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In this paper, we consider Galerkin and multi-Galerkin methods and their iterated versions for solving the nonlinear Hammerstein-type integral equation on the half-line with sufficiently smooth ker...

We consider in this paper the Galerkin and multi-Galerkin methods and their iterated versions to solve the linear Fredholm integral equation of the second kind on the real line with a sufficiently smooth kernels, using Hermite polynomials as basis functions. We obtain optimal convergence results in iterated Galerkin method in weighted L 2 − norm. We also discuss multi-Galerkin methods and we obtain the superconvergence results in both multi-Galerkin and iterated multi-Galerkin methods. Numerical results are presented to confirm the theoretical results.

In this article, Galerkin and collocation methods and their iterated versions are discussed to solve the nonlinear Hammerstein type integral equation on the half-line for both convolution and non-convolution kernels using the space of piecewise polynomial subspace. We prove that the approximate and iterated approximate solution in Galerkin and collocation methods converges with the order O ( n − r ) and O ( n − 2 r ) respectively in the infinity norm, where n − 1 is the maximum norm of the partition and r denotes the order of the piecewise polynomial employed in the approximation. We improve these results further by considering multi-projection methods. In fact we show that iterated approximate solution in multi-Galerkin, multi-collocation converges with the order O ( n − 4 r ) . Numerical results are presented to confirm the theoretical results.

In this paper, we study discrete projection methods for solving the Hammerstein integral equations on the half-line with a smooth kernel using piecewise polynomial basis functions. We show that discrete Galerkin/discrete collocation methods converge to the exact solution with order $${\mathcal {O}}(n^{-min\{r, d\}}),$$ whereas iterated discrete Galerkin/iterated discrete collocation methods converge to the exact solution with order $${\mathcal {O}}(n^{-min\{2r, d\}}),$$ where $$n^{-1}$$ is the maximum norm of the graded mesh and r denotes the order of the piecewise polynomial employed and $$d-1$$ is the degree of precision of quadrature formula. We also show that iterated discrete multi-Galerkin/iterated discrete multi-collocation methods converge to the exact solution with order $${\mathcal {O}}(n^{-min\{4r, d\}})$$ . Hence by choosing sufficiently accurate numerical quadrature rule, we show that the convergence rates in discrete projection and discrete multi-projection methods are preserved. Numerical examples are given to uphold the theoretical results.

In this paper, we consider the Galerkin and iterated Galerkin methods for solving Fredholm-Hammestein integral equations with a Green’s kernel, whose first derivative has singularity. We obtain error bounds and convergence rates for both the Galerkin and iterated Galerkin methods using graded mesh. In fact, by choosing the grading exponent appropriately, we obtain superconvergence results in iterated Galerkin method.

This paper considers the Galerkin and multi-Galerkin methods and their iterated versions to solve the linear Fredholm integral equation of the second kind on the half-line with sufficiently smooth kernels, using Laguerre polynomials as basis functions. Here we are able to prove that approximate solution in Galerkin method converges to the exact solution with order $$\mathcal {O}(n^{-\frac{r}{2}})$$ in weighted $$L^{2}-$$ norm. Also the approximate solution in iterated-Galerkin method converges with order $$\mathcal {O}(n^{-r})$$ in both infinity and weighted $$L^{2}-$$ norm, where r is the smoothness of the solution and n is the highest degree of the Laguerre polynomials employed in the approximation. We also show that multi-Galerkin and iterated multi-Galerkin methods gives superconvergence results using Laguerre polynomials. In fact, we are able to establish that the approximate solutions in multi-Galerkin and iterated multi-Galerkin methods converges to the exact solution with orders $$\mathcal {O}(n^{-\frac{3r}{2}})$$ and $$\mathcal {O}(n^{-2r}),$$ respectively, in weighted $$L^{2}-$$ norm. Numerical results are presented to confirm the theoretical results.