A Method for Solving the Fredholm Integral Equation of the First Kind

The paper considers a numerical method for solving the Fredholm integral equation of the first kind. The essence of the method is to replace the original equation with the corresponding regularized equation of the second kind, which is then solved by the modified spline collocation method. The solution in this case is represented as a linear combination of minimal splines. The coefficients at the splines are computed using local approximation (in some cases, quasi-interpolation) methods. Results of numerical experiments are presented and show that on model problems, the method proposed yields sufficiently accurate approximations, and the approximation accuracy can be improved by using minimal nonpolynomial splines and related functionals. © 2023, Springer Nature Switzerland AG.