# A Rayleigh–Ritz Method for Numerical Solutions of Linear Fredholm Integral Equations of the Second Kind

2022-01-01
Kaya, Ruşen
Taşeli, Hasan
A Rayleigh–Ritz Method is suggested for solving linear Fredholm integral equations of the second kind numerically in a desired accuracy. To test the performance of the present approach, the classical one-dimensional Schrödinger equation -y″(x)+v(x)y(x)=λy(x),x∈(-∞,∞) has been converted into an integral equation. For a regular problem, the unbounded interval is truncated to x∈ [ - ℓ, ℓ] , where ℓ is regarded as a boundary parameter. Then, the resulting integral equation has been solved and the results are compared with the very well known eigenvalues of the Schrödinger equation with several types of potential functions v(x). It is shown that the eigenvalues recorded to about 15 significant figures are in excellent agreement with the results that exist in the literature.
Journal of Mathematical Chemistry

# Suggestions

 An Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed Linear Delay Differential Equations Cengizci, Süleyman (Hindawi Limited, 2017) In thiswork, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, ...
 An Application of the rayleigh-ritz method to the integral-equation representation of the one-dimensional schrödinger equation Kaya, Ruşen; Taşeli, Hasan; Department of Mathematics (2019) In this thesis, the theory of the relations between differential and integral equations is analyzed and is illustrated by the reformulation of the one-dimensional Schrödinger equation in terms of an integral equation employing the Green’s function. The Rayleigh- Ritz method is applied to the integral-equation formulation of the one-dimensional Schrödinger equation in order to approximate the eigenvalues of the corresponding singular problem within the desired accuracy. The outcomes are compared with those r...
 Solution of initial and boundary value problems by the variational iteration method Altintan, D.; Uğur, Ömür (2014-03-15) The Variational Iteration Method (VIM) is an iterative method that obtains the approximate solution of differential equations. In this paper, it is proven that whenever the initial approximation satisfies the initial conditions, vim obtains the solution of Initial Value Problems (IVPs) with a single iteration. By using this fact, we propose a new algorithm for Boundary Value Problems (BVPs): linear and nonlinear ones. Main advantage of the present method is that it does not use Green's function, however, it...
 The Sturm-Liouville operator on the space of functions with discontinuity conditions Uğur, Ömür (2006-03-01) The Sturm-Liouville differential operators defined on the space of discontinuous functions, where the moments of discontinuity of the functions are interior points of the interval and the discontinuity conditions are linear have been studied. Auxiliary results concerning the Green's formula, boundary value, and eigenvalue problems for impulsive differential equations are emphasized.
 A DRBEM Approach for the STOKES Eigenvalue Problem Tezer, Münevver; Türk, Önder (2016-07-04) In this study, we propose a novel approach based on the dual reciprocity boundary element method (DRBEM) to approximate the solutions of various Steklov eigenvalue problems. The method consists in weighting the governing differential equation with the fundamental solutions of the Laplace equation where the definition of interior nodes is not necessary for the solution on the boundary. DRBEM constitutes a promising tool to characterize such problems due to the fact that the boundary conditions on part or all...
Citation Formats
R. Kaya and H. Taşeli, “A Rayleigh–Ritz Method for Numerical Solutions of Linear Fredholm Integral Equations of the Second Kind,” Journal of Mathematical Chemistry, pp. 0–0, 2022, Accessed: 00, 2022. [Online]. Available: https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85128815222&origin=inward.