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Analytic and numerical solutions of discrete Bagley–Torvik equation
Advances in Difference Equations volume 2021, Article number: 222 (2021)
Abstract
In this research article, a discrete version of the fractional Bagley–Torvik equation is proposed:
where \(0<\nu <1\) or \(1<\nu <2\), subject to \(u(0)=a\) and \(\nabla _{h} u(0)=b\), 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.
1 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 [1–7]. 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 \(\lambda _{i}\), \(i=0,1,2\) are constants, \(\lambda _{2}\neq 0\), f is a real-valued function and \(D^{\nu }\) is the fractional derivative \(\nu =\frac{1}{2}\) or \(\nu =\frac{3}{2}\). 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, 10–21].
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 [22–25]. 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 h-difference which as \(h\rightarrow 0\) 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.
2 Preliminaries
In this section, some basic definitions and results which will be used further are presented.
Definition 2.1
Let \(u(t),\ t\in [0,\infty )\), be a real- or complex-valued function and \(h>0\) be a fixed shift value. Then the forward difference operator on \(h\mathbb{Z}\) is defined as
and the backward difference operator on \(h\mathbb{Z}\) is defined as
For \(h=1\), this gives \(\Delta u(t) = u(t+1) - u(t)\) and \(\nabla u(t) = u(t) - u(t-1)\), respectively.
The forward jumping operator on the time scale \(h\mathbb{Z}\) is \(\sigma _{h}(t)=t+h\) and the backward jumping operator is \(\rho _{h}(t)=t-h\). For \(a,b\in R\) and \(h>0\), we use the notation \(\mathbb{N}_{a,h}=\{a, a+h, a+2h,\ldots,\}\) and \(_{b,h}\mathbb{N}=\{b,b-h,b-2h,\ldots\}\).
Definition 2.2
For \(h>0\) and \(\mu \in R\), the increasing h-polynomial factorial function is defined as
where \(t_{h}^{[{0}]}=1\), Γ is the Euler gamma function and \(\frac{t}{h}+\mu ,\frac{t}{h},\notin \{0,-1,-2,-3,\ldots\}\), 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)
-
(i)
\({}_{a}^{C}\nabla _{h}^{\nu }u(t)={}_{a}\nabla _{h}^{\nu }u(t)- \frac{(t-a)_{h}^{\overline{-\nu }}u(a)}{\Gamma (1-\nu )}\).
-
(ii)
\({}_{h}^{C}\nabla _{b}^{\nu }u(t)={}_{h}\nabla _{b}^{\nu }u(t)- \frac{(b-t)_{h}^{\overline{-\nu }}u(b)}{\Gamma (1-\nu )}\).
Definition 2.5
([29])
Assume that u is defined on \(\mathbb{N}_{a,h}\). 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.
Lemma 2.6
For the function \(u(t)\) defined on \(\mathbb{N}_{a,h}\) and \(n-1<\nu \leq n\), then
For the positive integer n,
Definition 2.7
(Nabla h-discrete Mittag-Leffler)
For \(\lambda \in R\), \(|\lambda |<1\) and \(\theta ,\beta ,\rho ,t\in C\) with \(\operatorname{Re}(\theta )>0\), the nabla h-discrete Mittag Leffler functions are defined by
For \(h=\beta =\rho =1\), one can write
where \((\rho )_{k}=\rho (\rho +1)\cdots (\rho +k-1)\) and \((1)_{k}=k!\).
Lemma 2.8
(Finite inverse principle law)
Let \(t>0\), \(h>0\) and m be a positive integer. Then, for the equation \(\nabla _{h} v(t)=u(t)\), \(v(t)=\nabla _{h}^{-1}u(t)\) obeys the finite inverse principle law as
Proof
Take \(\nabla _{h} v(t)=u(t)\), now applying Eq. (4), then
Replace \(v(t)\) by \(v(t-h)\) in (15) and resubstitute in (15), then
Continuing like this gives (14). □
Lemma 2.9
Let \(t\in R\), \(a,h>0\). Then
Proof
Take \(u(t)=(1-hs)^{\frac{t}{h}-1}\) in (4), which gives
Now, the proof of (17) follows by taking \(\nabla _{h}^{-1}\) on both sides. □
Corollary 2.10
Let \(t\in (-\infty ,\infty )\) 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 \(u(t)v(t)\) gives
Now considering \(w(t)=\nabla _{h} v(t)\) and taking \(\nabla _{h}^{-1}\) on both sides gives (20). □
3 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 \(\mathbb{N}_{a,h}\) modifying Lemma 2.6 using the closed form(inverse difference operator).
Definition 3.1
Assume that \(u(t)\) is defined on \(\mathbb{N}_{a,h}\). 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 \(t\in \mathbb{N}_{a,h}\), \(h,\mu >0\) and \(s\neq 0\),
Proof
Taking \(u(t)=t\) and \(w(t)=(1-hs)^{\frac{t}{h}-1}\) in (20), using (7) and (17), then
Again taking \(u(t)=t_{h}^{\overline{2}}\) and \(w(t)=(1-hs)^{\frac{t}{h}-1}\) 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). □
Lemma 3.4
Let \(\mu ,h>0\) and \(s\neq 0\), then
Proof
The proof follows by applying the limits 0 to ∞ in (25) and using (24). □
Example 3.5
which is verified by MATLAB for the particular values of \(h=2\), \(s=1/3\) and \(\mu =3\) 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 \(s\in R\), \(0<\nu <1\) and \(u,v:\mathbb{N}_{a,h}\rightarrow R\) be a function. The nabla h-discrete convolution of u with v is defined by
Theorem 3.8
([29] The h-convolution theorem)
For any \(\nu \in R\slash \{\ldots,-2,-1,0\}\), \(s\in R\) and u, v defined on \(\mathbb{N}_{a,h}\), we have
Lemma 3.9
For \(\lambda \in R\), \(|\lambda |<1\) and \(\theta ,\beta ,\nu ,t\in C\) with \(\operatorname{Re}(\theta )>0\),
Proof
Applying the Laplace transform in (12) and using (29) give
□
4 Solution of discrete Bagley–Torvik equation
In this section, we find the analytic solution of the discrete fractional Bagley–Torvik equation given as
where \(0<\nu <1\) or \(1<\nu <2\), subject to \(u(0)=a\) and \(\nabla _{h} u(0)=b\), with a and b being real numbers.
Here, one can solve the above fractional equation in the two cases of the particular values of \(\nu =\frac{1}{2}\) and \(\frac{3}{2}\) with numerical analysis.
4.1 Case 1: \(\nu =\frac{1}{2}\)
Here, the researchers solve the discrete fractional Bagley–Torvik equation for \(\nu =\frac{1}{2}\) by employing the Laplace transform.
Theorem 4.1
The discrete fractional Bagley–Torvik equation
where \(0<\nu <1\), subject to \(u(0)=a\) and \(\nabla _{h} u(0)=b\), has the solution
Proof
By applying the Laplace transform on (34) and using Lemma 2.6, one finds
simplifying and applying the initial conditions lead to
Now,
Now applying the inverse Laplace transform and using (32) give (35). □
4.2 Case 2: \(\nu =\frac{3}{2}\)
Theorem 4.2
The discrete fractional Bagley–Torvik equation
where \(1<\nu <2\), subject to \(u(0)=a\) and \(\nabla _{h} u(0)=b\), has the solution
Proof
By applying the Laplace transform on (40) and using Lemma 2.6,
and simplifying and applying the initial conditions,
Now,
Now applying the inverse Laplace transform and using (32) give (41). □
5 Results and discussion
In this section, we give numerical solutions and graphical illustrations for the considered Bagley–Torvik equation for particular values of some parameters.
For the particular values of \(a=b=A=B=h=0.1\) and \(\nu =1/2\), \(f(t)=1\), the solution (35) is graphically shown in Fig. 1.
Again for the particular values of \(a=b=0.0001\), \(A=B=0.001\), \(h=0.1\) and \(\nu =1/2\), \(f(t)=t\), the solution (35) is graphically shown in Fig. 2.
When the non-integer order is \(\nu =3/2\) and for the values \(a=0.01\), \(b=0.02\), \(A=0.1\), \(B=0.2\), \(h=0.15\) and \(f(t)=t^{2}+1\) the solution (35) graphically is shown in Fig. 3.
Finally, for the values \(a=0.125\), \(b=0.15\), \(A=B=0.25\), \(h=0.05\) and \(f(t)=0\) the solution (35) is graphically shown in Fig. 4,
6 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.
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References
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993)
Kilbas, A., Srivastava, H.M., Trujillo, J.J.: Theory and Application of Fractional Differential Equations. North Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Publishers (2006)
Hilfer, R.: Applications of Fractional Calculus in Physics. Word Scientific, Singapore (2000)
Qureshi, S., Atangana, A.: Fractal–fractional differentiation for the modeling and mathematical analysis of nonlinear diarrhea transmission dynamics under the use of real data. Chaos Solitons Fractals 136, 109812 (2020)
Atangana, A., Araz, S.İ.: A novel Covid-19 model with fractional differential operators with singular and non-singular kernels: analysis and numerical scheme based on Newton polynomial. Alex. Eng. J. 60(4), 3781–3806 (2021)
Bagley, R.L., Torvik, P.J.: Fractional calculus-a different approach to the analysis viscoelastically damped structures. AIAA J. 21, 741–748 (1983)
Torvik, P.J., Bagley, R.L.: On the appearance of fractional derivative in the behavior of real materials. J. Appl. Mech. 51, 741–748 (1983)
Ray, S.S., Bera, R.K.: Analytical solution of the Bagley–Torvik equation by Adomian decomposition method. Appl. Math. Comput. 168, 398–410 (2005)
El-Sayed, A.M.A., El-Kalla, I.L., Ziada, E.A.A.: Analytical and numerical solutions of multiterm nonlinear fractional orders differential equations. Appl. Numer. Math. 60, 788–797 (2010)
Hu, Y., Luo, Y., Lu, Z.: Analytical solution of the linear fractional differential equation by Adomian decomposition method. J. Comput. Appl. Math. 215, 220–229 (2008)
Karaaslan, M.F., Celiker, F., Kurulay, M.: Approximate solution of the Bagley–Torvik equation by hybridisable discontinuous Galerkin methods. Appl. Math. Comput. 219, 6328–6343 (2013)
Enesiz, Y.C., Keskin, Y., Kurnaz, A.: The solution of the Bagley–Torvik equation with the generalized Taylor collocation method. J. Franklin Inst. 347, 452–466 (2010)
Diethelm, K., Ford, N.J.: Numerical solution of the Bagley–Torvik equation. BIT Numer. Math. 4(3), 490–507 (2002)
Wang, Z.H., Wang, X.: General solution of the Bagley–Torvik equation with fractional-order derivative. Commun. Nonlinear Sci. Numer. Simul. 15, 1279–1285 (2010)
Bansal, M.K., Jain, R.: Analytical solution of Bagley–Torvik equation by generalize differential transform. Int. J. Pure Appl. Math. 110, 265–273 (2016)
Anjara, F., Solofoniaina, J.: Solution of general fractional oscillation relaxation equation by adomians method gen. Math. Notes 20, 1–11 (2014)
Fazli, H., Nieto, J.J.: An investigation of fractional Bagley–Torvik equation. Open Math. 17, 499–512 (2019)
Gamel, M., Abd-El-Hady, M., El-Azab, M.: Chelyshkov-tau approach for solving Bagley–Torvik equation. Appl. Math. 8, 1795–1807 (2017)
Uddin, M., Ahmad, S.: On the numerical solution of Bagley–Torvik equation via the Laplace transform. Tbil. Math. J. 10, 279–284 (2017)
Abdeljawad, T.: Dual identities in fractional difference calculus within Riemann. Adv. Differ. Equ. 2013 (2013) 16 pages
Abdeljawad, T., Baleanu, D.: Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels. Adv. Differ. Equ. 2016 (2016) 18 pages
Abdeljawad, T., Baleanu, D.: Fractional differences and integration by parts. J. Comput. Anal. Appl. 13, 574–582 (2011)
Atici, F.M., Sengul, S.: Modeling with fractional difference equations. J. Math. Anal. Appl. 369, 1–9 (2010)
Sakar, M.G., Saldır, O., Akgül, A.: A novel technique for fractional Bagley–Torvik equation. Proc. Natl. Acad. Sci. India Sect. A Phys. Sci. 89(3), 539–545 (2019)
Zafar, A.A., Kudra, G., Awrejcewicz, J.: An investigation of fractional Bagley–Torvik equation. Entropy 22(1), 28 (2020)
Suwan, I.: Shahd owies and thabet abdeljawad, monotonicity results for h-discrete fractional operators and application. Adv. Differ. Equ. 2018, 207 (2018)
Abdeljawad, T.: Different type kernel h-fractional differences and their fractional h-sums. Chaos Solitons Fractals 116, 146–156 (2018)
Acknowledgements
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research, King Saud University, for its funding through the Research Unit of the Common First Year Deanship.
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This research was funded by the Deanship of Scientific Research, King Saud University.
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Meganathan, M., Abdeljawad, T., Motawi Khashan, M. et al. Analytic and numerical solutions of discrete Bagley–Torvik equation. Adv Differ Equ 2021, 222 (2021). https://doi.org/10.1186/s13662-021-03371-3
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DOI: https://doi.org/10.1186/s13662-021-03371-3
MSC
- 47B39
- 39A70
- 65L05
- 65L06
- 26A33
Keywords
- Fractional calculus
- Difference operator
- Laplace transform
- Bagley–Torvik equation
- Caputo derivative