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On homogeneous second order linear general quantum difference equations.

Nashat Faried1, Enas M Shehata2, Rasha M El Zafarani1

  • 1Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt.

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Summary
This summary is machine-generated.

This study proves the existence and uniqueness of solutions for second-order β-difference equations using a quantum difference operator. It also establishes fundamental solutions for linear homogeneous equations and derives the Euler-Cauchy β-difference equation.

Keywords:
Euler-Cauchy general quantum difference equationa general quantum difference operatorgeneral quantum difference equations

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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.