Monte Carlo method for solving Fredholm integral equations of the second kind

Farnoosh, Rahman; Ebrahimi, Morteza

doi:10.1016/j.amc.2007.04.097

Cited by 47 publications

(24 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Consider the second-kind linear Fredholm integral equation of the form the f (x) and K(x, t) are known functions and y(x) is the unknown function that is to be determined. This type of equations has been solved in many papers with many different methods [1][2][3][4][5][6]. Wavelet bases have been used recently which, primarily because of their local supports and vanishing moment properties, lead to a sparse matrix.…”

Section: Introductionmentioning

confidence: 99%

Low memory requirement and computational time method for solving a class of integral equations

Maleknejad

Sahlan

Torabi

2011

Procedia Computer Science

View full text Add to dashboard Cite

In this work, we present a new low memory requirement and low computational time method for solving a class of integral equation of the second kind which is based on the use of B-Spline wavelets. Because of vanishing moments and compact support and semiorthogonality properties of these wavelets, operational matrix of the method is very sparse. Also, applying some appropriate thresholding parameter and GMRES method for solving sparse systems, we get a computationally attractive method with low CPU requirement. For showing the accuracy of the method some numerical method are presented.

show abstract

Section: Introductionmentioning

confidence: 99%

Low memory requirement and computational time method for solving a class of integral equations

Maleknejad

Sahlan

Torabi

2011

Procedia Computer Science

View full text Add to dashboard Cite

show abstract

“…Iterative methods based on discretization and projection techniques are presented in [Borzabadi, Fard, 2009;Javadi, 2014]. Random search algorithm is presented in [Farnoosh, Ebrahimi, 2008], Newton-Kantorovich-quadrature method is presented in [Saberi-Nadjafi, Heidari, 2010], etc.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of Urysohn type nonlinear second kind integral equations by successive quadratures using embedded Dormand and Prince scheme 5(4)

Maglevanny¹,

Karyakina²

2020

CRM

View full text Add to dashboard Cite

We present the iterative algorithm that solves numerically both Urysohn type Fredholm and Volterra nonlinear one-dimensional nonsingular integral equations of the second kind to a specified, modest user-defined accuracy. The algorithm is based on descending recursive sequence of quadratures. Convergence of numerical scheme is guaranteed by fixed-point theorems. Picard's method of integrating successive approximations is of great importance for the existence theory of integral equations but surprisingly very little appears on numerical algorithms for its direct implementation in the literature. We show that successive approximations method can be readily employed in numerical solution of integral equations. By that the quadrature algorithm is thoroughly designed. It is based on the explicit form of fifth-order embedded Runge-Kutta rule with adaptive step-size self-control. Since local error estimates may be cheaply obtained, continuous monitoring of the quadrature makes it possible to create very accurate automatic numerical schemes and to reduce considerably the main drawback of Picard iterations namely the extremely large amount of computations with increasing recursion depth. Our algorithm is organized so that as compared to most approaches the nonlinearity of integral equations does not induce any additional computational difficulties, it is very simple to apply and to make a program realization. Our algorithm exhibits some features of universality. First, it should be stressed that the method is as easy to apply to nonlinear as to linear equations of both Fredholm and Volterra kind. Second, the algorithm is equipped by stopping rules by which the calculations may to considerable extent be controlled automatically. A compact C++-code of described algorithm is presented. Our program realization is self-consistent: it demands no preliminary calculations, no external libraries and no additional memory is needed. Numerical examples are provided to show applicability, efficiency, robustness and accuracy of our approach.

show abstract

“…The one-dimensional Volterra type of integral equation has been solved by many numerical methods, such as collocation methods [1], Taylor-series expansion methods [2], Gausstype quadratures method [3], spectral methods [6], Chebyshev polynomial method [7], Tau method [8], sine-cosine wavelets method [9], Monte Carlo method [10], and Haar functions method [11]. But in two-dimensional cases, a small amount of work has been done (see, e.g., [12][13][14]).…”

Section: Introductionmentioning

confidence: 99%

Numerical Study of Two-Dimensional Volterra Integral Equations by RDTM and Comparison with DTM

Abazari

Kılıçman

2013

Abstract and Applied Analysis

View full text Add to dashboard Cite

The two-dimensional Volterra integral equations are solved using more recent semianalytic method, the reduced differential transform method (the so-called RDTM), and compared with the differential transform method (DTM). The concepts of DTM and RDTM are briefly explained, and their application to the two-dimensional Volterra integral equations is studied. The results obtained by DTM and RDTM together are compared with exact solution. As an important result, it is depicted that the RDTM results are more accurate in comparison with those obtained by DTM applied to the same Volterra integral equations. The numerical results reveal that the RDTM is very effective, convenient, and quite accurate compared to the other kind of nonlinear integral equations. It is predicted that the RDTM can be found widely applicable in engineering sciences.

show abstract

Monte Carlo method for solving Fredholm integral equations of the second kind

Cited by 47 publications

References 5 publications

Low memory requirement and computational time method for solving a class of integral equations

Low memory requirement and computational time method for solving a class of integral equations

Numerical solution of Urysohn type nonlinear second kind integral equations by successive quadratures using embedded Dormand and Prince scheme 5(4)

Numerical Study of Two-Dimensional Volterra Integral Equations by RDTM and Comparison with DTM

Contact Info

Product

Resources

About