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A Hybrid Method of Improved Block Pulse with Bernstein Polynomials for Numerical Solution of Linear Ill-Posed First Kind Fredholm Integral Equation
Abstract
This paper investigates the approximate solution of linear first-kind Fredholm integral equation (LFFIE). The LFFIE is an ill-posed problem (ILPP) and often requires solving linear system of equations with high condition number, which makes it difficult or impossible to solve. A novel approach is introduced, employing a mix of Benstein polynomials (BEPLs) and improved block-pulse functions (IBLPFs) within the domain [0,1) as hybrid functions. Various properties of these functions are used to convert the LFFIE in to algebraic equations. Analysis of the method's convergence together with numerical examples are provided to demonstrate how relevant the method is. The numerical results proves that the hybrid function of IBLPFs together with BEPLs solve the LFFIE even with large condition number of the matrix.