Numerical Approximations of Riemann-Integrable Solutions to Hammerstein's Equations

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This paper considers a numerical method for approximating solutions of Hammerstein's equation u(x)=∫^1_-1 K(x,s)f(s,u(s))ds+h(x),x∈[-1,1] in the functional space consisting of Riemann-integrable functions, where K, f, and h are given functions and u is the solution to be determined. The fundamental feature of the method is the numerical solution of a fixed-point problem concerning an operator defined in the functional space, which has the same solution as Hammerstein's equation.Two kinds of approximate solutions are constructed by employing Lagrange's interpolation and a natural projection from the viewpoint of collocation methods. For these approximate solutions, it is shown that the L_2 -norm converges and that the uniform norm also converges, especially when the functional space is restricted toa subspace of itself consisting of continuous functions.Error analysis is done for the approximate solutions, and a scheme of numerical computation for obtaining approximate solutions is proposed in which Hammerstein's equation has sufficiently smooth solutions. The scheme is especially applicable to problems that have one or more solutions. Several examples are included to verify the utility of the scheme and to illustrate the convergence of approximate solutions in problems with discontinuity.
一般社団法人情報処理学会の論文
1992-10-31