Analytic and numerical solutions of discrete Bagley–Torvik equation

Abstract

Abstract In this research article, a discrete version of the fractional Bagley–Torvik equation is proposed: 1 ∇ h 2 u ( t ) + A C ∇ h ν u ( t ) + B u ( t ) = f ( t ) , t > 0 , $$ \nabla _{h}^{2} u(t)+A{}^{C} \nabla _{h}^{\nu }u(t)+Bu(t)=f(t),\quad t>0, $$ where 0 < ν < 1 $0<\nu <1$ or 1 < ν < 2 $1<\nu <2$ , subject to u ( 0 ) = a $u(0)=a$ and ∇ h u ( 0 ) = b $\nabla _{h} u(0)=b$ , with a and b being real numbers. The solutions are obtained by employing the nabla discrete Laplace transform. These solutions are expressed in terms of Mittag-Leffler functions with three parameters. These solutions are handled numerically for some examples with specific values of some parameters.

Keywords

• Fractional calculus
• Difference operator
• Laplace transform
• Bagley–Torvik equation
• Caputo derivative