Opuscula Math. 27, no. 2 (2007), 205-220

Opuscula Mathematica

# The use of integral information in the solution of a two-point boundary value problem

Tomasz Drwięga

Abstract. We study the worst-case $$\varepsilon$$-complexity of a two-point boundary value problem $$u^{\prime\prime}(x)=f(x)u(x)$$, $$x \in [0,T]$$, $$u(0)=c$$, $$u^{\prime}(T)=0$$, where $$c,T \in \mathbb{R}$$ ($$c \neq 0$$, $$T \gt 0$$) and $$f$$ is a nonnegative function with $$r$$ ($$r\geq 0$$) continuous bounded derivatives. We prove an upper bound on the complexity for linear information showing that a speed-up by two orders of magnitude can be obtained compared to standard information. We define an algorithm based on integral information and analyze its error, which provides an upper bound on the $$\varepsilon$$-complexity.

Keywords: boundary value problem, complexity, worst case setting, linear information.

Mathematics Subject Classification: 68Q25, 65L10.

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