Abstract
In this article, the iterative transformation method and homotopy perturbation transformation method are applied to calculate the solution of time-fractional Cauchy-reaction diffusion equations. In this technique, Shehu transformation is combined of the iteration and the homotopy perturbation techniques. Four examples are examined to show validation and the efficacy of the present methods. The approximate solutions achieved by the suggested methods indicate that the approach is easy to apply to the given problems. Moreover, the solution in series form has the desire rate of convergence and provides closed-form solutions. It is noted that the procedure can be modified in other directions of fractional order problems. These solutions show that the current technique is very straightforward and helpful to perform in applied sciences.
1. Introduction
In recent centuries, fractional partial differential equation (FPDE) fundamental signification is well-known in different engineering fields. Numerous physical phenomena can be modelled using FPDEs in various engineering and science fields such as physics, finance, aerospace, mechanics, biology, signal processing, biochemistry, and polymers [1, 2]. It is challenging to create computational models for any natural phenomenon using a traditional differential operator since it can describe those phenomena, primarily hereditary properties. They have a memory effect and the integrated capacity to describe and explain physical phenomena that cannot be adequately explained using an integer-order differential equation. The nonlocality of fractional operators in partial differential equations is one purpose for their importance in modeling different areas of chemical, biological, psychological, thermoplasticity, physical, and mechanical systems [3–5].
Reaction-diffusion (RD) equations define various nonlinear schemes in chemistry, physics, biology, ecology, and other field sciences. RD equations are commonly used in ecology as a model for geographic impacts. They agree on three significant forms of environmental concepts: a minimum patch size required to maintain a population, the establishment of spatial trends in population distributions in homogeneous ecosystems, and the propagation of wavefronts associated with biological invasions. RD equations can be studied using techniques from the concept of PDEs and dynamical schemes [6–9].
The equation of reaction-diffusion write of the following form:
The term of is diffusion term and is the reaction function. Moreover, the diffusion general term is , where is a 2nd order nonlinear elliptic operators. In this article, we consider the fractional-order one-dimensional, reaction–diffusion equation
If , then, it becomes classical reaction-diffusion equation, where is the concentration, is the reaction parameter, and is the diffusion coefficient, with the initial and boundaries conditions
The model defined by equations (2) and (3) is called the characteristic Cauchy model in the domain , and the model given by equations (2) and (4) is called the noncharacteristic Cauchy equation in the domain , using different analytical and numerical methods to solve reaction diffusion equation, such as Picard technique [10], homotopy perturbation technique [3], differential transformation technique and variation iteration technique [11], Adomian decomposition method [12], homotopy analysis method [13], fractional iteration algorithm I [14], new Sumudu transform iterative method [15], and finite-difference discretization scheme [16].
Daftardar-Gejji and Jafari in 2006 [17] suggested DJM solves linear and nonlinear differential equations. The DJM is simple and easy to comprehend and use, and it provides faster numerical solutions than the variational iterative technique [18] and the Adomian decomposition technique [19, 20]. The new iterative transformation method (NITM) is a combination of Shehu transform and the new iterative approach that provides the solution in the form of convergent series in an easy way. Another technique is the homotopy perturbation transform method (HPTM) to some applicable fractional models arising in real-life problems. The HPTM is applied directly to fractional models without any linearization, discretization, or variable transforms. The HPTM is an iterative technique that converges to solutions in closed form or approximate solutions. The nonlinear terms are decomposed successfully via He’s polynomials, and the fractional derivatives are computed in the Caputo sense. Applications of three fractional models are demonstrated, and the analytical and numerical simulations of the three fractional models are provided to bolster the efficiency, simplicity, and high accuracy of the HPTM. Many researchers use this method to solve different fractional-order partial differential equations, such as fractional order gas dynamic equation [21], convection-diffusion problems [22], and Klein-Gordon equations [23].
This paper uses the Iterative transform method and homotopy perturbation transform method to solve fractional Cauchy-reaction diffusion equation equations using the fractional operator of Caputo type. The fractional calculus fundamental definitions are defined in Section 2, writing the general methodologies in Sections 3 and 4, many test models show the effectiveness of suggested techniques in Sections 5 and 6. Finally, the conclusion is given in Section 7.
2. Basic Definitions
2.1. Definition
The Riemann fractional integral is given as [24, 25]
2.2. Definition
The Caputo fractional derivative of is defined as [24, 25]
2.3. Definition
The integral transform of Shehu transform in Set A, the function is defined by [26–28]
The transformation of Shehu is defined as for a function is given as
The Shehu transform of a function is : then, is called the inverse of which is given as
2.4. Definition
The th derivative of Shehu transformation is defined as [27–29]
2.5. Definition
The Shehu transform of fractional derivative is given as [26–28].
3. The HPTM General Implementation
Consider the general fractional partial differential equation solve to HPTM:
Applying Shehu transformation of equation (13), we get
Now, using inverse Shehu transformation, we achieve aswhere
The parameter show the producer of perturbation is given aswhere the perturbation is parameter and .
The nonlinear components can be defined aswhere He’s polynomials in terms of and can be calculated aswhere
Substituting equations (18) and (19) in equation (15), we obtain as
The coefficients comparison on both sides, we have
The components simply calculated to the convergence series form. We can achieve
4. The NITM General Implementation
Consider the general fractional partial differential equation solve to NITM:where linear and nonlinear functions represent by and .
With initial condition
Apply the Shehu transformation of equation (23), we obtain as
Using the differentiation property of Shehu transform is given as
The Shehu inverse transform applies in equation (26), we have
In the iterative method, we getthe nonlinear term is expressed as
Substituting equations (28), (29), and (30) in equation (27), we can obtain the solution
The iterative method can be defined as
Finally, equations (23) and (24) provide the -terms solution in series form is achieved as
Similarly, the convergence proof of the NITM, see [17].
4.1. Example
Consider the fractional Cauchy-reaction diffusion equation is given as:with the initial and boundaries conditions
Applying Shehu transform in equation (36), we get
Using inverse Shehu transform in the above equation, we get
First, we implement the NITM, we get the following
The series form solution is given as
The approximate solutions is achieved as
Now we implement the HPTM, we obtain as
Where the nonlinear function of the polynomial is shown by . For instance, the He’s polynomial component is obtained through the recursive relation , . Now, on both sides, the coefficient of corresponding power is calculated, and the following solution is obtained as follows:
Then the series form result is defined as
The approximate solution, we can achieve as
The actual solution is
Figure 1 shows the exact and the analytical solution figures at of Example 1. Figure 1 shows that both the analytical and actual solutions are in close contact with each other. Figure 2 shows that the first graph analytical solution of fractional-order at , and the second graph fractional order . In Figure 3, the first graph shows the approximate solution of fractional-order at , and the second graph shows the different fractional-order of Example 1. In Figure 4, the 2D graphs show that the first actual and approximate solution and second graph show the different fractional-order of Example 1. The solution is very rapidly convergent by using the homotopy perturbation method and the iterative method by modifying the Shehu transformation. It may be concluded that both methodologies are efficient and very powerful to find the analytical result as well as approximate results of various physical fractional problems.
4.2. Example
Consider the fractional Cauchy-reaction diffusion equation is given as:with initial conditionand the exact solution is given as
First, we implement the NITM, we obtain as
The series form of iterative transform method is defined as
The approximate solution is given as
Now, we implement HPTM, we get
On both sides Comparisons coefficient, we have
The series form solution is express as
Now, , then the exact result of equation (52) is .
Figure 5 shows the exact and the analytical solution figures at of Example 2. Figure 5 shows that both the analytical and actual solutions are in close contact with each other. Figure 6 shows the first graph analytical solution of fractional-order at and the second graph fractional order . In Figure 7, the first graph shows the approximate solution of fractional-order at , and the second graph shows the different fractional-order of Example 2. In Figure 8, the 2D graphs show that the first actual and approximate solution and second graph show the different fractional-order of Example 2. The solution is very rapidly convergent by using the homotopy perturbation method and the iterative method by modifying the Shehu transformation. It may be concluded that both methodologies are efficient and very powerful to find the analytical solution as well as approximate results of various physical fractional problems.
4.3. Example
Consider fractional-order Cauchy-reaction diffusion equation:with initial conditionthe actual solution is
Apply the Shehu transform, we obtain as
First, we implement the NITM
The series form of proposed technique can be defined as
The approximate solution is given as
Now, we apply HPTM, we get
The coefficients of Comparison on both sides, we achieve as
Finally, we obtain the analytical result to the truncate series.
Now, for , the actual result of the above series is
Figure 9 shows the exact and the analytical solution figures at of Example 3. Figure 9 shows that both the analytical and actual solutions are in close contact with each other. Figure 10 shows that the first graph analytical solution of fractional-order at and second graph fractional order . In Figure 11, the first graph shows the approximate solution of fractional-order at , and the second graph shows the different fractional-order of Example 3. The solution is very rapidly convergent by using the Homotopy perturbation method and the iterative method by modifying the Shehu transformation. It may be concluded that both methodologies are efficient and very powerful to find the analytical result as well as approximate results of various physical fractional problems.
4.4. Example
Consider the fractional Cauchy-reaction diffusion equation is given as:with initial conditionthe exact result is
First, we apply the NITM
The series form result is defined as
The approximate solution is
Now, implement the HPTM, we obtain as
The comparison of coefficient of on both sides, we get
Finally, we obtain the approximate result by truncate series
The special case for , the close form above equation is expressed as
Figure 12 shows the exact and the analytical solution figures at of Example 4. Figure 12 shows that both the analytical and actual solutions are in close contact with each other. Figure 13 shows the first graph analytical solution of fractional-order at and second graph fractional order . In Figure 14, the first graph shows the approximate solution of fractional-order at , and the second graph shows the different fractional-order of Example 4. The solution is very rapidly convergent by apply the homotopy perturbation method and the iterative method by modifying the Shehu transformation. It may be concluded that both methodologies are efficient and very powerful in finding the analytical result as well as approximate results of various physical fractional problems.
5. Conclusion
This paper develops an effective and new coupling method of the homotopy perturbation method, the iterative method, and Shehu transform approach for the time-fractional Cauchy-reaction diffusion equations. With the help of the Shehu transform, the calculation of this method is very straightforward and simple. A very close contact of the achieved solutions with the exact solution of the problem is analyzed. It is also investigated that the rate of convergence of the proposed techniques is accurate the solution of fractional-order partial differential equations. Thus, we further this technique can be applied to solve other schemes of linear and nonlinear fractional-order partial differential equations.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Authors’ Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgments
One of the co-authors (A. M. Zidan) extends their appreciation to the Deanship of Scientific Research at King Khalid University, Abha 61413, Saudi Arabia, for funding this work through research groups program under grant number R.G.P.1/30/42.