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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.

With the study of extensive literature on the Laplace transform with one and two variables and its properties, applications are available, but there is no work on n-dimensional Laplace transform. In this research article, we define n-dimensional fractional frequency Laplace transform with shift values. Several theorems are derived with properties of the Laplace transform. The results are numerically analyzed and discussed through MATLAB.

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Our main goal in this work is to derive the frequency Laplace transforms of the products of two and three functions with tuning factors. We propose the Laplace transform for certain types of multiseries of circular functions as well. For use in numerical results, we derive a finite summation formula and m-series formulas. Moreover, we discuss various explanatory examples.