A method for the approximate solution of the second-order linear differential equations in terms of Taylor polynomials. (English) Zbl 0887.65084

Sezer, Mehmet

Int. J. Math. Educ. Sci. Technol. 27, No. 6, 821-834 (1996).

Given a second-order linear differential equation for \(y:x \mapsto y(x)\) \[ P(x)y'' +Q(x)y' +R(x)y= f(x) \tag{1} \] where \(P,Q,R,f\) are elements of \(C^n([a,b], \mathbb{R})\) with \(P(x)\neq 0\) \(\forall x\in[a,b]\) and associated conditions of the form \[ \sum^1_{i=0} \bigl[a_i y^{(i)} (a)+b_iy^{(i)} (b)+c_iy^{(i)} (c)\bigr]= \lambda \tag{2} \]
\[ \sum^1_{i=0} \bigl[\alpha_iy^{(i)} (a)+ \beta_iy^{(i)} (b)+ \gamma_i y^{(i)} (c) \bigr]= \mu \] where \(a\leq c\leq b\) and \(a_i,b_i,c_i, \alpha_i, \beta_i, \gamma_i, \lambda\) and \(\mu\) are suitable real coefficients.
The author is interested in an approximate solution of problem (1), (2) in terms of Taylor polynomials about any point \[ y(x)= \sum^N_{n=0} {1\over n!} y^{(n)} (c)(x-c)^n \] with \(a\leq c\leq b\) and \(y^{(n)} (c)\), \(n=0,1,\dots,N\) are the coefficients to be determined. A matrix method is introduced to determine the Taylor polynomial of order \(N\) of a solution to (1), (2). The method is illustrated by considering four examples.

Reviewer: H.Ade (Mainz)

MSC:

65L10	Numerical solution of boundary value problems involving ordinary differential equations
34A45	Theoretical approximation of solutions to ordinary differential equations
34B05	Linear boundary value problems for ordinary differential equations

Keywords:

second-order linear differential equation; Taylor polynomials; matrix method

References:

[1]	DOI: 10.1080/0020739890200310 · Zbl 0683.45001
[2]	DOI: 10.1080/0020739940250501 · Zbl 0823.45005
[3]	Fox L., Chebyshev Polynomials in Numerical Analysis, (1968)
[4]	Sezer M., Int. J. Math. Educ. Sci. TechnoL (1996)
[5]	SCHEID F., Schaum’s Solved problem series, in: 2000 Solved Problems in Numerical Analysis, (1990)
[6]	DOI: 10.1080/0020739890200101 · Zbl 0684.34005