Linear Fredholm Integro-differential Equations Research Articles

A performance assessment of an HDG method for second-order Fredholm integro-differential equation: existence-uniqueness and approximation

In this paper, we obtain the existence--uniqueness of solution to the second-order linear Fredholm integro-differential equation (FIDE) with Dirichlet boundary condition by hybridizable discontinuous Galerkin (HDG) method. A key property of these methods is to eliminate all internal degrees of freedom and to construct a linear system that only includes globally coupled unknowns at the element interfaces. After designing and implementing HDG algorithm, we provide some necessary and sufficient conditions based on the stabilization parameter and kernel function to guarantee the existence-uniqueness of the approximate solution. Then, some numerical examples are carried out to assess the performance of the present method. When comparing with existing some methods in literature, the experimental studies verify the reliability and feasibility of the HDG method for the problem under consideration.

The Fractional Residual Power Series Method for Solving a System of Linear Fractional Fredholm Integro-differential Equations

In this manuscript, the fractional residual power series (FRPS) method is employed in solving a system of linear fractional Fredholm integro-differential equations. The significant role of this system in various fields has attracted the attention of researchers for a decade. The definition of fractional derivative here is described in the Caputo sense. The proposed method relies on the generalized Taylor series expansion as well as the fact that the fractional derivative of stationary is zero. The process starts by constructing a residual function by supposing the finite order of an approximate power series solution that prescribes the initial conditions. Then, utilizing some conditions, the residual functions are converted to a linear system for the power series coefficients. Solving the linear system reveals the coefficients of the fractional power series solution. Finally, by substituting these coefficients into the supposed form of a solution, the approximate fractional power series solutions are derived. This technique has the advantage of being able to be applied directly to the problem and spending less time on computation. It is not only an easy method for implementation of the problem, but also provides productive results after a few iterations. Some problems with known solutions emphasize the procedure's simplicity and reliability. Moreover, the obtained exact solution demonstrated the efficiency and accuracy of the presented method.

Solving linear Fredholm integro-differential equation by Nyström method

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

AN IMPROVED APPROACH FOR SOLUTIONS OF SYSTEMS OF LINEAR FREDHOLM INTEGRO DIFFERENTIAL EQUATIONS

In this paper, a numerical matrix method is used to solve the systems of high-order linear Fredholm integro-differential equations with variable coefficients under mixed conditions. The technique consists of collocation points and the Morgan-Voyce polynomials. The residual error functions of numerical solutions of the method are also presented. Firstly, the approximate solutions are formed and secondly, an error problem is constituted in favor of the residual error function. The numerical solutions are computed for this error problem by using the present method. The approximate solutions of the original problem and the error problem are the corrected Morgan-Voyce polynomial solutions. When the exact solutions of the problem are not known, the absolute errors can be approximately constructed through the approximate solutions of the error problem. Numerical examples are included to demonstrate the validity and the applicability of the technique, and also the results are compared with the different methods. All numerical computations have been performed using MATLAB v7.11.0 (R2010b).

Numerical Solution for Linear Fredholm Integro-Differential Equation Using Touchard Polynomials

A new method based on the Touchard polynomials (TPs) was presented for the numerical solution of the linear Fredholm integro-differential equation (FIDE) of the first order and second kind with condition. The derivative and integration of the (TPs) were simply obtained. The convergence analysis of the presented method was given and the applicability was proved by some numerical examples. The results obtained in this method are compared with other known results.

The Numerical Solutions of FIEs of the Second Kind of Degenerated Type Using Bownd’s Methods

In this work, we consider the numerical solutions of some types of Fredholm integro-differential equations (FIDEs). Namely, Bownd’s, successive, Adomian’s and the improved methods are used to solve linear Fredholm integro-differential equations (FIDEs). In addition, some numerical examples are given to show the efficiency and accuracy of the proposed methods. The numerical results show that using the improved methods is the best technique to solve such types of problems.

Theoretical model for parallel SQUID arrays with fluxoid focusing

We have developed a comprehensive theoretical model for predicting the magnetic field response of a parallel SQUID array in the voltage state. The model predictions are compared with our experimental data from a parallel SQUID array made of a YBCO thin-film patterned into wide tracks, busbars and leads, with eleven step-edge Josephson junctions. Our theoretical model uses the Josephson equations for resistively shunted junctions as well as the second Ginzburg-Landau equation to derive a system of coupled first-order nonlinear differential equations to describe the time-evolution of the Josephson junction phase differences which includes Johnson noise. Employing the second London equation and Biot-Savart's law, the supercurrent density distribution is calculated, using the stream function approach, which leads to a 2D second-order linear Fredholm integro-differential equation for the stream function with time-dependent boundary conditions. The novelty of the model is that it calculates the stream function everywhere in the thin-film structure to determine during the time-evolution the fluxoids for each SQUID array hole. Our numerical model calculations are compared with our experimental data and predict the bias-current versus voltage and the voltage versus magnetic field response with unprecedented accuracy. The model elucidates the importance to fully take Meissner shielding and current crowding into account in order to properly describe fluxoid focussing and bias-current injection. Furthermore, our model illustrates the failure of the simple lumped-element approach to describe a parallel SQUID array with a wide thin-film structure.

Numerical Solution of Fractional-Order Fredholm Integrodifferential Equation in the Sense of Atangana–Baleanu Derivative

In the present article, our aim is to approximate the solution of Fredholm-type integrodifferential equation with Atangana–Baleanu fractional derivative in Caputo sense. For this, we propose a method based on Laplace transform and inverse LT. In our numerical scheme, the given equation is transformed to an algebraic equation by employing the Laplace transform. The reduced equation will be solved in complex plane. Finally, the solution of the given problem is obtained via inverse Laplace transform by representing it as a contour integral. Then, the trapezoidal rule is used to approximate the integral to high accuracy. We have considered linear and nonlinear fractional Fredholm integrodifferential equations to validate our method.

A Robust Numerical Method for a Singularly Perturbed Fredholm Integro-Differential Equation

In this paper, we deal with a fitted second-order homogeneous (non-hybrid) type difference scheme for solving the singularly perturbed linear second-order Fredholm integro-differential equation. The numerical method represents the exponentially fitted scheme on the Shishkin mesh. Numerical example is presented to demonstrate efficiency of proposed method.

Runge-Kutta Fehlberg Method for Solving Linear and Nonlinear Fuzzy Fredholm Integro-Differential Equations

Runge-Kutta Fehlberg Method for Solving Linear and Nonlinear Fuzzy Fredholm Integro-Differential Equations

BOOLE COLLOCATION METHOD BASED ON RESIDUAL CORRECTION FOR SOLVING LINEAR FREDHOLM INTEGRO-DIFFERENTIAL EQUATION

In this study, a Boole polynomial based method is presented for solving the linear Fredholm integro-differential equation approximately. In this method, the given problem is reduced to a matrix equation. The solution of the obtained matrix equation is found by using Boole polynomial, its derivatives and collocation points. This solution is obtained as the truncated Boole series which are defined in the interval [a,b]. In order to demonstrate the validity and applicability of the method, numerical examples are included. Also, the error analysis related with residual function is performed and the found approximate solutions are calculated.

Efficient spectral-collocation methods for a class of linear Fredholm integro-differential equations on the half-line

In this paper, an extension of the Legendre spectral collocation method has been proposed for the numerical solution of a class of linear Fredholm integro-differential equation on the half-line. The properties of mapped Legendre functions are first presented. These properties together with the Legendre–Gauss points are then utilized to reform the Fredholm integro-differential equation in semi-infinite interval into a singular equation in finite interval and to reduce it to the solution of a simple matrix equation. Besides, in order to show the efficiency and accuracy of the proposed method, some numerical examples are considered and solved through a survey of three approaches, namely: Exponential, rational and logarithmic Legendre functions collocation methods. Furthermore, a comparison of the results, shows that using exponential functions, leads to more accurate results and faster convergence.

Exponential spline for the numerical solutions of linear Fredholm integro-differential equations

In this paper, we introduce a new scheme based on the exponential spline function for solving linear second-order Fredholm integro-differential equations. Our approach consists of reducing the problem to a set of linear equations. We prove the convergence analysis of the method applied to the solution of integro-differential equations. The method is described and illustrated with numerical examples. The results reveal that the method is accurate and easy to apply. Moreover, results are compared with the method in (J. Comput. Appl. Math. 290:633–640, 2015).

Bell Polynomial Approach for the Solutions of Fredholm Integro-Differential Equations with Variable Coefficients

In this article, we approximate the solution of high order linear Fredholm integro-differential equations with a variable coefficient under the initial-boundary conditions by Bell polynomials. Using collocation points and t... | Find, read and cite all the research you need on Tech Science Press

On solving linear Fredholm integro-differential equations via finite difference-Simpson's approach

On solving linear Fredholm integro-differential equations via finite difference-Simpson's approach

A fast multiscale Galerkin method for solving second order linear Fredholm integro-differential equation with Dirichlet boundary conditions

In this paper, a fast multiscale Galerkin method is developed for solving second order linear Fredholm integro-differential equation with Dirichlet boundary conditions. The method is based on a matrix truncation strategy which leads to generating coefficient matrix rapidly. We prove that the method is stable and has an optimal convergence order and nearly linear computational complexity (up to a logarithmic factor). Numerical examples are presented to illustrate its computational efficiency, approximation accuracy and theoretical results, and to compare the computed results with those of the original multiscale Galerkin method proposed recently by the same authors.

Numerical computation of fractional Fredholm integro-differential equation of order 2β arising in natural sciences

This article investigates the approximate solutions to a class of fractional linear Fredholm integro-differential equations arising in physical phenomena based upon the use of an effective treatment technique, called the residual power series (RPS) technique. This approach relies on the generalized Taylor formula as well as the residual error function under the sense of Caputo, with the aim of deriving a supportive analytical solution in the form of a rapidly convergent fractional power series with easily computable components. The RPS algorithm is easy to implement without linearization, limitations or restrictions on the problem’s nature and the number of grid points. To justify the efficiency and accuracy of the proposed technique, an illustrative example is given. The results obtained indicate that the algorithm is simple, accurate, and powerful to solve such equations.

Usage of the Variational Iteration Technique for Solving Fredholm Integro-Differential Equations

Integral and integro-differential equations are one of the most useful mathematical tools in both pure and applied mathematics. In this article, we present a variational iteration method for solving Fredholm integro-differential equations. This study provides an analytical approximation to determine the behavior of the solution. To show the efficiency of the present method for our problems in comparison with the exact solution we report the absolute error. From the computational viewpoint, the variational iteration method is more efficient, convenient and easy to use. The method is very powerful and efficient in nding analytical as well as numerical solutions for wide classes of linear and nonlinear Fredholm integro-differential equations. Moreover, It proves the existence and uniqueness results and convergence of the solution of Fredholm integro-differential equations. Finally, some examples are included to demonstrate the validity and applicability of the proposed technique. The convergence theorem and the numerical results establish the precision and efficiency of the proposed technique.

Solving Fractional Fredholm Integro-Differential Equations by Laguerre Polynomials

The main purpose of this study was to present an approximation method based on the Laguerre polynomials to obtain the solutions of the fractional linear Fredholm integro-differential equations. This method transforms the integro-differential equation to a system of linear algebraic equations by using the collocation points. In addition, the matrix relation for Caputo fractional derivative of Laguerre polynomials is also obtained. Besides, some examples are presented to illustrate the accuracy of the method and the results are discussed.

Laguerre approach for solving system of linear Fredholm integro-differential equations

A numerical scheme has been developed for solving the system of linear Fredholm integro-differential equations subject to the mixed conditions using Laguerre polynomials. Using collocation method, the system of Fredholm integro-differential equations has been transformed to the system of linear equations in unknown Laguerre coefficients, which leads to the solution in terms of Laguerre polynomials. Moreover, the accuracy and applicability of the scheme have been compared with Tau method and Adomian decomposition method that reveals the proposed scheme to be more efficient.