reaction equation with variable coefficients.

The strictly positive definite radial basis functions collocation method together with the decomposition of the interpolation matrix is used to turn the problem into a system of nonlinear first-order differential equations. Then a numerical solution of this system is computed by changing in the classical fourth-order Runge–Kutta method as well.

Several test problems are provided to confirm the validity and efficiently of the proposed method.

For the first time, some famous examples are solved by using the proposed high-order technique.

The purpose of this paper is to develop pseudospectral meshless radial point Hermit interpolation (PSMRPHI) for applying to the Motz problem.

The author aims to propose a kind of PSMRPHI method.

Based on the Motz problem, the author aims also to compare PSMRPHI and PSMRPI which belong to more influence type of meshless methods.

Although the PSMRPHI method has been infrequently used in applications, the author proves it is more accurate and trustworthy than the PSMRPI method.

This paper discusses an electrostatic, homogeneous field in a uniform two‐dimensional domain with Neumann's boundary conditions. The boundary conditions are known only at some segments of the boundary. The synthesis is understood as the computation of the remaining boundary conditions which would ensure the required potential distribution in some subdomains within the boundary. The introduction of a single‐layer potential leads to Fredholm's equation of the second order. Stepwise approximation of the source distribution along the boundary rearranges Fredholm's equation and the requirements concerning the single layer potential distribution. It leads to a matrix equation with a rectangular coefficient matrix. In order to solve approximately this equation, in the sense of the least squares minimization, the singular value decomposition (SVD) method is used. The choice of subdomains with determined potential distribution influences significantly the conditioning of the equation. Easy selection of an acceptable solution among all possible solutions proves the suitability of the SVD method in the above problem. The numerical experiments reported in the paper are a good illustration of this.

The main objective is to develop an efficient BEM scheme for the numerical solution of two‐dimensional heat problems. Our scheme will be of the re‐initialization type, in which the domain integrals are computed by a recursion relation which depends only on the boundary temperature and flux at previous time step. To obtain the re‐initialization approach, we will use in the integral representation formula a Green function corresponding to zero temperature in a box containing the original domain, instead of using the classical free space fundamental solution. This Green function is given in terms of the original fundamental solution plus a regular solution of the heat equation inside the domain under consideration. It can therefore be used in the integral representation formula of the heat equation (direct formulation) to obtain the solution of a heat problem in such a domain. The Green function mentioned can be obtained by the images method, and the resulting source series can also be rewritten in terms of a double Fourier series, that we will use in the domain integral of the integral representation formula to transform such integral into equivalent surface integrals.

The purpose of this study is to obtain a scheme for the numerical solution of Volterra integro-differential equations with time periodic coefficients deduced from the charged particle motion for certain configurations of oscillating magnetic fields.

The method reduces the solution of these types of integro-differential equations to the solution of two-dimensional Volterra integral equations of the second kind. The new method uses the discrete collocation method together with thin plate splines constructed on a set of scattered points as a basis.

The scheme can be easily implemented on a computer and has a computationally attractive algorithm. Numerical examples are included to show the validity and efficiency of the new technique.

The author uses thin plate splines as a type of free-shape parameter radial basis functions which establish an effective and stable method to solve electromagnetic integro-differential equations. As the scheme does not need any background meshes, it can be identified as a meshless method.