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A rational approximation based on Bernstein polynomials for high order initial and boundary values problems. (English) Zbl 1217.65150

Summary: We introduce a new method to solve high order linear differential equations with initial and boundary conditions numerically. In this method, the approximate solution is based on rational interpolation and collocation method. Since controlling the occurrence of poles in rational interpolation is difficult, a construction which was found by M. S. Floater and K. Hormann [Numer. Math. 107, No. 2, 315–331 (2007; Zbl 1221.41002)] is used with no poles in real numbers. We use the Bernstein series solution instead of the interpolation polynomials in their construction. We find that our approximate solution has better convergence rate than the one found by using collocation method. The error of the approximate solution is given in the case of the exact solution \(f\in C^{d+2}[a, b]\).

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations

Citations:

Zbl 1221.41002
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References:

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