XVI.—Dual Series Relations. III. Dual Relations Involving Trigonometric Series

1963 ◽  
Vol 66 (3) ◽  
pp. 173-184 ◽  
Author(s):  
R. P. Srivastav
Keyword(s):  
Integral Equation ◽  
Numerical Methods ◽  
Integral Equations ◽  
Trigonometric Series ◽  
Analytical Solutions ◽  
Fredholm Integral Equation ◽  
Auxiliary Function

SynopsisThe methods developed in I, II of this series of papers are applied to a solution of a variety of dual series relations involving trigonometric series. In general the problem is reduced to one of solving (usually by numerical methods) a Fredholm integral equation of the second kind for an auxiliary function g(t), but for certain values of the parameters it is possible to obtain analytical solutions of the integral equations and these cases are considered in detail.

scholarly journals A new Model for Solving Three Mixed Integral Equations with Continuous and Discontinuous Kernels

2021 ◽  
pp. 29-38
Author(s):  
M. A. Abdou ◽  
M. I. Youssef
Keyword(s):  
Integral Equation ◽  
Numerical Methods ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Algebraic System ◽  
Linear Algebraic System ◽  
New Model ◽  
Different Types ◽  
Discontinuous Kernel ◽  
Discontinuous Kernels

In this paper, we discuss a new model to obtain the answer to the following question: how can we establish the different types of mixed integral equations from the Fredholm integral equation? For this, we consider three types of mixed integral equations (MIEs), under certain conditions.  The existence of a unique solution of such equations is guaranteed. Using analytic and numerical methods, the three MIEs formulas yield the same Fredholm integral equation (FIE) formula of the second kind. For continuous kernel, the solution of these three MIEs, via the FIEs, is discussed analytically. In addition, for a discontinuous kernel, the Toeplitz matrix method (TMM) and Product Nyström method (PNM) are used to obtain, in each method, a linear algebraic system (LAS). Then, the numerical results are obtained, the error is computed in each case, and compared as well.

scholarly journals Numerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials

2010 ◽  
Vol 2 (2) ◽  
pp. 264-272 ◽  
Author(s):  
A. Shirin ◽  
M. S. Islam
Keyword(s):  
Integral Equation ◽  
Galerkin Method ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Numerical Solutions ◽  
Bernstein Polynomials ◽  
Fredholm Integral Equations ◽  
Matrix Formulation ◽  
Piecewise Polynomials ◽  
The Galerkin Method

In this paper, Bernstein piecewise polynomials are used to solve the integral equations numerically. A matrix formulation is given for a non-singular linear Fredholm Integral Equation by the technique of Galerkin method. In the Galerkin method, the Bernstein polynomials are used as the approximation of basis functions. Examples are considered to verify the effectiveness of the proposed derivations, and the numerical solutions guarantee the desired accuracy.  Keywords: Fredholm integral equation; Galerkin method; Bernstein polynomials. © 2010 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v2i2.4483               J. Sci. Res. 2 (2), 264-272 (2010) 

THE ADOMIAN DECOMPOSITION METHOD APPLIED TO THE FUZZY SYSTEM OF FREDHOLM INTEGRAL EQUATIONS OF THE SECOND KIND

2006 ◽  
Vol 14 (01) ◽  
pp. 101-110 ◽  
Author(s):  
S. ABBASBANDY ◽  
T. ALLAHVIRANLOO
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Fuzzy System ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Fredholm Integral Equations ◽  
Extension Principle ◽  
Numerical Example ◽  
Adomian Decomposition

In this work, the Adomian decomposition(AD) method is applied to the Fuzzy system of linear Fredholm integral equations of the second kind(FFIE). First the crisp Fredholm integral equation is solved by AD method and then the crisp solution is fuzzified by extension principle. The proposed algorithm is illustrated by solving a numerical example.

scholarly journals On the Fredholm integral equation associated with pairs of dual integral equations

1969 ◽  
Vol 16 (3) ◽  
pp. 185-194 ◽  
Author(s):  
V. Hutson
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Bessel Function ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Unknown Function ◽  
Sufficient Conditions ◽  
Neumann Series ◽  
Dual Integral Equations ◽  
Image Position

Consider the Fredholm equation of the second kindwhereand Jv is the Bessel function of the first kind. Here ka(t) and h(x) are given, the unknown function is f(x), and the solution is required for large values of the real parameter a. Under reasonable conditions the solution of (1.1) is given by its Neumann series (a set of sufficient conditions on ka(t) for the convergence of this series is given in Section 4, Lemma 2). However, in many applications the convergence of the series becomes too slow as a→∞ for any useful results to be obtained from it, and it may even happen that f(x)→∞ as a→∞. It is the aim of the present investigation to consider this case, and to show how under fairly general conditions on ka(t) an approximate solution may be obtained for large a, the approximation being valid in the norm of L2(0, 1). The exact conditions on ka(t) and the main result are given in Section 4. Roughly, it is required that 1 -ka(at) should behave like tp(p>0) as t→0. For example, ka(at) might be exp ⌈-(t/ap)⌉.

scholarly journals Analytical Solution of System of Volterra Integral Equations Using OHAM

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Muhammad Akbar ◽  
Rashid Nawaz ◽  
Sumbal Ahsan ◽  
Dumitru Baleanu ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Numerical Methods ◽  
Analytical Solution ◽  
Integral Equations ◽  
Function Method ◽  
Auxiliary Function ◽  
Volterra Integral Equations ◽  
Large Parameter ◽  
Reliable Technique ◽  
Present Technique ◽  
Optimal Homotopy Asymptotic Method

In this work, a reliable technique is used for the solution of a system of Volterra integral equations (VIEs), called optimal homotopy asymptotic method (OHAM). The proposed technique is successfully applied for the solution of different problems, and comparison is made with the relaxed Monto Carlo method (RMCM) and hat basis function method (HBFM). The comparisons show that the present technique is more suitable and reliable for the solution of a system of VIEs. The presented technique uses auxiliary function containing auxiliary constants, which control the convergence. Moreover, OHAM does not require discretization like other numerical methods and is also free from small or large parameter.

scholarly journals Numerical Methods for Solving Fredholm Integral Equations of Second Kind

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
S. Saha Ray ◽  
P. K. Sahu
Keyword(s):  
Integral Equation ◽  
Numerical Methods ◽  
Integral Equations ◽  
Applied Mathematics ◽  
Fredholm Integral Equations ◽  
Nonlinear Fredholm Integral Equations ◽  
Linear And Nonlinear ◽  
Interdisciplinary Effort

Integral equation has been one of the essential tools for various areas of applied mathematics. In this paper, we review different numerical methods for solving both linear and nonlinear Fredholm integral equations of second kind. The goal is to categorize the selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study on numerical methods for solving integral equations.

scholarly journals The Study of Triple Integral Equations with Generalized Legendre Functions

2008 ◽  
Vol 2008 ◽  
pp. 1-12
Author(s):  
B. M. Singh ◽  
J. Rokne ◽  
R. S. Dhaliwal
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Triple Integral Equations

A method is developed for solutions of two sets of triple integral equations involving associated Legendre functions of imaginary arguments. The solution of each set of triple integral equations involving associated Legendre functions is reduced to a Fredholm integral equation of the second kind which can be solved numerically.

scholarly journals Solution of system of the Fredholm integral equation of the first kind

2021 ◽  
Vol 1 (4) ◽  
pp. 1-7
Author(s):  
Vladimir Uskov
Keyword(s):  
Integral Equation ◽  
Unique Solution ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Production Costs ◽  
Fredholm Integral Equations ◽  
System Of Equations ◽  
Blurred Image ◽  
Basic Functions ◽  
Image Production

The article is devoted to the study of a system of two inhomogeneous Fredholm integral equations of the first kind with two required functions depending on one variable. Integral equations describe the restoration of a blurred image, production costs, etc. Fredholm integral equations with one desired function have been considered in many works, but relatively few works have been devoted to systems of such equations. The questions of stability for the solution of systems and the construction of a regularizing system of equations were investigated, but the solution was not constructed in an explicit form. In this paper, the kernels depend on two variables. The case is considered: in the kernels and inhomogeneities, the variables are separated in the equations; these functions are decomposed on the basis of two functions on the interval of integration. Examples of basic functions are given. A condition is determined under which the system has a unique solution in the chosen basis, formulated as a theorem. The solution is found in the form of an expansion in this basis. To illustrate the results obtained, an example is considered

scholarly journals A Triangle Algorithm of Padé-Type Approximant for Two-Dimensional Fredholm Integral Equations of the Second Kind

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Meilan Sun ◽  
Chuanqing Gu
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Recursive Algorithm ◽  
High Order ◽  
Fredholm Integral Equations ◽  
Two Dimensional ◽  
Initial Value ◽  
Numerical Examples ◽  
Type Approximation

The function-valued Padé-type approximation (2DFPTA) is used to solve two-dimensional Fredholm integral equation of the second kind. In order to compute 2DFPTA, a triangle recursive algorithm based on Sylvester identity is proposed. The advantage of this algorithm is that, in the process of calculating 2DFPTA to avoid the calculation of the determinant, it can start from the initial value, from low to high order, and gradually proceeds. Compared with the original two methods, the numerical examples show that the algorithm is effective.

Modified Adomian Decomposition Method for Solving Fuzzy Volterra-Fredholm Integral Equation

2018 ◽  
Vol 85 (1-2) ◽  
pp. 53 ◽  
Author(s):  
Ahmed A. Hamoud ◽  
Kirtiwant P. Ghadle
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Fredholm Integral Equations ◽  
Modified Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Uniqueness Of The Solution ◽  
Fuzzy Solution

In this paper, a modied Adomian decomposition method has been applied to approximate the solution of the fuzzy Volterra-Fredholm integral equations of the first and second Kind. That, a fuzzy Volterra-Fredholm integral equation has been converted to a system of Volterra-Fredholm integral equations in crisp case. We use MADM to find the approximate solution of this system and hence obtain an approximation for the fuzzy solution of the Fuzzy Volterra-Fredholm integral equation. A nonlinear evolution model is investigated. Moreover, we will prove the existence, uniqueness of the solution and convergence of the proposed method. Also, some numerical examples are included to demonstrate the validity and applicability of the proposed technique.

Sign in / Sign up

Export Citation Format

Share Document