Hyers–Ulam Stability of Order <i>k</i> for Euler Equation and Euler–Poisson Equation in the Calculus of Variations

Abstract

In this paper, we define and study Hyers–Ulam stability of order 1 for Euler’s equation and Hyers–Ulam stability of order m−1 for the Euler–Poisson equation in the calculus of variations in two special cases, when these equations have the form y″(x)=f(x) and y(m)(x)=f(x), respectively. We prove some estimations for Jyx−Jy0x, where y is an approximate solution and y0 is an exact solution of the corresponding Euler and Euler-Poisson equations, respectively. We also give two examples.

Keywords

• Euler equation
• Euler–Poisson equation
• calculus of variations
• Hyers–Ulam stability