Kind Linear Fredholm Research Articles

Numerical Solution of the Fredholm and Volterra Integral Equations by Using Modified Bernstein–Kantorovich Operators

The main aim of this paper is to numerically solve the first kind linear Fredholm and Volterra integral equations by using Modified Bernstein–Kantorovich operators. The unknown function in the first kind integral equation is approximated by using the Modified Bernstein–Kantorovich operators. Hence, by using discretization, the obtained linear equations are transformed into systems of algebraic linear equations. Due to the sensitivity of the solutions on the input data, significant difficulties may be encountered, leading to instabilities in the results during actualization. Consequently, to improve on the stability of the solutions which imply the accuracy of the desired results, regularization features are built into the proposed numerical approach. More stable approximations to the solutions of the Fredholm and Volterra integral equations are obtained especially when high order approximations are used by the Modified Bernstein–Kantorovich operators. Test problems are constructed to show the computational efficiency, applicability and the accuracy of the method. Furthermore, the method is also applied to second kind Volterra integral equations.

Analysis of Polynomial Collocation and Uniformly Spaced Quadrature Methods for Second Kind Linear Fredholm Integral Equations – A Comparison

This work aims to compare the polynomial collocation method against the uniformly spaced quadrature methods (Trapezoidal, Simpson’s and Weddle’s rules) to solve the non-homogeneous linear Fredholm integral equations of second kind with non-singular kernel. Nystrom’s interpolation technique is used to attain closed form approximations of the solutions obtained by quadrature methods. The formulation and implementation of the methods along with application on various test problems are presented for comparison. The results obtained, in general, highlight some cases where polynomial collocation yields ill-conditioned systems after discretization with same basis set, and reflects the suitability of quadrature rules. The main focus has been to extract exact closed form solutions by the used methods which can be used as replacement of analytical solutions.

Erratum to: Fast iterative refinement method for mixed systems of integral and fractional integro-differential equations

The authors’ new method of iterative refinement has been successfully applied to the linear fractional Fredholm integro-differential equation. In this work, we adapt our method to obtain approximate analytical solutions for linear Volterra equations of both the first and the second kind of fractional type. A detailed convergence analysis is presented for each kind. The authors also apply their method to the challenging first kind linear Fredholm integral equation. Three different cases are considered to test the efficacy of our method. These include mixed systems of various forms of linear integral and fractional integro-differential equations. We compare our method with six well-established methods: Picard’s successive approximations, an accelerated form of the latter, the Adomian decomposition, the variational iteration, the homotopy perturbation and the homotopy analysis. The results show the versatility of the new method in solving a wide variety of equations, its high convergence speed and accuracy and its capability of directly dealing with equations of the first kind. We also prove that Picard’s method for linear equations is a special case of our method.

Numerical performance of Half-Sweep Geometric Mean (HSGM) iterative method for solving third order Newton–Cotes quadrature system

The main aim of this paper is to examine the performance of Half-Sweep Geometric Mean (HSGM) iterative method for solving large, dense, third order Newton–Cotes quadrature system associated with numerical solutions of second kind linear Fredholm integral equations. The formulation and implementation of the method are presented. Some numerical analysis are included to verify the efficiency of the method.

Solving second kind linear Fredholm integral equations via Quarter-Sweep SOR iterative method

In this paper, we consider the numerical solutions of linear Fredholm integral equations of the second kind. The Quarter-Sweep Successive Over-Relaxation (QSSOR) iterative method is applied to solve linear systems generated from discretization of the second kind linear Fredholm integral equations using quadrature method. In addition, the formulation and implementation of the proposed method to solve the problem are also presented. Numerical tests and comparisons with other existing methods are given to illustrate the effectiveness of the proposed method.

A Jacobi-collocation method for solving second kind Fredholm integral equations with weakly singular kernels

In this work, we propose a Jacobi-collocation method to solve the second kind linear Fredholm integral equations with weakly singular kernels . Particularly, we consider the case when the underlying solutions are sufficiently smooth. In this case, the proposed method leads to a fully discrete linear system. We show that the fully discrete integral operator is stable in both infinite and weighted square norms. Furthermore, we establish that the approximate solution arrives at an optimal convergence order under the two norms. Finally, we give some numerical examples, which confirm the theoretical prediction of the exponential rate of convergence.

Utilizing feed-back neural network approach for solving linear Fredholm integral equations system

This paper intended to offer an architecture of artificial neural networks (NNs) for finding approximate solution of a second kind linear Fredholm integral equations system. For this purpose, first we substitute the N-th truncation of the Taylor expansion for unknown functions in the origin system. By applying the suggested neural network for adjusting the real coefficients of given expansions in resulting system. The proposed NN is a two-layer feed-back neural network such that it can get a initial vector and then calculates it’s corresponding output vector. In continuance, a cost function is defined by using output vector and the target outputs. Consequently, the reported NN using a learning algorithm that based on the gradient descent method, will adjust the coefficients in given Taylor series. Eventually, we have showed this method in comparison with existing numerical methods such as trapezoidal quadrature rule provides solutions with good generalization and high accuracy. The proposed method is illustrated by several examples with computer simulations.

Half-Sweep Geometric Mean Iterative Method for the Repeated Simpson Solution of Second Kind Linear Fredholm Integral Equations

In previous studies, the effectiveness of the Half-Sweep Geometric Mean (HSGM) iterative method has been shown in solving first and second kind linear Fredholm integral equations using repeated trape­zoidal (RT) discretization scheme. In this work, we investigate the efficiency of the HSGM method to solve dense linear system gener­ated from the discretization of the second kind linear Fredholm in­tegral equations by using repeated Simpson's 1/3(RS1) scheme. The formulation and implementation ofthe proposed method are also pre­sented. In addition, several numerical simulations and computational complexity analysis were also included to verify the efficiency of the proposed method.

Computational methods based on complexity reduction approach for first kind linear Fredholm integral equations with semi-smooth kernel

The main aim of this paper is to investigate the performance a family of Gauss-Seidel (GS) iterative method consists of Full-Sweep Gauss-Seidel (FSGS), Half-Sweep Gauss-Seidel (HSGS) and Quarter-Sweep Gauss-Seidel (QSGS) methods in solving linear systems associated with the first kind linear Fredholm integral equations. The details of the proposed methods are explained. Some numerical analyses are given in order to verify the performance of the proposed methods.

Numerical Solutions of Linear Fredholm Integral Equations Using Half-Sweep Arithmetic Mean Method

The main aim of this paper is to examine the effectiveness of the Half-Sweep Arithmetic Mean (HSAM) iterative method in solving dense linear systems generated from the discretization of the first and second kind linear Fredholm integral equations. The formulation and implementation of the proposed method is also presented. Numerical tests and comparisons with other existing methods are given to illustrate the effectiveness of the proposed method.

Comparisons of Quadrature Schemes With Arithmetic Mean Iterative Method For Second Kind Linear Fredholm Integral Equations

In this paper, we investigate the applications of two different quadrature schemes that is repeated trapezoidal (RT) and repeated modified trapezoidal (RMT) schemes via Arithmetic Mean iterative method to solve second kind linear Fredholm integral equations. Furthermore, the derivation and implementation of the proposed method are also included. Numerical tests and comparisons are given to illustrate the effectiveness of the proposed method.