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Abstract
Diagonal cubic Hermite-Padé approximation to the exponential function with coefficient polynomials of degree at most m is considered. Explicit formulas and differential equations are obtained for the coefficient polynomials. An exact asymptotic expression is obtained for the error function and it is also shown that these generalized Padé-type approximations can be used to asymptotically minimize the expressions on the unit disk. As an application, a class of local analytical difference schemes based on diagonal cubic Padé approximation for diffusion-convection equation with constant coefficients is proposed.
Keywords: Padé-type approximant; Cubic Hermite-Padé approximation; Asymptotic formula; Differential equation
(ProQuest: ... denotes formulae omitted.)
1 Introduction
The Padé approximation theory has been widely used in problems of theoretical physics[1][3][4][5], numerical analysis[9][10], and electrical engineering, especially in modal analysis model[2], order reduction of multivariable systems[6][11][13][14].
We consider approximations of e-x generated by finding polynomials Pm,Qm, Rm and Sm so that ..., (1) where Pm,Qm,Rm,Sm ∈ πm (the vector space of all algebraic polynomials of degree at most m), and Pm has leading coefficient 1. The approximation of e-x is given by one of the following three functions (j = 0,1,2) ..., which is real and closest to 0, where ..., ...; ....
Obviously, δjm(x) is the natural cubic generalization of the main diagonal Padé approximant -Qm/Pm satisfying ... and the diagonal quadratic Hermite-Padé approximant [7] ... satisfying ... (2)
Our primary aim is to derive the exact asymptotic formula for {Em}, the explicit formulae of {Pm}, {Qm}, {Rm}, {Sm}, {Em} and to treat some minimization problems concerning related approximations on the unit disk in C.
Exact results concerning best rational approximation to the exponential function, particularly the Meinardus conjecture, have attracted much attention ([8]). Theorem 4 can be viewed as a cubic version of this conjecture on the disk. A linear version, due to Trefethen appeared in [15]; a quadratic version on the disk given by Borwein can be found in [7].
As an application, this paper proposes a class of local analytical difference schemes based on cubic Padé approximation to e-x for the following diffusion-convection equation ... (3) with the initial condition ...; and the boundary condition ...; where ... are all real constants.
2 Explicit Formulae of the Polynomial Coefficients
Let ..., ..., ..., with ...;...