Home » Pub-Topic: Integral Equations and Linear Functional Analysis, Pub-Topic: Partial Differential Equations, Pub-Type: Articles in International Journals

PAPER: A reproducing kernel Hilbert discretization method for linear PDEs with nonlinear right-hand side

REF. DATE:18 November 2014 CREATED/MODIFIED: 18 January 2016 9 views No Comment

A reproducing kernel Hilbert discretization method for linear PDEs with nonlinear right-hand side, Lib. Math. 34:2(2014), 91-104.


ABSTRACT: In this paper, the main problem of study concerns to find a suitable discretization method to (numerically represent) the solution u\in\mathcal{H}, in a Hilbert space, of the general equation

(1)   \begin{equation*}  L[u(x)]=f(x,u(x)),\quad x\in\Omega\subset\bkR^N, \end{equation*}

satisfying a given boundary condition (e.g. Dirichlet), where L is a linear differential (or integral) operator and f is a nonlinear function with enough regularity. The method Aveiro Discretization Method in Mathematics (ADMM), introduced by Saitoh et al. (2014; doi:10.1007/978-1-4939-1106-6 3), can deal with problem (1) when f is a function that do not depends on u. In fact, ADMM is a general method for solving by discretization, in a specific optimal sense and when a priori some data may be missing, a wide class of linear mathematical problems by using some key ideas of reproducing kernels and Tikhonov regularization theory. Here, we aim to extend the ADMM method to a more general situation where the nonlinearity may depend on u. Then, we apply the scheme to find the (optimal) discretization solution of the problem

    \[\vartheta\frac{\partial^2 u}{\partial y^2}+\alpha \frac{\partial^2 u}{\partial x^2}+\gamma u= u^3,\quad \mbox{ on } \bkR^2,\]

for arbitrary coefficient functions \vartheta, \alpha, \gamma.


Download: Preprint version (RIA |  download) / Publisher version (doi:10.14510/lm-ns.v34i2.1306)
Extra info: (P105/2014)

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