New homotopy analysis transform method for solving multidimensional fractional diffusion equations

Abstract In this paper, we introduce a new semi-analytical method called the homotopy analysis Shehu transform method (HASTM) for solving multidimensional fractional diffusion equations. The proposed technique is a combination of the homotopy analysis method and the Laplace-type integral transform called the Shehu transform which is a generalization of the Laplace and the Sumudu integral transforms. Shehu transform is user-friendly, and its visualization is easier than the Sumudu, and the natural transforms. The convergence analysis of the method is proved, and we provide some applications of the fractional diffusion equations to validate the efficiency and the high accuracy of the technique. The results obtained using the HASTM are in complete agreement with the results of the existing techniques.

In 2001, Ford and Simpson discovered that the Caputo fractional derivative (Caputo, 1967) is the most suitable tool for solving fractional models, because the Riemann-Liouville fractional derivative always requires the initial conditions which are not available in many physical models (Ford & Simpson, 2001).In 1968, Oldham and Spanier discovered that the half-order derivatives and the integrals can be used to model many chemical problems that are more useful than the classical techniques (Oldham & Spanier, 1974).Besides, more recent contributions on the theory and applications of fractional calculus including fractal calculus are available in the literature.Interested readers are refer to, for example, (Losada & Nieto, 2015;Atangana & Baleanu, 2016;Atangana & Khan, 2019;Atangana & Koca, 2016;Belgacem et al., 2019;Bokhari, Baleanu, & Belgacem, 2019;Caputo & Fabrizio, 2015;Fan, Yang, & Liu, 2019;He, 2018;Wang, An, & Wang, 2019;Wang, Yao, & Yang, 2019;Wang & Deng, 2019;Wang & Yao, 2019, He & Ji, 2019).The most important property of the fractional order derivatives is their nonlocal behavior, hence can easily be applied to many problems, since they have the memory effect (Ray & Sahoo, 2015) and the higher degree of freedom in the mathematical model compared with the classical order derivatives.Moreover, the concept of fractional calculus was used to model the anomalous behavior of diffusion equations by replacing ordinary order derivatives with fractional order derivatives (see Elsaid, Shamseldeen, & Madkour, 2016).
Recently, Elsaid et al. study the fractional diffusion with the spatial Riesz-Feller fractional derivative and the Caputo fractional derivative using the optimal homotopy analysis method (Elsaid, 2011).Fractional diffusion equations gained considerable popularity in synchronization, mechanical system, control, plasma physics, quantum mechanics, chaos, and dynamic system.In the literature, diffusion equations correspond to the Markov process and the well-known Brownian motion (Markov, 1960;Feynman, 1964).
In recent years, many analytical and numerical algorithms have been utilized for solving the fractional diffusion equations such as the homotopy analysis method (Jafari & Seifi, 2009), the Adomian decomposition method (Jafari & Daftardar-Gejji, 2006), the variational iteration method (Ganji, Tari, & Jooybari, 2007;Wazwaz, 2007), the homotopy perturbation method (He, 1999;Jafari & Momani, 2007), the homotopy analysis Sumudu transform method (Kumar, Singh, & Kumar, 2015), the reduce differential transform method (Singh & Srivastava, 2018;Rawashdeh & Obeidat, 2015), and so on.Among these algorithms is the homotopy analysis method (HAM) which was first introduced by the Chinese mathematician Liao in 1992 (Liao, 1992).The wellknown perturbation technique in the literature which is based on the existence of small or large parameter called the perturbation quantity (Nayfeh, 2000), was significantly improved by the idea of the homotopy analysis method.
Motivated by the existing techniques, the main objective of this paper is to study the multidimensional fractional diffusion equations using the Laplace-type integral transform called Shehu transform (Maitama & Zhao, 2019b) and the homotopy analysis method.The fractional derivatives are considered in Caputo's sense.The Shehu transform become Laplace's transform (Spiegel, 1965) when the variable u ¼ 1, and becomes the Yang's integral transform (Yang, 2016) when the variable s ¼ 1, and it is a generalization of the Laplace and the Sumudu integral transform (Watugala, 1998).Besides, the proposed integral transform is similar to natural transform (Khan & Khan, 2008).The duality properties of the proposed integral transform with other existing integral transforms are given in Table 1.
Some properties and theorems of the proposed integral transform, such as the derivative property, the convolution theorem, and the inverse property are studied.The applications of the HASTM are illustrated, and the results obtained are compared with the Adomian decomposition method, the homotopy analysis method, the homotopy analysis Sumudu transform method, the homotopy perturbation Sumudu transform method, and the homotopy perturbation Laplace transform technique.
The remaining sections of this article are arranged as follows.In Section 2, we present the new integral transform and some preliminaries of fractional calculus.In Section 3, we discuss the analysis of the HASTM and its convergence.In Section 4, the He's Laplace method is given.In Section 5, applications of the new homotopy analysis transform method are presented.Results and discussion are given in Section 6.Finally, in Section 7 some conclusions are presented.

Shehu transform and fractional calculus
Definition 1.A real function f ðtÞ, t > 0, is said to be in the space C s s 2 R if there exists a real number pð> sÞ, such that f ðtÞ ¼ t p f 1 ðtÞ, where f 1 ðtÞ 2 C½0, 1Þ, and it is said to be in the space C s iff f ðmÞ 2 C m s , m 2 N: Definition 2. Riemann-Liouville fractional derivative.The Riemann-Liouville fractional derivative of order a > 0, of a function f(t) is defined as (Podlubny, 1999;Diethelm & Ford, 2002).
In 2016, a new fractional derivative with the nonlocal and no-singular kernel was proposed by Atangana and Baleanu (2016 (9) Before we define the suggested integral transform, let us review some important development on fractal calculus which gives a physical understanding rather than mathematical understanding.Definition 5. A fractal derivative (Hausdorff derivative) of a function f with order a is defined as (Chen, 2006;Atangana & Khan, 2019) The generalized version of Equation ( 10) is Additionally, if the function f is differentiable in Equation (10), we deduce and the corresponding integral is However, in 2011, He (2011) discovered that Equation ( 10) is much simpler among the definitions of fractal derivatives, but lack physical understanding.As a result, a new fractal derivative was introduced as Definition 6.He's fractal derivative of a function f with order a is defined as (He, 2011(He, , 2018) ) Df ðtÞ Dx a ¼ lim ᭝x!f 0 f ðAÞ À f ðBÞ The distance between two points ðA and BÞ where k is a constant, f 0 is the smallest measure, f(t) is the distance between the two pints (A and B) on fractal space, and a is the fractal dimension.
Equivalently, the general fractal derivative of Equation ( 14) can be written as In the following definition, we define the Shehu transform and some of its properties based on fractional calculus.
Definition 7. Shehu transform (Maitama & Zhao, 2019b).The Shehu transform of the function f(t) of exponential order is defined over the set of functions, ' , by the following integral S½f ðtÞ ¼ Fðs, uÞ ¼ (16) It converges if the limit of the integral exists, and diverges if not.
The following useful results follows directly from Equation ( 16) Where f Ã w is the convolution of two functions f(t) and w(t) which is defined as Proof.See Maitama and Zhao (2019a).Proof.See Maitama and Zhao (2019a).
In the next theorem, we define the inverse of the suggested integral transform.Theorem 6. Inverse Shehu transform.Let F(s, u) be the Shehu transform of the function f(t), then its inverse transform is given by Fðs, uÞds: (23) Proof.The proof of Equation ( 23) follows directly from the duality with the natural transform (see Table 1).w In the next section, we illustrate the basic idea of the HASTM.

Homotopy analysis Shehu transform method
To formulate the basic idea of the HASTM, we consider the following nonlinear fractional partial differential equation ðD a vÞðx, tÞ þ Rvðx, tÞ þ @vðx, tÞ ¼ gðx, tÞ; where @ is the nonlinear operator, ðD a vÞðx, tÞ is the Caputo fractional derivative, Rv(x, t) is the remaining linear operator, and g(x, t) is the source term.
Theorem 7. Convergence of the series solutions.
If series of solutions is converges to wðx, tÞ, where v m ðx, tÞ is generated by the Mth-order deformation Equation (35) based on Equations ( 36) and (39), then wðx, tÞ must be the solution of the original problem Equation (24).
In the next theorem, we provide the absolute error analysis of the proposed algorithm Theorem 8. Absolute error analysis.Let v(x, t) be the approximate solution of the truncated finite series P m n¼0 v n ðx, tÞ.Suppose it is possible to obtain a real number j 2 ð0, 1Þ such that kv nþ1 ðx, tÞk jkv n ðx, tÞk, for 8n, then the maximum absolute error is vðx, tÞ À X m n¼0 v n ðx, tÞ Proof.Let the series P m n¼0 v n ðx, tÞ be finite, then vðx, tÞ À The proof is complete. w In the next section, we provide the He's homotopy perturbation Laplace transform method and later compare its results with our suggested technique (HASTM).

Homotopy perturbation Laplace transform technique (HPLTT)
Let consider the nonlinear fractional partial differential equation Applying the Laplace transform and its properties on both sides of Equation ( 46) gives Computing the inverse Laplace transform of Equation ( 47), we get vðx, tÞ ¼ Gðx, tÞ where G(x, t) is a term arising from the source term and the prescribed initial condition.
According to the homotopy perturbation method which was first introduced by He (1999), the unknown solution v(x, t) can be express as vðx, tÞ ¼ and the nonlinear term @ðvðx, tÞÞ is decomposed as where H n ðvÞ is the He's polynomial which can be computed with the following formula Substituting Equations ( 49) and (50) into Equation (48), we obtain Comparing the coefficient of likes powers of f in Equation ( 52), we obtain the following approximations: . .
Then the series solutions of the HPLTT is obtained as

Applications of the HASTM
In this section, the HASTM is efficiently applied to fractional diffusion equation to validate its efficiency and high accuracy.

Results and discussion
Figure 1(a-c) show the diffusion results for a ¼ 2, B ¼ a ¼ 1:7, C ¼ a ¼ 1:5, and t ¼ 1 respectively.Figure 1(a) show a clear combination of wave and diffusion behavior for a ¼ 2 (exact solution).Besides, the results show that the exact solution exhibits fast diffusion behavior.The surface solutions behavior of Equation ( 54) for different values of B ¼ a ¼ 1:7, C ¼ a ¼ 1:5 are plotted in Figure 1(b) and 1(c) respectively.The result show that decrease in a result decreased in wave behavior.In Figure 1(d), the exact and approximate solutions for the 10th-order approximations 54) for different value A¼a ¼ 1:9, B¼a ¼ 1:7 and C¼a ¼ 1:5 are plotted at different time intervals.The graph show that increase in the value of a result increase in the diffusion behavior and vise verse.Figure 1(d) show that the increase in the number of terms accelerate the convergence of the series solutions.In Figure 1(e, f), the absolute error analysis of the 10th and 15th-order approximations are presented.The numerical solutions of vð0:25, 0:25Þ using the 4th-order approximations for different values of h are depicted in Figure 1(g).In Figure 1(h), the h-curves of vð0:25, tÞ using 4th-order iterations are presented.In Figure 1(i), when a ¼ 2, h ¼ À1 in Equation ( 54), the result of HASTM is the same with HPLTT presented in Section 4. The comparison of exact solution and seven orderapproximations of Equation ( 54) when a ¼ 2, t ¼ 1, Hðx, tÞ ¼ exp À 1 10 x þ 1 10 t À Á : is presented in Figure 1(j).The result of Equation ( 61) is in excellent agreement with the Adomian decomposition method (ADM) (Jafari & Daftardar-Gejji, 2006), the homotopy analysis method (HAM) (Jafari & Seifi, 2009), and the modified homotopy perturbation method (MHPM) (Kumar et al., 2015).
In Figure 2(h), the h-curves of vð0:5, 0:5, tÞ using 4thorder approximations are depicted.The evolution results show a fast wave and diffusion behavior.The solutions of Equation ( 73) is in complete agreement with the Adomian decomposition method (ADM) (Jafari & Daftardar-Gejji, 2006) when h ¼ À1: The surface solutions behavior for different a in Equation ( 77) is presented in Figure 3(a-d) respectively.The HASTM solution for a ¼ 1 is depicted in Figure 3(a).The solution of the diffusion Equation ( 77) for H ¼ a ¼ 0:8, K ¼ a ¼ 0:5, and t ¼ y ¼ 1 are presented in Figure 3(b) and 3(c), respectively.We observe that the diffusion solution behavior is obtained for different values of a.In Figure 3(d), we obtain an exact and approximate solutions for the 10th-order approximations 77) for different values of a ¼ 1, H ¼ a ¼ 0:9, J ¼ a ¼ 0:8, and K ¼ a ¼ 0:7 at different time intervals.The absolute error analysis of the 10th and 15th-order approximations are presented in Figure 3(e,  f).Numerical simulations of vð0:1, 0:1, 0:1, 0:1Þ using the 4th-order approximations for different values of h are presented in Figure 3(g).In Figure 3(h), using the 4th-order approximations, the h-curves of vð0:1, 0:1, 0:1, tÞ are depicted for different h: The results show a clear differences for various values of a.The solutions of Equation ( 84) is in complete agreement with the homotopy perturbation Sumudu transform method (HPSTM) (Kumar et al., 2015), and the reduce differential transform method (RDTM) (Singh & Srivastava, 2018) when h ¼ À1: The classical wave solution (a ¼ 1) is plotted in Figure 4(a) and 4(b) respectively.The absolute error analysis of the 10th and 15th-order approximations are presented in Figure 4(c) and 4(d), respectively.In Figure 4(e), a comparison of the 15th-order approximations approximate and the exact solutions of the diffusion Equation ( 88) for a ¼ 1, and 0.5 are presented.The 10th-order approximations (v of the diffusion Equation ( 88) for a ¼ 0:5 is plotted in Figure 4(f).We observe that the diffusion solution behavior is maintain for a ¼ 0:5: The numerical   66) for a ¼ 1, 0.9, 0.7, 0.5.(e) Absolute error of the 10thorder approximations of Equation ( 66).(f) Absolute error of the 15th-order approximations of Equation ( 66).(g) 4th-order approximations of Equation ( 66) for different h.(h) h-curves of Equation (66) using 4th-order approximations.77) for a ¼ 1, 0.8, 0.7, 0.5.(e) Absolute error of the 10th-order approximations of Equation ( 77).(f) Absolute error of the 15th-order approximations of Equation ( 77).(g) 4thorder approximations of Equation ( 77) for different h.(h) h-curves of Equation ( 77) using 4th-order approximations.88).(d) Absolute error of the 15th-order approximations of Equation ( 88).(e) Comparison of approximate and exact solutions of Equation ( 88) for a ¼ 1, 0.5.(f) Approximate solution of 15th-order approximations of Equation ( 88) when a ¼ 0.5.(g) 4th-order approximations of Equation ( 88) for different h.(h) h-curves of Equation ( 88) using 4th-order approximations solutions of vð0:025, 0:025Þ using the 4th-order approximations for different values of h are presented in Figure 4(g).In Figure 4(h), using the 4th-order approximations, the h-curves of vð0:025, tÞ are presented.The same result is obtained using the variational iteration method (VIM) (Wazwaz, 2007), and the homotopy analysis method (HAM) (Dehghan, Manafian, & Saadatmandi, 2009) as a special cases.Finally, the mathematical formulations and findings of the proposed semi-analytical method proved to have the following features advantages.It can easily be applied to highly nonlinear problems.Unlike the explicit and implicit numerical methods, the proposed method can be used as numerical or analytical method which requires only the initial conditions.It avoids the cumbersome of some computational methods and the series solutions are always convergent with the help of the nonzero convergence-control parameter.Based on the numerical simulations of Equations (54) (66) (77), and (88) it is clear that the ADM (Jafari & Daftardar-Gejji, 2006), the VIM (Wazwaz, 2007), the RDTM (Singh & Srivastava, 2018), and the MHPM (Jafari & Momani, 2007) are special cases of the HASTM when the nonzero convergence-control parameter h ¼ À1:

Conclusion
In this paper, we proposed a novel computational method called the homotopy analysis Shehu transform method (HASTM) for solving fractional diffusion equations.The HASTM is based on the homotopy analysis method, and an integral transform which generalizes the Laplace transform and the Sumudu integral transform.The proposed semi-analytical method reduces the computational size, avoids round-off errors, and the series solutions converge rapidly within a few iterations with the help of the nonzero convergence-control parameter.We discussed many useful properties of the proposed technique including convergence analysis and the basic idea of the method.The multidimensional fractional diffusion equations are successfully solved using the HASTM, and the results obtained are compared with the results from previous techniques.The HASTM can be considered as a refinement of the pre-existing methods.

Table 1 .
Duality of the Shehu transform with some integral transforms in the literature.