A residual method for solving nonlinear operator equations and their application to nonlinear integral equations using symbolic computation
Introduction
The mathematical formulation of a nonlinear operator equation reads as follows:where F maps the real Hilbert space into itself and F is continuous in . The Hilbert space is equipped with the inner product 〈·,·〉 and the associated norm .
We are interested in solving (1) with a method that does not use F′ (the Fréchet derivative), but shares some features in common with the method of minimum residuals. To be precise, we discuss a specific method for solving (1), namely, Residual Algorithm for Nonlinear Operator Equations (RANOE) which is a variation and generalization of the methods SANE [8] and DF-SANE [7] for systems of nonlinear equations on . These algorithms have a characteristic in common, that is, they use in a systematic way the residual Fx as search direction. For the new method we also present a convergence analysis.
An interesting particular case of (1) is the solution of nonlinear integral equations, that arise in important areas of science and engineering. Different methods for solving integral equations have been developed in the last few years [1], [2], [4], [5], [6], [9], [10], [11], [12], [13].
In this work we also describe the application of RANOE to the resolution of nonlinear integral equations. To be precise, we use a symbolic implementation of RANOE for solving nonlinear integral Hammerstein’s equations given by symmetric and continuous kernels. We show numerical results for some test nonlinear integral equations using the Symbolic Toolbox of Matlab for the implementation of RANOE.
Section snippets
The method and its properties
The iterations of any method of minimum residuals can be defined bywhere is a given initial approximation to the solution of (1), λk are real numbers, and {dk} is a sequence of elements in [14]. Usually these methods use F′ to define the numbers λk or the sequence {dk}. Under certain hypothesis, the methods given by (2) converge locally to the solution of (1). For the general theory behind the methods of minimum residuals, the reader is conveyed to Ref. [14].
We previously
Applying the RANOE method to nonlinear integral equations
Let us apply the RANOE method to solve the nonlinear integral Hammerstein’s equation. For this part we consider the Hilbert space of the squared integrable functions in the interval [a, b], denoted by L2[a, b], whit inner product given by
The mathematical formulation of the nonlinear integral Hammerstein’s equation iswhere the function φ(t, y) is continuous, the kernel k(x, t) is continuous and symmetric, k(x, t) = k(t, x), g is
Numerical experience
We are interested in illustrating the performance of RANOE for solving some nonlinear Hammerstein’s equations. We use the symbolic computation of Matlab 6.0 for the implementation of RANOE (Matlab codes shown in Appendix A). All the runs were carried out on a Pentium Centrino Duo Computer at 2.4 GHz with 2 GB of RAM.
We implemented RANOE with the following parameters: α0 = 1, αmin = 10−10, αmax = 1010, λmin = 10−10, λmax = 1, and γ = 10−4. Also, we take ηk = βρk, where β = ∥Fy0∥2 and ρk = 4−k, for all k ⩾ 0. We
Final remarks
Our main comments on the RANOE are as follows:
- 1.
The RANOE method generates a sequence {yk} such that all its limit points are solutions of the nonlinear Eq. (1). If the initial point is close enough to an isolated solution y∗, then the whole sequence {yk} converge to y∗. Now, if F is a strongly monotone operator, the sequence {yk} converge to the unique solution of (1).
- 2.
The main conclusions of Section 3 are as follows. Theoretically, if the abstract form of the Hammerstein’s equation has an
Acknowledgements
We would like to thank Prof. Marcos Raydan, from Universidad Central de Venezuela, for many suggestions which greatly improved the quality of this paper. We would also like to thank two anonymous referees for helpful suggestions. The author was supported by CDCH-UCV project PI-08-7276-2008/2.
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