Quasi-Exactly Solvable Differential Models: A Canonical Polynomials Approach

• 作者: Mohamad K. El-Daou Nadia S. Al-Mutawa Talal H. AlZanki
• 来源期刊: 美国计算数学期刊(英文)
• 学科: 数学
• 关键词: POLYNOMIAL Solutions CANONICAL POLYNOMIALS Tau Method Quasi-Exactly SOLVABLE Models
• 年，卷（期）: 2019,（2）
• 页码范围: 48-60
• 页数: 13页
• 分类号: O1

摘要

A quasi-exactly solvable model refers to any second order differential equation with polynomial coefficients of the form A(x)y’’(x)+B(x)y’(x)+C(x)y(x)=0 where a pair of exact polynomials {y(x), C(x)} with respective degrees {deg[y]=n, deg[C]=p} are to be found simultaneously in terms of the coefficients of two given polynomials {A(x), B(x)}. The existing methods for solving quasi-exactly solvable models require the solution of a system of nonlinear algebraic equations of which the dimensions depend on n, the degree of the exact polynomial solution y(x). In this paper, a new method employing a set of polynomials, called canonical polynomials, is proposed. This method requires solving a system of nonlinear algebraic equations of which the dimensions depend only on p, the degree of C(x), and do not vary with n. Several examples are implemented to testify the efficiency of the proposed method.