# Analytical solution of Bagley-Torvik equations using Sumudu transformation method

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## Abstract

Improvement in some aspect of ecology and financial mathematics is strongly dependent on the analytical solution of Bagley-Torvik equations. The aim of this manuscript is to find the analytical solution of Bagley-Torvik equations which belongs to a class of fractional differential equation by the use of Sumudu transformation method (STM). Here the fractional derivatives are well-defined in Caputo sense. First, some fundamental properties of STM are given, and then STM is applied to the Bagley-Torvik equation which gives an exact solution. The proposed method is an easy, highly efficient and robust method for finding the exact solution.

## Keywords

Sumudu transform method Bagley-Torvik equation Caputo fractional derivative## 1 Introduction

In current years, fractional calculus (FC) has found to be important in various fields viz fluid dynamics, ecology, financial mathematics [1, 2, 3, 4, 5, 6]. It is sometimes difficult to obtain the analytical solution of the fractional differential equations (FDEs). Various important research on FC has been deliberated in the past years, and a lot of books have been written by various authors namely Miller and Ross [7], Oldham and Spanier [8], Podlubny [9]. General ideas about FC are introduced in these books which may help the readers to understand the basic concepts of FC. Recently some analytical and numerical techniques have been developed for the solution of physical problems viz. homotopy perturbation method by Wu and He [10], modified homotopy perturbation method by Jena and Chakraverty [11], Adomian decomposition method by Momani and Odibat [12] and Yavuz and Ozdemir [13], and modified decomposition method by Edeki et al. [14].

In the above regard, STM has been found to be a novel method to handle FDEs. The STM was first introduced by Watugala in 1993. This method was implemented to solve various types of engineering control problems by Watugala [15, 16] too. Later, this method was extended to solve two-dimensional engineering problem by Watugala [17]. The significant applications to partial differential equations and inversion formulae were established in two papers by Weerakoon [18, 19] in 1994 and 1998. The Sumudu transform was also first defined by Weerakoon against Deakin’s definition who claimed that there is no difference between the Sumudu and the Laplace and who reminded Weerakoon that the Sumudu transform is really the S-multiplied transform disguised in Deakin [20] and Weerakoon [21]. The solutions of integral equations and discrete dynamical systems of convolution type using STM were later achieved by Asiru [22, 23, 24]. The Sumudu transform was also used to solve many ordinary differential equations with integer order and although Belgacem’s reasonable advantages for implementing to fractional differential equations commenced in 2008 with various teams of researchers in Katatbeh and Belgacem [25]. It is worth mentioning that novel STM has not been used in solving the Bagley-Torvik equation. So to the best of the present authors’ knowledge, this is the first time that STM has been implemented for solving fractional order Bagley-Torvik equation.

*m*,

*c*,

*k*,

*f*(

*x*) and

*u*(

*x*) denote the mass, damping, stiffness coefficients, external force, and displacement function, respectively. \( \frac{{d^{\alpha } u}}{{dx^{\alpha } }} \) is the FD of order \( \alpha \in (0,2) \). Here \( \delta_{0} \) and \( \delta_{1} \) are real constants.

The rest of the manuscript are arranged as follows: some essential definitions related to fractional calculus are included in Sect. 2. Some basic features and theorems of STM are presented in Sect. 3. In Sect. 4, STM is applied to Bagley-Torvik Equation. Finally, a conclusion is illustrated in Sect. 6.

## 2 Basic features of fractional calculus

### **Definition 2.1**

### **Definition 2.2**

### **Definition 2.3**

### **Definition 2.4**

- (a)$$ D_{t}^{\alpha } J_{t}^{\alpha } f\left( t \right) = f\left( t \right) $$(8)
- (b)$$ J_{t}^{\alpha } D_{t}^{\alpha } f\left( t \right) = f\left( t \right) - \sum\limits_{k = 0}^{m} {f^{\left( k \right)} \left( {0^{ + } } \right)} \frac{{t^{k} }}{k!},\;{\text{for}}\;t > 0\;{\text{and}}\;m - 1 < \alpha \le m,\;m \in N. $$

## 3 Basic properties of STM

### **Definition 3.1**

### **Theorem 1**

*If*\( F\left( u \right) \)

*is the ST of*\( y\left( t \right) \)

*, then the ST of*\( n^{th} \)

*order derivative is defined in Belgacem et al.*[27]

*as follows*

*By using Theorem*1

*, the ST of*\( \frac{dy\left( t \right)}{dt} \)

*and*\( \frac{{d^{2} y\left( t \right)}}{{dt^{2} }} \)

*are given by*

### **Theorem 2**

*The ST of Caputo fractional derivative is well-defined in Chaurasia and Singh*[28]

*as*

## 4 STM implementation to Bagley-Torvik equations

In this section, STM is implemented to Bagley-Torvik equations of fractional order in the following examples.

### *Example 1*

### *Example 2*

### *Example 3*

### *Example 4*

## 5 Conclusion

In this paper, STM is successfully applied to solve Bagley-Torvik equations. Four examples are solved by STM which show that it is a very useful and highly effective technique in term of yielding an analytical solution. Due to its properties, it is mostly used for solving a different kind of linear and nonlinear fractional differential equation for finding the exact solution.

## Notes

### Acknowledgements

The first author would like to thank Department of Science and Technology, Government of India for giving INSPIRE fellowship (IF170207) to carry out the present work.

### Compliance with ethical standards

### Conflict of interest

All authors declare that they have no conflict of interest.

## References

- 1.Deng R, Davies P, Bajaj AK (2004) A case study on the use of fractional derivatives: the low frequency viscoelastic uni-directional behavior of polyurethane foam. Nonlinear Dyn 38:247–265CrossRefGoogle Scholar
- 2.Rossikhin YA, Shitikova MV (2004) Analysis of the viscoelastic rod dynamics via models involving fractional derivatives or operators of two different orders. Shock Vib Dig 36(1):3–26CrossRefGoogle Scholar
- 3.Agrawal OP (2004) Analytical solution for stochastic response of a fractionally damped beam. ASME J Vib Acoust 126(4):561–566CrossRefGoogle Scholar
- 4.Jena RM, Chakraverty S (2019) A new iterative method based solution for fractional Black-Scholes option pricing equations (BSOPE). SN Appl Sci 1(1):95CrossRefGoogle Scholar
- 5.Shah NA, Hajizadeh A, Zeb M, Ahmad S, Mahsud Y, Animasaun IL (2018) Effect of magnetic field on double convection flow of viscous fluid over a moving vertical plate with constant temperature and general concentration by using new trend of fractional derivative. Open J Math Sci 2:253–265CrossRefGoogle Scholar
- 6.Shah NA, Elnaqeeb T, Animasaun IL, Mahsud Y (2018) Insight into the natural convection flow through a vertical cylinder using caputo time-fractional derivatives. Int J Appl Comput Math 4:80–99MathSciNetCrossRefGoogle Scholar
- 7.Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New YorkzbMATHGoogle Scholar
- 8.Oldham KB, Spanier J (1974) The fractional calculus. Academic Press, New YorkzbMATHGoogle Scholar
- 9.Podlubny I (1999) Fractional differential equations. Academic Press, New YorkzbMATHGoogle Scholar
- 10.Wu Y, He JH (2018) Homotopy perturbation method for nonlinear oscillators with coordinate dependent mass. Results Phys 10:270–271CrossRefGoogle Scholar
- 11.Jena RM, Chakraverty S (2019) Solving time-fractional Navier-Stokes equations using homotopy perturbation Elzaki transform. SN Appl Sci 1(1):16CrossRefGoogle Scholar
- 12.Momani S, Odibat Z (2006) Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method. Appl Math Comput 177:488–494MathSciNetzbMATHGoogle Scholar
- 13.Yavuz M, Ozdemir N (2018) A quantitative approach to fractional option pricing problems with decomposition series. Konuralp J Math 6(1):102–109MathSciNetzbMATHGoogle Scholar
- 14.Edeki SO, Motsepa T, Khalique CM, Akinlabi GO (2018) The Greek parameters of a continuous arithmetic Asian option pricing model via Laplace Adomian decomposition method. Open Phys 16:780–785CrossRefGoogle Scholar
- 15.Watugala GK (1993) Sumudu transform: a new integral transform to solve differential equations and control engineering problems. Int J Math Educ Sci Technol 24(1):35–43MathSciNetCrossRefGoogle Scholar
- 16.Watugala GK (1998) Sumudu transform—a new integral transform to solve differential equations and control engineering problems. Math Eng Ind 6(4):319–329MathSciNetzbMATHGoogle Scholar
- 17.Watugala GK (2002) The Sumudu transform for functions of two variables. Math Eng Ind 8(4):293–302MathSciNetzbMATHGoogle Scholar
- 18.Weerakoon S (1994) Application of Sumudu transform to partial differential equations. Int J Math Educ Sci Technol 25(2):277–283MathSciNetCrossRefGoogle Scholar
- 19.Weerakoon S (1998) Complex inversion formula for Sumudu transform. Int J Math Educ Sci Technol 29(4):618–621MathSciNetzbMATHGoogle Scholar
- 20.Deakin MAB (1997) The Sumudu transform” and the Laplace transform. Int J Math Educ Sci Technol 28(1):159–160MathSciNetCrossRefGoogle Scholar
- 21.Weerakoon S (1997) The Sumudu transform and the Laplace transform—reply. Int J Math Educ Sci Technol 28(1):160zbMATHGoogle Scholar
- 22.Asiru MA (2001) Sumudu transform and the solution of integral equations of convolution type. Int J Math Educ Sci Technol 32(6):906–910MathSciNetCrossRefGoogle Scholar
- 23.Asiru MA (2002) Further properties of the Sumudu transform and its applications. Int J Math Educ Sci Technol 33(3):441–449MathSciNetCrossRefGoogle Scholar
- 24.Asiru MA (2003) Application of the Sumudu transform to discrete dynamical systems. Int J Math Educ Sci Technol 34(6):944–949MathSciNetCrossRefGoogle Scholar
- 25.Katatbeh QD, Belgacem FBM (2011) Applications of the Sumudu transform to fractional differential equations. Nonlinear Stud 18(1):99–112MathSciNetzbMATHGoogle Scholar
- 26.Chakraverty S, Tapaswini S, Behera D (2016) Fuzzy arbitrary order system: fuzzy fractional differential equations and applications. Wiley, New YorkCrossRefGoogle Scholar
- 27.Belgacem FBM, Karaballi AA, Kalla SL (2003) Analytical investigations of the Sumudu transform and applications to integral production equations. Math Probl Eng 2003(3):103–118MathSciNetCrossRefGoogle Scholar
- 28.Chaurasia VBL, Singh J (2010) Application of Sumudu transform in Schödinger equation occurring in quantum mechanics. Appl Math Sci 4(57):2843–2850MathSciNetzbMATHGoogle Scholar
- 29.Pedas A, Tamme E (2011) On the convergence of spline collocation methods for solving fractional differential equations. J Comput Appl Math 235:3502–3514MathSciNetCrossRefGoogle Scholar
- 30.Belgacem FBM, Karaballi AA (2006) Sumudu transform fundamental properties investigations and applications. J Appl Math Stoch Anal 2006:23MathSciNetCrossRefGoogle Scholar
- 31.Parisa R, Yadollah O (2018) Application of Müntz-Legendre polynomials for solving the Bagley-Torvik equation in a large interval. seMA 75:517–533MathSciNetCrossRefGoogle Scholar
- 32.Mohammadi F, Mohyud-Din ST (2016) A fractional-order Legendre collocation method for solving the Bagley-Torvik equations. Adv Differ Equ 2016:269–283MathSciNetCrossRefGoogle Scholar
- 33.Gülsu M, Öztürk Y, Anapali A (2017) Numerical solution the fractional Bagley-Torvik equation arising in fluid mechanics. Int J Comput Math 94(1):173–184CrossRefGoogle Scholar
- 34.Ford NJ, Connolly JA (2009) Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations. J Comput Appl Math 229:382–391MathSciNetCrossRefGoogle Scholar