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This paper extends the homotopy perturbation Sumudu transform method (HPSTM) to solve linear and nonlinear fractional Klein-Gordon equations. To illustrate the reliability of the method, some examples are presented. The convergence of the HPSTM solutions to the exact solutions is shown. As a novel application of homotopy perturbation sumudu transform method, the presented work showed some essential difference with existing similar application four classical examples also highlighted the significance of this work.

Nonlinear phenomena that appear in many areas of scientific fields such as solid state physics, plasma physics, fluid dynamics, mathematical biology and chemical kinetics are modeled in terms of nonlinear partial differen- tial equations and in many scientific and engineering applications one of the corner stones of modeling are partial differential equations. For example, the Klein-Gordon equation which is of the form

with initial conditions

appears in modeling of problems in quantum field theory, relavistic physics, dispersive wave phenomena, plasma physic, nonlinear optics and applied physical sciences. The complexity of the equations though requires the use of numerical and analytical methods in most cases. A broad class of analytical solution and numerical solution methods were used to handle these problems. The topic of fractional partial differential equations has attracted a great atteation in the recent years. There are several analytical have been presented in the literature to solve fractional partial differential equations (FPDEs), such as the Fourier transform method [

Recently, several numerical methods have been introduced for this purpose, such as: the homotopy pertur- bation method (HPM) has first proposed by He [

The homotopy perturbation method (HPM) is extended to drive the exact solutions for linear (nonlinear) ordinary (partial) differential equations of fractional order. The homotopy perturbation method is also combined with the vartional iteration method [

The homotopy perturbation method (HPM) was also investigated by many researchers to handle partial differential equations arising in science and engineering [

In this paper, we applied homotopy perturbation Sumudu transform method (HPSTM) to obtain the analytical exact and approximate solutions for the fractional Klein-Gordon equation with time-fractional derivatives of the form:

where

The paper is organized as follows: in Section 2, we recall some definitions of fractional calculus theory. In Section 3, we describe the homotopy perturbation Sumudu transform method. In Section 4, contains the main results and an examples to show the efficiency of using HPSTM to solve fractional-time Klein-Gordon equa- tions. Conclusions are given in Section 5.

In this section, we mention the following basic definitions and properties of the fractional calculus theory and Sumudu transform.

Definition 1 The Riemann-Liouville fractional integral operator of order

Definition 2 The fractional derivative of

for

Definition 3 The Mittag-Leffler function which is ageneralization of exponential function (see [

Some special cases of the Mittag-Leffler function are as follows:

1.

2.

Definition 4 The Sumudu transform is defined over the set of functions:

by the following formula:

Some special properties of the Sumudu transform are as follows:

1.

2.

Other properties of the Sumudu transform can be found in [

Definition 5 The Sumudu transform of the Caputo fractional derivative is defined as follows [

To illustrate the basic idea of this method, we consider a general fractional partial differential equation with the initial condition of the form:

with

where

Applying the Sumudu transform (denoted in this paper by

Using the differentiation property of the Sumudu transform and the initial conditions in Equation (12), we have

Operating with the Sumudu inverse on both sides of Equation (14) gives

where

and the nonlinear term can be decomposed as

for some Adomian’s polynomials

Substituting Equation (16) and Equation (17) in Equation (15), we get

Equating the terms with identical powers of

proceeding in the same manner, the rest of the components

In this section, in order to asses the applicability and the accuracy of the fractional homotopy Sumudu transform method the following four examples.

Example 1 Consider the time-fractional partial differential Klein-Gordon equation

subject to the initial conditions

Taking the Sumudu transform on both sides of Equation (22), thus we get

and

Using the property of the Sumudu transform and the initial condition in Equation (23), we have

Operating with the Sumudu inverse on both sides of Equation (24) we get

By applying the homotopy perturbation method, and substituting Equation (16) in Equation (25) we have

Equating the terms with identical powers of

Thus the solution of Equation (22) is given by

If we put

Which is in full agreement with the result in Reference [

Example 2 Consider the inhomogeneous linear time-fractional partial differential Klein-Gordon equation

subject to the initial conditions

Taking the Sumudu transform on both sides of Equation (28), thus we get

and

Using the property of the Sumudu transform and the initial condition in Equation (29), we have

Operating with the Sumudu inverse on both sides of Equation (30) we get

By applying the homotopy perturbation method, and substituting Equation (16) in Equation (31) we have

Equating the terms with identical powers of

Thus the solution of Equation (36) is given by

If we put

Which is in full agreement with the result in Reference [

Example 3 Consider the non-linear time-fractional partial differential Klein-Gordon equation

subject to the initial conditions

Taking the Sumudu transform on both sides of Equation (34), thus we get

and

Using the property of the Sumudu transform and the initial condition in Equation (35), we have

Operating with the Sumudu inverse on both sides of Equation (36) we get

By applying the homotopy perturbation method, and substituting Equations (16) in (37) we have

Equating the terms with identical powers of

Thus the solution of Equation (34) is given by

If we put

we obtain the exact solution

Which is in full agreement with the result in Reference [

Example 4 Consider the one-dimensional linear inhomogeneous fractional Klein-Gordon equation

subject to the initial conditions

Taking the Sumudu transform on both sides of Equation (40), thus we get

and

Using the property of the Sumudu transform and the initial condition in Equation (41), we have

Operating with the Sumudu inverse on both sides of Equation (42) we get

By applying the homotopy perturbation method, and substituting Equation (16) in Equation (43) we have

Equating the terms with identical powers of

Thus the solution of Equation (40) is given by

If we put

Which is in full agreement with the result in Reference [

As it is presented above in Example 4 we obtained homotopy perturbation Sumudu transform solution of Equation (40) for values of

method (HPSTM). The values of

In this paper, we have introduced a combination of the homotopy perturbation method and the Sumudu transform method for time fractional problems. This combination builds a strong method called the HPSTD. This method has been successfully applied to one-dimensional fractional equations and also for problems of linear and nonlinear partial differential equations. The HPSTD is an analytical method and runs by using the initial conditions only. Thus, it can be used to solve equations with fractional and integer order with respect to time. An important advantage of the new approach is its low computational load.