Numerically solving a nonlinear integral equation when the reciprocal of the solution lies in the integrand

Abstract

The objective of the present work is to develop a numerical method for solving a nonlinear integral equation given by y ( t ) = f ( t ) + ∫ 0 1 k ( t , s ) 1 [ y ( s ) ] α d s , where y ( t ) ∈ C 2 [ 0 , 1 ] with y ( t ) > 0 , f ( t ) ∈ C [ 0 , 1 ] with f ( t ) ≥ 0 , the kernel function k ( t , s ) is non-negative and continuous on [ 0 , 1 ] × [ 0 , 1 ] and α > 0 . The existence of a continuous positive solution of this equation is well-established in the literature. However, there is no reported method to solve it numerically for any α > 0. To attain the desired objective, the renowned Chebyshev collocation method is used. Having unknown Chebyshev coefficients, this method converts the integral equation into a matrix equation that produces a set of nonlinear algebraic equations. To solve these equations, computationally, the well-established Newton’s method is employed. To validate the effectiveness and precision of the method, various numerical examples with well-defined exact solutions are examined. Obtained numerical solutions confirm the accuracy and validity of the numerical method.