Analytic and numerical solutions of discrete Bagley–Torvik equation

• Published in: Advances in difference equations Vol. 2021; no. 1; pp. 1 - 12
• Main Authors: Meganathan, Murugesan, Abdeljawad, Thabet, Motawi Khashan, M., Britto Antony Xavier, Gnanaprakasam, Jarad, Fahd
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 29.04.2021; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Bagley–Torvik equation; Caputo derivative; Difference and Functional Equations; Difference operator; Fractional calculus; Functional Analysis; Laplace transform; Laplace transforms; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Parameters; Partial Differential Equations; Real numbers; 65L05; 65L06; Bagley–Torvik equation; 47B39; Difference operator; Caputo derivative; 26A33; Laplace transform; 39A70; Fractional calculus
• ISSN: 1687-1847; 1687-1839; 1687-1847
• DOI: 10.1186/s13662-021-03371-3

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 , where 0 < ν < 1 or 1 < ν < 2 , subject to u ( 0 ) = a and ∇ 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.