In recent years, fractional calculus has been increasingly used for numerous applications in many scientific and technical fields such as medical sciences, biological research, as well as various chemical, biochemical and physical fields. Fractional calculus can be, for instance, employed to solve a lot of problems within the biomedical research field. Such an important application is studying membrane biophysics and polymer viscoelasticity  .
Numerous methods and approaches have been presented in recent years. Some of these methods are analytical such as Fourier transform method  , the fractional Green function method  , the popular Laplace method   , the iteration method  , the Mellin transform method and the method of the orthogonal polynomial  .
Numerical methods and approaches are also popular and used to obtain approximate solutions of FPDEs. Examples of such a numerical method for solving FPDEs are the Homotopy Perturbation Method (HPM)      , the Differential Transform Method (DTM)  , the Variational Iteration Method (VIM)  , the New Iterative Method (NIM)   , the Homotopy Analysis Method (HAM)    and the Adomian Decomposition Method (ADM)  .
Among these numerical methods, the VIM and the ADM are the most popular ones that are used to solve differential and integral equations of integer and fractional order. The HPM is a universal approach which can be used to solve both fractional and ordinary differential equations FODs as well as fractional partial differential equations FPDEs. The HPM method was originally proposed by He   . The HPM is a coupling of homotopy in the topology and the perturbation method. The method is used to solve various types of equations such us the Hellmholtz equation, the fifth order KdV  , the Kleein-Gorden Equation  , the Fokker-Plank equation   , the nonlinear Kolmogorov- Petrovskii-Piskunov Equation  as well as other types of equations as proposed and used in   .
In this paper, we present two approaches to derive the exact solution of various types of FPDEs. Those methods are: The Modified Homotopy Perturbation Method (MHPM) and the Homotopy Perturbation and Sumudu Transform Method (HPSTM). The MHPM is a fast approach that’s based on designing and utilizing a proper initial approximation which satisfies the initial condition of the HPM. However, the HPSTM is a combination of the homotopy perturbation method and Sumudu transform. The paper is structured in six sections. In Section 2, we begin with an introduction to some necessary definitions of fractional calculus theory. In Section 3, we describe the basic idea of the HPM. In Section 4, we describe the MHPM. In Section 5, we describe the HPSTM. In Section 6, we present two examples to show the efficiency of using the MHPM and HPSTM to solve the fractional Black-Scholes equation. Finally, relevant conclusions are drawn in Section 7.
2. Basic Definitions of Fractional Calculus
In this section, we present the basic definitions and properties of the fractional calculus theory, which are used further in this paper.
A real function is said to be in the space , if there exists a real number such that where and it is said to be in the space if .
The left sided Riemann-Louiville fractional integral of order , of a function is defined as:
where , , and is the Gamma function. Properties of the operator for , , , , can be founded in  , and are defined as follow
Let , . The left-Sided Caputo fractional of in the Caputo sense is defined by  as follows
According to the  , Equations (1) and (2) becomes, for
The single parameter and the two parameters variants of the Mitting-Leffler function are denoted by and , respectively, which are relevant for their connection with fractional calculus, and are defined as:
the k-th derivatives are
some special cases of the Mitting-Leffler function are as follows
Other properties of the Mitting-Leffler functions can be found in  . These functions, are generalizations of the exponential function, because, most linear differential equations of fractional order have solutions that are expressed in terms of these functions.
Sumudu transform over the following set of functions
is defined by
where . Some special properties of the Sumudu transform are as follows
Other properties of the Sumudu transform can be found in  .
is the Sumudu transform of . And according to  we have:
1) is a meromorphic function, with singularities having .
2) There exists a circular region with radius R and positive constants, M and k, with
then the function is given by
The Sumudu transform , of the Caputo functional integral is defined as 
then it can be easily understood that
Consider the following n-term liner fractional differential equation  theorem
with the constant initial condition
Then we see that the analytical general solution of Equation (14) is
where is the Green function and it’s defined by
And is the m-th derivative of the Mitting-Leffler function.
In a special case of the latter theorem, the following relaxation-oscillation  is solved:
where are real constants, and . By utilizing theorem 2.1, we obtain the solution of Equation (14) as follows
where . It is easy to see that if , then the solution of the Equation (14) is given as follows
which will be used in the coming examples, discussed in this work.
3. The Basic Idea of the HPM
In this section, we will briefly present the main idea of the HPM. At first, we will consider the following nonlinear differential equation
where and Gamma are a general differential function operator, a boundary operator, a known analytical function and the boundary of the domain , respectively. The operator A can be decomposed into a linear operator denoted by L, and nonlinear operator denoted by N. Hence, the equation can be written as follows
As a result, we construct a homotopy that satisfies:
which equivalent to
when the value of p is varied from zero to one, we can see easily that
If the parameter p is assumed as small, then the solution of the Equation (23) can be expressed as a power series in p as follows
The best approximation for the solution of Equation (23) is
4. The Modified Homotopy Perturbation Method (MHPM)
Meanwhile, we are able to apply the HPM to solve the class of time-fractional partial differential equations defined by
subject to the initial condition
where . The constructed homotopy satisfies
By simplifying Equation (30) we get
where the embedding parameter p is considered to be small, and applied to the classical perturbation technique. The next step is to use the homotopy parameter p to expand the solution into the following form
For the sake of clarifying the solution procedure of this method, we consider a general nonlinear time-fractional partial differential equation:
By substituting Equation (32) into Equation (31) and equating the terms with identical power of p, we can obtain a series of equations as the following
Applying the Riemann-Louiville fractional integral of order in the both sides
Substitute the results of the Equation (35) into the Equation (32), and applying the Perturbation technique   we obtain an authentic n-th approximation of the exact solution, given by
If there exist some terms , then the exact solution can be written in the following form
We Assume that the initial approximation of Equation (28) is given by
where is the initial condition of the Equation (28), and .
The aim of this algorithm is to find the terms and . We assume that , , which means that the exact solution is given by
As satisfies also the initial condition, we get
and , on the other part we have,
By substitution the Equation (38) into the Equation (28), we obtain
Therein, the FPDEs are changed to a fractional order differential equation, which simplifies the problem. When is an integer, the equation is transformed to an ordinary differential equation.
5. The Homotopy Perturbation Sumudu Transform Method (HPSTM)
We illustrate the basic idea of HPSTM by considering the general time-fractional nonlinear non-homogeneous partial differential equation with the initial condition of the general form defined by:
where and .
With the initial condition
Here by applying Sumudu Transform on the both sides of the Equation (43), we get
which, upon using a property of the Sumudu transform, yields
where f is a function of x.
Taking the inverse of Sumudu on the both sides, we obtain
Now, we apply the HPM to the Equation (47) to get the solution
where is the embedding (or homotopy) parameter. is an initial approximation which satisfies the boundary conditions, and are the nth order approximations which are functions yet to be determined. It is noted that setting in the Equation (48) gives an approximate solution of the given nonlinear differential equation.
The nonlinear term can be decomposed as follows
where are the He’s polynomials given by,
By substituting Equation (48) and Equation (49) into Equation (47), we obtain the following equation
In the meantime, we have completed the coupling of the Sumudu transform and the HPM using He’s polynomials. By equating the terms of the same power of p in the Equation (51), we obtain the given approximation for .
Finally, the approximate and analytical solution of the Equation (43) is given by truncating the following series
In order to apply the MHPM and the HPSTM in Sections 4 and 5, respectively, consider the following fractional Black-Scholes option pricing equations as follows:
6.1. Example 1
6.1.1. Use the MHPM
By applying the MHPM, the initial approximation will be given by
We obtain the fractional system
Solving the Equation (60) and Equation (61) by applying the theorem 2.1, we obtain
and the exact solution is
If we put in Equation (63) or solving Equation (60) and Equation (61) with , we obtain the exact solution
6.1.2. Use the HPSTM
Applying the Sumudu transform to the both sides of the Equation (54), we get
Operating the inverse Sumudu transform on both sides in the Equation (65), we have
Now, applying the HPM
Equating the corresponding power where, of p on both sides in Equation (67), we obtain
So that the solution of the problem is given by
where is the Mittag-Leffler function in one parameter. For special case, , we get
6.2. Example 2
Subject to the initial condition,
6.2.1. Use the MHPM
By applying the MHPM, the initial approximation will be given by
We obtain the final system
Solving the Equation (49) and Equation (50), we get
This is the exact solution of the given option pricing Equation (72). The solution at the special case , is given as follows
6.2.2. Use the HPSTM
Applying the Sumudu transform on both sides of the Equation
Operating the inverse Sumudu transform on both sides in the Equation (83), we have
Now, applying the homotopy perturbation method we have
Equating the corresponding power of p on both sides in Equation (85), we obtain
So that the solution of the problem is given by
This is the exact solution of the given option pricing equation Equation (72). The exact solution at the special case is
It is widely known that various scientific models in the life sciences end up with a PDEs or FPDEs. Solving these equations is very important to understand many phenomena in life sciences. In this work, two methods are applied: MHPM and HPSTM, in order to find the solution of the Fractional Black-Scholes model. The both methods are based on the HPM. However, they are effectively employed for getting the solution. To conclude, MPHM and HPSTM are exceptionally effective and productive methods to discover the approached solution, also the numerical solution for the Time-Fractional Partial Differential Equations.
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