Some numerical solutions of integral equations

Integral equations appear in may fields such as engineering and statistics, as well as applied mathematics and physics. While the analytical solution of some of these equations can be easily done, it may not always be easy to obtain the analytical solution of some of them. For this reason, various approaches can be made for the solution of the equation can be obtained. Equations with known exact solutions are used to check the accuracy and consistency of these methods In this study, Fredholm and Volterra integral equations were solved using the Taylor polynomial method and Bernstein polynomial method on the same samples, and the approximate solutions obtained were compared with the exact solution and the effectiveness of the methods was discussed. According to the results obtained, it was seen that both methods gave very successful results, but the Bernstein polynomial method converged to the solution faster and showed computational superiority. On the other hand, in approximations made with Taylor polynomials, if the Taylor expansions of the functions in the integral equation can be written, it is found to be more effective than Bernstein polynomials in reaching the exact solution. Keywords: Integral Equation, Fredholm Integral Equation, Volterra Integral Equation , Taylor Expansion, Bernstein Polynomials