Abstract
The task of present research is to establish an enhanced version of residual power series (RPS) technique for the approximate solutions of linear and nonlinear space-time fractional problems with Dirichlet boundary conditions by introducing new parameter . The parameter allows us to establish the best numerical solutions for space-time fractional differential equations (STFDE). Since each problem has different Dirichlet boundary conditions, the best choice of the parameter depends on the problem. This is the major contribution of this research. The illustrated examples also show that the best approximate solutions of various problems are constructed for distinct values of parameter . Moreover, the efficiency and reliability of this technique are verified by the numerical examples.
1. Introduction
The well-established tool fractional differential equations (FDEs) attract growing attention of scientists in a wide range of scientific areas such as biology, physics, and engineering since it is very effective and accurate for modelling many processes in real life [1–11].
In FDEs, noninteger derivatives are taken into account to include memory of the system. Fractional mathematical models have results which are very close to the experimental data [12]. Since FDEs are applicable and variable, the progress of them is very rapid in diverse fields of science and engineering. Therefore, studies on solving FDEs produce efficient and reliable techniques to construct numerical and analytical solutions of them [13–20].
Diverse methods such as a semianalytical approach, a modified wavelet approach, linearized novel operational matrix-based scheme, hybrid spectral linearized scheme, finite difference/spectral algorithm, innovative operational matrix-based computational scheme, and a Chelyshkov polynomial-based algorithm are utilized to construct the solutions of FDEs [20–22].
RPS technique, which can be applied directly to the problem, is one of the most convenient ones to establish approximate solutions with high precision for linear and nonlinear FDEs [21–28]. Moreover, RPS technique allows us to construct the exact solution of the initial value problems whose solutions are polynomial.
Various mathematical models including differential equations such as model of vibration equation of large membranes, fractional Black–Scholes option pricing equations, and time–fractional fuzzy vibration equation of large membranes are analyzed by the RPS technique [27, 28].
Modified versions of the RPS technique are developed to acquire better approximate solutions of FDEs such as the solutions of STFDE in [29–34].
The motivation of this study is to employ a new version of RPS technique to establish better approximate solutions of fractional mathematical models including STFDE in Caputo sense. In the RPS technique, determining a suitable estimation for the initial guess approximations plays an important role. Therefore, in this study, we try to construct best initial guess approximation by introducing a new parameter. The novelty of this method is that new parameter is introduced to get the best approximate solution for the fractional mathematical problems including STFDE with Dirichlet boundary conditions. When we are given Dirichlet boundary conditions, we utilize the parameter to determine the initial rate of change which enables us to acquire the best approximate solution by the RPS technique.
2. Preliminaries
Now, the basic concepts of fractional calculus are presented [1–5].
Definition 1. The Riemann-Liouville (R-L) fractional derivative operator of order is described as [2, 5] where and .
Definition 2. The R-L fractional-order integration operator is described as [2, 5] Following Podlubny [2], we may have
Definition 3. The -order derivative of in Caputo sense is defined as [2, 5]
Definition 4 (see [2, 5]). (1)(2)
Definition 5. -truncated series of the RPS method is given as
3. RPS Technique for Fractional Mathematical Models with Dirichlet Boundary Conditions
We provide an efficient and reliable method for the solution of STFDE with Dirichlet boundary conditions. Let us take the following initial-boundary value problem into consideration:
The -truncated series is established by taking Equations (6) and (7) into account as follows:
The use of Equations (1) and (3) leads to the -truncated series as follows:
An approximation for (6)–(8) is achieved by employing the new RPS technique as follows:
Notice that we introduce a new parameter in this approximate solutions to get a better approximation of the problems (6)–(8). Hence, the -truncated series of of is achieved as follows:
Equation (13) leads to the approximate solution to become
The RPS technique leads to the following definition of the residual function as
In order to establish the coefficients , plunge Equation (12) into Equation (14), and solve the following equation
To acquire , taking and in Equation (14) leads to where
Employing Equation (15) allows us to have
Similarly, we take and in Equation (14) to establish the coefficient , where
Taking Equation (15) into account leads to
As in the previous steps, taking and in Equation (14) leads to where
Using Equation (15) when and , we get
To determine , substituting and into Equation (14), then where
Using Equation (15) when and , we get and so on.
Now, the next step is to obtain the coefficients , by Equation (9).
The equation allows us to establish the required coefficients . Taking into in Equation (14) leads to where which allows us to determine the coefficients as follows:
In a similar form, is acquired by substituting into Equation (14) then where
Hence, we have the coefficient in the following form: and so on. By substituting the coefficients , into Equations (18), (21), (24), and (27) lead to the coefficients .
As a verification of the new RPS technique, we observe that we obtain as a classical exact solution of the problems (6)–(8) with integer order as and .
4. Convergence Analysis
Theorem 6. Let , and , where , and . Moreover, can be differentiated times with respect to on and can be differentiated times with respect to on . Then, where and .
Moreover, for values and such that and , the error term has the term as follows:
Proof. The proof of this theorem follows from the proof 4.3 [27].
5. Numerical Examples
In this section, we provide illustrative examples to prove how effective and reliable the RPS technique proposed in this study is.
Example 1. Consider the STFDE with initial-boundary conditions for which represents the exact solution for and . Based on the RPS technique and above problem, we have the following coefficients:
In Table 1, the approximate solutions are given for various values of with and . Table 2 provides the values of exact solution and an approximate solution for the parameter and various orders of and . Based on Table 1, we conclude that approximate solutions have a higher accuracy as adding more components. Moreover, Figure 1 supports our conclusion.
Example 2. Take the initial-boundary value problem including STFDE with source function into consideration for which denotes the exact solution for and . Based on the RPS technique and above problem, the following coefficients are obtained:
In Table 3, the approximate solution is given for various values of with and . Table 4 provides the values of exact solution and the approximation with the parameter and various orders and . Based on Table 3, we conclude that the approximate solution has a higher accuracy as adding more components. Moreover, Figure 2 confirms our conclusion.
Example 3. Consider the initial-boundary value problem including fractional Burger equation for which represents the exact solution for integer order derivatives. Based on the RPS technique and above problem, the following coefficients are established:
In Table 5, the approximate solution is given for various values of with and . Table 6 provides the values of exact solution and the approximation with the parameter and various orders and . Based on Table 5, we conclude that the approximate solution has a higher accuracy as adding more components. Moreover, our conclusion is supported by Figure 3.
6. Conclusion
In this study, the enhanced version of RPS technique is proposed to establish better approximate solutions of linear and nonlinear space-time fractional problems with Dirichlet boundary conditions by introducing new parameter . The illustrated examples present that the best approximate solutions are achieved for specific values of the parameter . Moreover, the value of parameter which leads to the best approximate solution depends on the fractional mathematical problem. The numerical examples also prove that this new effective and reliable technique is easy to implement.
Data Availability
There is no data available.
Conflicts of Interest
The authors declare that they have no conflicts of interest.