Second kind Chebyshev polynomials for solving logarithmic Volterra- Fredholm integral equations

It is known that the logarithmic Volterra integral equations are so hard to find its exact analytic solution, so we are obliged to find a new numerical method strong and reliable.. In this work, we develop a technical approximation to obtain a numerical solution of the logarithmic Volterra and Fredholm integral equations, This new approximation is based on the Shifted second Chebyshev polynomials the unknown function will be approximated by a truncated sum of shifted Chebyshev Polynomials. our spectral method sends the given equation into an algebraic system of equations with the unknown expansion coefficients. The convegence, the efficiency and the high accuracy of this method are realized by many examples.