Computational method for solving weakly singular Fredholm integral equations of the second kind using an advanced barycentric Lagrange interpolation formula

In this study, we applied an advanced barycentric Lagrange interpolation formula to find the interpolate solutions of weakly singular Fredholm integral equations of the second kind. The kernel is interpolated twice concerning both variables and then is transformed into the product of five matrices; two of them are monomial basis matrices. To isolate the singularity of the kernel, we developed two techniques based on a good choice of different two sets of nodes to be distributed over the integration domain. Each set is specific to one of the kernel arguments so that the kernel values never become zero or imaginary. The significant advantage of thetwo presented techniques is the ability to gain access to an algebraic linear system equivalent to the interpolant solution without applying the collocation method. Moreover, the convergence in the mean of the interpolant solution and the maximum error norm estimation are studied. The interpolate solutions of the illustrated four examples are found strongly converging uniformly to the exact solutions.

Computational Method for Solving Weakly Singular Fredholm Integral Equations of the Second Kind using an Advanced Barycentric Lagrange Interpolation Formula

In this study, we applied an advanced barycentric Lagrange interpolation formula to find the interpolate solutions of weakly singular ‎Fredholm integral equations of the ‎second kind. The kernel is ‎interpolated twice concerning ‎both variables and then is transformed into the product of five ‎matrices; two of them are monomial basis ‎matrices. To isolate the singularity of the kernel, we ‎developed two techniques based on a good choice of ‎different two sets of nodes to be distributed ‎over the integration domain. Each set is specific to one of the ‎kernel arguments so that the kernel ‎values never become zero or imaginary. The significant advantage of the ‎two presented ‎techniques is the ability to gain ‎‎access to an algebraic linear system equivalent to the interpolant solution without applying the collocation method. Moreover, the convergence in the ‎mean of the interpolant solution ‎and the maximum error norm estimation are studied. The ‎interpolate solutions of the illustrated four ‎examples are found strongly converging uniformly to the ‎exact solutions.

Bivariate Interpolation in Triangular Form

The purpose of this paper is to derive lagrange interpolation formula for a single variable and two independent variables in triangular form. Firstly, single variable interpolation is derived and then two independent variable interpolation derived in triangular form. In this paper, we derived the formula of three points of fit approximation, six points of fit approximation and ten points of fit approximation with their examples respectively. And also derived the approximation using a general triangular form of points with examples.

Bivariate Interpolation in Rectangular Form

The purpose of this paper is to derive lagrange interpolation formula for a single variable and two independent variables. Firstly, single variable interpolation is derived and then two independent variable interpolation derived. Some practical problems are computed by virtue of these interpolation formula.

Approximate Solution Technique for Singular Fredholm Integral Equations of the First Kind with Oscillatory Kernels

An efficient quadrature formula was developed for evaluating numerically certain singular Fredholm integral equations of the first kind with oscillatory trigonometric kernels. The method is based on the Lagrange interpolation formula and the orthogonal polynomial considered are the Legendre polynomials whose zeros served as interpolation nodes. A test example was provided for the verification and validation of the rule developed. The results showed the convergence of the solution and can be improved by increasing n.

Bivariate uniform approximation via bivariate Lagrange interpolation polynomials

In the present note, we extend some univariate uniform approximation results by means of Lagrange interpolating polynomials [Ivan, M., Elements of Interpolation Theory, Mediamira Science Publisher, Cluj-Napoca (2004)] to the bivariate case. It is well known that generally, in the univariate case, the sequence of Lagrange interpolation polynomials does’t converges to the approximated function. This fact was first observed by G. Faber (see [9]), which constructed an example when the sequence of Lagrange interpolation polynomials diverges. The result of G. Faber was more generalized by I. Muntean (see [12]). M. Ivan established first sufficient conditions for the uniform convergence of the sequence of Lagrange interpolation polynomials associated to a univariate real valued function. First, we represent the remainder term of bivariate Lagrange interpolation formula in terms of bivariate divided difference. Using this representation we establish sufficient conditions for the uniform convergence of the sequence of bivariate Lagrange interpolation polynomials to the approximated function.