Analytic and numerical solutions of discrete Bagley–Torvik equation

Abstract

In this research article, a discrete version of the fractional Bagley–Torvik equation is proposed: $$ \nabla _{h}^{2} u(t)+A{}^{C} \nabla _{h}^{\nu }u(t)+Bu(t)=f(t),\quad t>0, $$ ∇ h 2 u ( t ) + A C ∇ h ν u ( t ) + B u ( t ) = f ( t ) , t > 0 , where $0<\nu <1$ 0 < ν < 1 or $1<\nu <2$ 1 < ν < 2 , subject to $u(0)=a$ u ( 0 ) = a and $\nabla _{h} u(0)=b$ ∇ 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.

The Generalized Fractional Proportional Delta Operator and New Generalized Transforms in Discrete Fractional Calculus

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Existence of solutions for a class of nonlinear fractional difference equations of the Riemann–Liouville type

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