In this paper we will see that, under certain conditions, the techniques of generalized moment problem will apply to numerically solve an Volterra integral equation of first kind or second kind. Volterra integral equa...In this paper we will see that, under certain conditions, the techniques of generalized moment problem will apply to numerically solve an Volterra integral equation of first kind or second kind. Volterra integral equation is transformed into a one-dimensional generalized moment problem, and shall apply the moment problem techniques to find a numerical approximation of the solution. Specifically you will see that solving the Volterra integral equation of first kind f（t） = {a^t K（t, s）x（s）ds a ≤ t ≤ b or solve the Volterra integral equation of the second kind x（t） =f（t）＋{a^t K（t,s）x（s）ds a ≤ t ≤ b is equivalent to solving a generalized moment problem of the form un = {a^b gn（s）x（s）ds n = 0,1,2… This shall apply for to find the solution of an integrodifferential equation of the form x＇（t） = f（t） ＋ {a^t K（t,s）x（s）ds for a ≤ t ≤ b and x（a） = a0 Also considering the nonlinear integral equation： f（x）= {fa^x y（x-t）y（t）dt This integral equation is transformed a two-dimensional generalized moment problem. In all cases, we will find an approximated solution and bounds for the error of the estimated solution using the techniques ofgeneralized moment problem.展开更多
文摘In this paper we will see that, under certain conditions, the techniques of generalized moment problem will apply to numerically solve an Volterra integral equation of first kind or second kind. Volterra integral equation is transformed into a one-dimensional generalized moment problem, and shall apply the moment problem techniques to find a numerical approximation of the solution. Specifically you will see that solving the Volterra integral equation of first kind f（t） = {a^t K（t, s）x（s）ds a ≤ t ≤ b or solve the Volterra integral equation of the second kind x（t） =f（t）＋{a^t K（t,s）x（s）ds a ≤ t ≤ b is equivalent to solving a generalized moment problem of the form un = {a^b gn（s）x（s）ds n = 0,1,2… This shall apply for to find the solution of an integrodifferential equation of the form x＇（t） = f（t） ＋ {a^t K（t,s）x（s）ds for a ≤ t ≤ b and x（a） = a0 Also considering the nonlinear integral equation： f（x）= {fa^x y（x-t）y（t）dt This integral equation is transformed a two-dimensional generalized moment problem. In all cases, we will find an approximated solution and bounds for the error of the estimated solution using the techniques ofgeneralized moment problem.
