# 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.

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