The solving linear one-dimemsional Volterra integral equations of the second kind in reproducing kernel space

Abstract

_{<span style="font-size: x-large;">In this paper, to solve a linear one-dimensional Volterra </span><span style="font-size: x-large;">integral equation of the second kind. For this purpose using the equation form, we have defined a linear transformation and by using it's conjugate and reproducing kernel functions, we obtain a basis for the functions space.Then we obtain the solution of </span><span style="font-size: x-large;"> </span><span style="font-size: x-large;">integral equation in terms of the basis functions. The examples presented in this paper show validity of the method. But this method does not provide results for nonlinear one-dimensional Volterra integral equations of the second kind. In this case for calculation Fourier cofficients the new method should be given. Thus the next focus on providing a method for calculating </span><span style="font-size: x-large;"> </span><span style="font-size: x-large;">Fourier </span><span style="font-size: x-large;"> </span><span style="font-size: x-large;">cofficients in the nonlinear mode. Also we think that this method can be generalized to linear two-dimensional Volterra integral equations of the second kind and we worked on this in the another paper</span>.}