Vol. 24, No. 2, 1968

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The approximation solution of y = F(x, y)

R. G. Huffstutler and Frederick Max Stein

Vol. 24 (1968), No. 2, 283–289

In general the exact solution of the differential system

y′ = F(x,y),y(0) = 0,

is either unattainable or is impractical to handle, even though a solution is known to exist and may even be obtained in certain cases. Thus some method of approximation is often employed. After the choice of approximating functions has been made, there still remains the questions of goodness of approximation and, if infinite processes are employed, the question of convergence.

The system described above is restricted in this paper to those cases in which F(x,y) is an analytic function of x and y for 1 x 1 and all y. Then F(x,y) can be written as a convergent power series

∑      i j
F (x,y) = i,g=0 aijx y .

By considering a sequence of n-th degree polynomials {Pnk(x)} which are 𝜖-approximate solutions of the truncated system

Lk(y) ≡ y′ −     aijxiyj = F (x,0),y(0) = 0,

the solution of the original system can be uniformly approximated by polynomials which satisfy Pnk(0) = 0 and which minimize

             k                         k
∥F(x,0)− Lk[Pn(x)]∥ = s0≦uxp≦1 |F (x,0)− Lk [Pn (x)]|.

Mathematical Subject Classification
Primary: 65.61
Received: 6 June 1967
Published: 1 February 1968
R. G. Huffstutler
Frederick Max Stein