The New Sumudu Transform Iterative Method For Studying The Random Component Time-Fractional Klein-Go

Abstract

Keywords: Expected value; random component time-fractional Klein-Gordon equation; the new Sumudu transform iterative method; variance.

1. Introduction

Fractional calculus is a quite important topic in several scientific areas [19, 25, 30, 31, 32, 35]. Methodologies of fractional calculus have been widely used in the modeling of a lot of real matters in applied mathematics. Specially, fractional partial differential equations (FPDEs) describe certain a lot of phenomena in several scientific areas such as damping laws, diffusion equations, heat transfer modeling, electrostatics, fluid flow, elasticity and many others [1, 2, 3, 4, 5, 21, 22, 23, 24, 27, 28]. In the literature, there are very few studies on random fractional partial differential equations (RFPDEs). Random fractional partial differential equations (RFPDEs) are described as fractional partial differential equations with random inputs that can be a random variable or a stochastic process. These equations have a enormous significance in a lot of applications in engineering, biology, physics, mathematics and many other applied sciences. It is generally not possible to find

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M. Merdan, H. Anaç, T. Kesemen / Sigma J Eng & Nat Sci 10 (3), 343-354, 2019

random nonlinear fractional partial differential equations analytically. Thus, several numerical methods and approximation schemes for RFPDEs and FPDEs have been improved. There are a lot of numerical methods and approximation schemes such as adomian decomposition method (ADM) [41], homotopy perturbation method (HPM) [17], differential transformation method (DTM) [29], variational iteration method (VIM) [18], fractional variational iteration method (FVIM) [45], random finite difference scheme [9, 14] and many other methods. The main motivation in writing this paper is to analyze random nonlinear fractional partial differential equations using the new Sumudu transform iterative method (NSTIM) which is the reliable computational method. The nonlinear Klein–Gordon equations are used to model a lot of problems in physics and these equations has been solved by different numerical methods in mathematics [6, 7, 13, 38, 42]. Recently, the fractional nonlinear Klein-Gordon equations have been solved by many numerical methods [15, 16, 20, 36]. For example, Golmankhaneh et al. successfully implemented the HPM for obtaining approximate analytical solutions of these equations [16]. This paper studies the random component time-fractional Klein-Gordon equation solve numerically by NSTIM. Wang and Liu established this method. They successfully applied this method to acquire the solutions of time-fractional Cauchy reaction-diffusion equations approximately and analytically [39]. There aren’t enough research and articles on the power series transformation like Sumudu transform in the literature. The Sumudu transform method (STM) proposed by G. K. Watugala, was applied to solve engineering problems [40]. The method was applied to partial differential equations by Weerakoon [43]. Weerakoon found the inverse formula of this transform [44]. Demiray et al. used the Sumudu transform method (STM) to find the solutions of fractional differential equations exactly [11]. Kumar and Daftardar-Gejji extended Sumudu transform iterative method (STIM) to solve various both FPDEs and systems of FPDEs [26]. Prakash et al. suggested a new iterative Sumudu transform method (NISTM) to acquire solutions of the nonlinear time fractional Zahkarov-Kuznetsov equations numerically [33]. In this paper, the solutions of this Klein-Gordon equation are approximately obtained with NSTIM and VIM. Projected technique is used for error analysis. In addition, approximate solutions obtained by two methods with exact solution of this equation are shown in comparison tables. The difference of this work from Prakash et al. (2018) and Wang and Liu (2016) studies is to examine the random component fractional partial differential equation. The aim of this study is to present the application of NSTIM for obtaining the approximate analytical solution of the random component time-fractional Klein-Gordon equation with Caputo derivative and for calculating the expected value and variance of this solution. It is observed that the numerical solution obtained by NSTIM for this Klein-Gordon equation is almost similar to exact solution for this Klein-Gordon equation. Tables indicate that absolute error is negligible. It is observed that NSTIM is superior than VIM for this Klein-Gordon equation.

2. The Basic Definitions

In this section, a few main definitions of fractional calculus, Sumudu transform and Gamma distribution are presented. Definition 2.1. For a real number the space if

is said to be in the space if there exists where and it is said to be in

(1) (2) Definition 2.3. The Riemann-Liouville fractional integral operator of order is given in the following [19, 39]

(2) where is the Gamma function. Some features of the operator that will be used in this study are given below: For (3) (4) Definition 2.4. For

is given as follows [8, 39] (4) Definition 2.6. The Sumudu transform of the Caputo fractional derivative is as follows [39] (5) Definition 2.7. If the probability density function of a random variable has the following form, then this random variable has the Gamma distribution and is called a Gamma random variable. For , (6) If the random variable has a Gamma distribution with parameters and expected value and variance of the random variable are given as follows [34]

In this study, Gamma distribution in the example is chosen as Gamma

3. The New Sumudu Transform Iterative Method

Consider the following equation with the initial condition (8) where

function, is linear operator and is nonlinear operator [39]. If Sumudu transform is implemented to both sides of Eq. (8), then it can be found that 345

M. Merdan, H. Anaç, T. Kesemen / Sigma J Eng & Nat Sci 10 (3), 343-354, 2019

(9) The following equation is obtained with the feature of the Sumudu transform [39] (10) From Eq. (10), it can be found that (11) If the inverse Sumudu transform is implemented to Eq. (11), then Eq. (12) is obtained (12) Assume the following equalities hold:

So Eq. (12) becomes Eq. (13): (13) where is a known function, is a linear operator of and The solution of Eq. (13) is given by the series form as follows [39]

Since K is a linear operator, the following equation is written as (15) The nonlinear operator

Thus, Eq. (13) is given as [39] (17) If the following recurrence is defined (18) then the Eq. (19) is obtained [39] (19) Thus, Eq. (20) is obtained as (20) The m-term solution of Eq. (13) is approximately obtained as (21) In this work, the approximate analytical solution of random component time-fractional KleinGordon equation is obtained with the NSTIM. It is illustrated in the numerical experiment.

4. Error Analysis Of Projected Technique

The error analysis of used technique acquired by NSTIM is given as follows. 346

Theorem 4.1. For all values, a real number satisfies Furthermore, if the truncated series is used to be an approximate solution then maximum absolute truncated error is found with [33]

5. Numerical Experiment

Consider the random component time-fractional Klein-Gordon equation (22) where

are Gamma distributed random variable with parameters and , i.e. . If Sumudu transform is applied to Eq. (22) and the differential feature of Sumudu transform is used, then the Eq. (23) is found (23) If the inverse Sumudu transform is applied to Eq. (23), then Eq. (24) is obtained (24) From Eq. (24), it is obtained as (25) For NSTIM, Eq. (26) holds: (26) By iteration, the results are obtained as follows

Thus, the approximate solution of Eq. (22) is found as follows

(27) The form is the approximate solution of the Eq. (22) for this form is the exact solution of this equation for .

M. Merdan, H. Anaç, T. Kesemen / Sigma J Eng & Nat Sci 10 (3), 343-354, 2019

Table 1. Comparison of the exact solution, approximate solution obtained with the sixth-order NSTIM and the VIM solution for α=1, 0.5 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.0 1.5 1.5 1.5 1.5 1.5

Exact Sol. 4.623059351 5.434325171 6.425207477 7.635473861 9.113696561 5.347150244 6.158416064 7.149298370 8.359564754 9.837787454 5.659198247 6.470464067 7.461346373 8.671612757 10.149835460

Nstim

VIM 4.623059077 5.434307078 6.424995080 7.634243080 9.108851080 5.347149970 6.158397970 7.149085970 8.358333970 9.832941970 5.659198241 6.470445970 7.461133970 8.670381970 10.144989970

Table 2. Comparison of the sixth- order NSTIM and VIM solution for α=0.9, 0.5 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.0 1.5 1.5 1.5 1.5 1.5

Nstim

VIM 4.798719554 5.734086881 6.849157273 8.189855214 9.804132230 5.522810447 6.458177768 7.573248163 8.913946104 10.528223110 5.834858450 6.770225768 7.885296163 9.225994106 10.840271110

6. Conclusion

In this study, this Klein-Gordon equation is analyzed by NSTIM. The approximate analytical solution of the random component time-fractional Klein-Gordon equation has been quickly and successfully obtained with NSTIM. NSTIM is more efficent than VIM as shown in Tables 3. Thus, it is concluded that NSTIM is quickly, effective and superior in obtaining the approximate solutions for random component nonlinear fractional partial differential equations.

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