Abstract
For the numerical solution of the initial value problemy′=f(x,y), −1≦x≦1;y(−1)=y 0 a global integration method is derived and studied. The method goes as follows.
At first the system of nonlinear equations
is solved. The matrix (A (n) i,k ) of quadrature coefficients is “nearly” lower left triangular and the pointsx k,n ,k=1,2,...,n are the zeros ofP n −P n−2, whereP n is the Legendre polynomial of degreen. It is showed that the errors
From the valuesf(x i,n ,y i,n ),i=1,2,...,n an approximation polynomial is constructed. The approximation is Chebyshevlike and the error at the end of the interval of integration is particularly small.
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Axelsson, O. Global integration of differential equations through Lobatto quadrature. BIT 4, 69–86 (1964). https://doi.org/10.1007/BF01939850
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DOI: https://doi.org/10.1007/BF01939850