We establish improved mean value estimates associated with the number of integer solutions certain systems diagonal equations, in some instances attaining sharpest conjectured conclusions. This is first occasion on which bounds this quali

‎the subject of this paper is the solution of the fredholm integral equation with toeplitz, hankel and the toeplitz plus hankel kernel. the mean value theorem for integrals is applied and then extended for solving high dimensional problems and finally, some example and graph of error function are presented to show the ability and simplicity of the ‎method.