The Exact Solution For The System Of The Volterra Integral Equations

In this paper, we mainly discuss how to solve the system of the Volterra integralequations, then we use the reproducing kernel method to give the representation of theexact solution . We will see that It has important theory significance and applicationvalue.First chapter : We introduce the history of the linear operator equation and it'snumerical solutions. Meanwhile, introduce the history of the development of the re-producing kernel theory.Second chapter : We mainly discuss the definition of the reproducing kernel andit's elementary properties. After that , we introduce the reproducing kernel space.Third chapter : We introduce the relative theory of linear bounded operator andadjoint operator. Then we will use the projective operator give the representation ofthe system of linear equations in W₂¹ space .Forth chapter : An exact representation of the solution u(x) for the system of theVolterra integral equationsu_i(x)+(n|∑|(j=1))∫_a^xk_ij(x,t)u_j(t)=f_i(x) i=1,2,…,nhas been given in the reproducing kernel space W₂¹ . When {f_i(x_l)}_l=1ⁿ are known , theapproximate solution um can be constructed by the exact solution directly . If {x_l}_l=1^∞is dense on [a,b],u_m will uniformly converge to u(x).