Analytic and numerical solutions of discrete Bagley–Torvik equation

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                </mml:math></jats:alternatives></jats:disp-formula> where <jats:inline-formula><jats:alternatives><jats:tex-math>$0<\nu <1$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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                </mml:math></jats:alternatives></jats:inline-formula>, subject to <jats:inline-formula><jats:alternatives><jats:tex-math>$u(0)=a$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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                </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$\nabla _{h} u(0)=b$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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                </mml:math></jats:alternatives></jats:inline-formula>, with <jats:italic>a</jats:italic> and <jats:italic>b</jats:italic> 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.</jats:p>


Introduction
There has been a great deal of interest in the fractional calculus for the last decades. The reasons of this interest are the findings which came to hand when some of the researchers modeled some dynamic systems making use of fractional operators [1][2][3][4][5][6][7]. One of the most interesting results among them is the one obtained by Bagley and Tovirk [8,9] who studied the viscoelasticity structures and the behavior of materials using fractional derivatives. The equation used by these two scientists is called the Bagley-Torvik equation and has the form [3] where λ i , i = 0, 1, 2 are constants, λ 2 = 0, f is a real-valued function and D ν is the fractional derivative ν = 1 2 or ν = 3 2 . Because of the applications of the Bagley-Torvik equation, many researches tackled the problem of finding the analytic and numerical solution of this equation [3,[10][11][12][13][14][15][16][17][18][19][20][21].
On the other side, discrete fractional calculus has also attracted the attention of many researchers. This type of calculus dealing with the sums and differences with non-integer quantities has also many applications in variety of fields [22][23][24][25]. Motivated by the above, we intended to find the analytic and the numerical solution of a certain discrete version of the Bagley-Torvik equation in this article. The considered equation contains the hdifference which as h → 0 gives the classical derivative. To the best of our knowledge no one [26,27] has discussed this before.
The article is arranged as follows.
The second and third sections discuss the theory needed to handle the equation under consideration. The fourth section proposes the solutions of the discrete Bagley-Torvik equation. The fifth section present numerical solutions of some special cases of the mentioned equation. The sixth section is devoted to the conclusion.

Preliminaries
In this section, some basic definitions and results which will be used further are presented.
, be a real-or complex-valued function and h > 0 be a fixed shift value. Then the forward difference operator on hZ is defined as and the backward difference operator on hZ is defined as For h = 1, this gives u(t) = u(t + 1)u(t) and ∇u(t) = u(t)u(t -1), respectively.
The forward jumping operator on the time scale hZ is σ h (t) = t + h and the backward jumping operator is ρ h (t) = th. For a, b ∈ R and h > 0, we use the notation N a,h = {a, a + h, a + 2h, . . . , } and b,h N = {b, bh, b -2h, . . .}.

Definition 2.2
For h > 0 and μ ∈ R, the increasing h-polynomial factorial function is defined as where t [0] h = 1, is the Euler gamma function and t h + μ, t h , / ∈ {0, -1, -2, -3, . . .}, as the division at a pole yields zero. If μ is a positive integer, then Remark 2.3 Applying the nabla operator on (5), then Proposition 2.4 ([28] The relation between nabla h -RL fractional difference and h-Caputo fractional difference) Definition 2.5 ([29]) Assume that u is defined on N a,h . Then the h-discrete Laplace transform of u is defined by When a = 0, this gives The following results are the h-discrete Laplace transform for the Caputo fractional difference and also for the integer difference operator.

Corollary 2.10
Let t ∈ (-∞, ∞) and h, s > 0, then Proof The proof follows by using the finite inverse principle law in (17).
Example 2.11 For the particular values of h = 2, s = 5, t = 6 and m = 100, Eq. (19) is verified by MATLAB and it turns out that its numerical value is 145.8.

Lemma 2.12 Let h > 0 and u, w be real-valued bounded functions. Then
Proof Applying the nabla operator on the function Now considering w(t) = ∇ h v(t) and taking ∇ -1 h on both sides gives (20).

Generalized nabla discrete h-Laplace transform and its convolution
Following the time scale calculus, one gave the following definition for the nabla discrete Laplace transform on N a,h modifying Lemma 2.6 using the closed form(inverse difference operator).
Definition 3.1 Assume that u(t) is defined on N a,h . Then the generalized nabla discrete Laplace transform of u is defined by Using the closed and summation form solution, the above equation can be written as Remark 3.2 (i) In the case a = 0, (ii) In the special case h = 1, Theorem 3.3 For t ∈ N a,h , h, μ > 0 and s = 0, Proof Taking u(t) = t and w(t) = (1hs) t h -1 in (20), using (7) and (17), then Again taking u(t) = t 2 h and w(t) = (1hs) t h -1 in (20) gives Now applying (26) and simplifying give which can be rewritten as By proceeding the above process up to μ times one finds (25).

Example 3.5 By equating
which is verified by MATLAB for the particular values of h = 2, s = 1/3 and μ = 3 being numerically equal as regards both closed and summation form solution as 54.
Remark 3.6 For the fraction ν, one can write (28) as Definition 3.7 ([29]) Let s ∈ R, 0 < ν < 1 and u, v : N a,h → R be a function. The nabla h-discrete convolution of u with v is defined by

Solution of discrete Bagley-Torvik equation
In this section, we find the analytic solution of the discrete fractional Bagley-Torvik equation given as 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.
Here, one can solve the above fractional equation in the two cases of the particular values of ν = 1 2 and 3 2 with numerical analysis.

Results and discussion
In this section, we give numerical solutions and graphical illustrations for the considered

Conclusion
In this article, the authors discussed a certain version of the discrete Bagley-Torvik equation involving a nabla h-fractional Caputo difference. The researchers obtained the analytical solutions favorably associated to the discrete Laplace transform and the discrete Mittag-Leffler functions. The researchers presented the numerical solutions for specific values of initial values, parameters and the right hand side of the equation. The nabla difference considered can be replaced by the delta difference operator. In this case, one should not think that the analytical solutions can be easily obtained. On the other hand, researchers may also replace the h-fractional Caputo difference by newly defined fractional differences involving non-singular kernels.