On homogeneous second order linear general quantum difference equations

Area of Science:

• Mathematics
• Numerical Analysis
• Difference Equations

Background:

• Introduces the β-Cauchy problem and second-order β-difference equations.
• Defines the general quantum difference operator and its properties.
• Highlights the significance of studying these equations in a neighborhood of a fixed point.

Purpose of the Study:

• To prove the existence and uniqueness of solutions for the β-Cauchy problem.
• To construct a fundamental set of solutions for linear homogeneous β-difference equations with constant coefficients.
• To derive the Euler-Cauchy β-difference equation.

Main Methods:

• Utilizes the general quantum difference operator for analysis.
• Applies fixed-point theory to establish solution existence and uniqueness.
• Analyzes characteristic equations for homogeneous β-difference equations.

Main Results:

• Existence and uniqueness of solutions for the β-Cauchy problem are proven.
• A fundamental set of solutions is constructed for linear homogeneous β-difference equations.
• The Euler-Cauchy β-difference equation is derived.

Conclusions:

• The study provides a comprehensive analysis of second-order β-difference equations.
• The findings contribute to the theory of difference equations and their applications.
• The derived Euler-Cauchy β-difference equation offers a new tool for analysis.