Analytical Simulation of Natural Convection Between Two Concentric Horizontal Circular Cylinders: A Hybrid Fourier Transform-Homotopy Perturbation Approach

In recent years, the study of heat transfer theory due to natural convection has garnered significant interest among scientists and engineers, owing to its crucial applications in various scientific and technological fields. These applications encompass heat storage systems, cooling of electronic components, solar collectors, nuclear reactors, and aircraft cabin insulation, among others [1-3]. One particularly important problem in natural convection, which has intrigued numerous researchers, involves the phenomenon between two concentric horizontal circular cylinders maintained at varying uniform temperatures.

Early contributions to this area include the experimental solutions provided by Crawford and Lemlich [4], who conducted a numerical study to approximate the steady-state differential equations with suitable difference equations. Their work explored the effect of diameter ratios at 2, 8, and 57, in the case of a Prandtl number of 0.7. Subsequent researchers have built upon this foundation, developing numerical and analytical solutions to the problem. For instance, Mack and Bishop [5] performed an analytical investigation of natural convection between two horizontal circular concentric cylinders with small temperature differences. Their approach utilized Rayleigh number power series to solve the governing mathematical equations, and they discussed the characteristics of their solutions, such as local and global heat transfer rates, streamline formation, velocity, and temperature distributions. Additionally, they analyzed the effects of radius ratio, Prandtl number, and Rayleigh number.

Kuehn and Goldstein [1] employed the finite difference method in a numerical study to obtain experimental solutions for natural convection between two horizontal circular concentric cylinders. Tsui and Tremblay [6] conducted a theoretical-numerical study for Grashof numbers ranging from 7×102 to 9×104, with a fixed Prandtl number of 0.7, and diameter ratio differences of 1.2, 1.5, and 2. Pop et al. [7] derived analytical solutions for transient natural convection between two concentric horizontal circular porous cylinders using the method of matched asymptotic expansions. Their approach divided the problem into three regions (inner boundary layer, core region, and outer boundary layer) and determined the analytical solutions for each region separately. They observed that their solutions markedly differed from steady-state solutions.

Sano and Kuribayashi [8] studied transient natural convection around a horizontal circular cylinder using the method of matched asymptotic expansions, resulting in new analytical solutions. Their research aimed to fill a gap in previous work by considering the displacement effect. Hassan and Al-Lateef [9] presented a numerical study to find the solution for two-dimensional transient natural convection heat transfer from isothermal horizontal cylindrical annuli. They employed the alternating direction implicit (ADI) method to solve both the vorticity and energy equations, while the stream function equation was solved using the successive over relaxation (SOR) method. Their results were summarized by Nussult number versus Grashof number curves, with diameter ratios and Prandtl number as parameters. Touzani et al. [10] numerically investigated natural convection in horizontal annulus with two heating blocks, noting that heat transfer was more significant in the upper region of the annulus and that the presence of the block improved overall heat transfer. Al-Saif and Al-Griffi [11] proposed a new algorithm that combined the homotopy perturbation method and the Yang transform to conduct an analytical study on the problem of two-dimensional transient natural convection between two concentric horizontal cylindrical annuli. Their work explored the effects of Grashof and Prandtl numbers, as well as the radius ratio, on heat transfer and fluid flow (air) at various values.

Despite the merits of the aforementioned methods and their capacity to find numerical and analytical solutions for non-linear problems, particularly in the context of heat transfer by natural convection, several challenges persist. These include high computational operations, time-consuming procedures, and the difficulty of using integral transformations (such as Laplace transform, Fourier transform, and Yang transform) to solve non-linear problems. To address these limitations, this study proposes a new algorithm combining the homotopy perturbation method (HPM) with the Fourier transform (FT), supported by the convolution theory, to create a hybrid procedure denoted as FT-HPM. This method leverages the convolution theory to reduce computational complexity related to integrative operations when using the homotopy perturbation method alone while also mitigating the difficulty of employing the Fourier transform to solve non-linear differential equations. The newly developed FT-HPM algorithm is applied to obtain approximate analytical solutions for the two-dimensional natural convection between two horizontal concentric cylinders. Tabular and graphical results of the new analytical approximate solutions demonstrate the usefulness, importance, and necessity of using this method. The accuracy and efficiency of this approach are validated by its agreement with the results of previous methods [10, 12, 13]. Furthermore, the convergence analysis of the solutions is studied both theoretically and experimentally by proving two theorems and deducing the necessary condition for the convergence of the method. The study also investigates the effect of Grashof number, Prandtl number, and radius ratio on the heat transfer and fluid flow in the annular space.

The main idea in this part is to improve the homotopy perturbation method by using the Fourier transform to get a new, more advanced procedure. To introduce the basic algorithm for this procedure, we rewrite Eq. (1) to get the following formula:

$L(U)+R(U)+N(U)=q(r)$                  (9)

where, $L=\partial^n / \partial r^n$  represents to the linear differential operator, R and N refer to the linear differential operator such that its order is less than L, the general non-linear differential operator, respectively, and q(r) denotes the source term. Moreover, the main steps of this method can be stated as follows:

By using the HPM, we have:

$(1-p)\left[L(U)-L\left(u_0\right)\right]+p[A(U)-q(r)]=0$                  (10)

Postulate that A(U)=L(U)+R(U)+N(U) and $L=\frac{\partial^n}{\partial r^n}$ , then, we deduce:

$\frac{\partial^n}{\partial r^n}(U)=\frac{\partial^n u_0}{\partial r^n}-p \frac{\partial^n u_0}{\partial r^n}-p[R(U)+N(U)-q(r)]$                  (11)

By the effect of the Fourier transform on both sides of the Eq. (11), we get:

$\mathcal{F}\left[\frac{\partial^n}{\partial r^n}(U)\right]=\mathcal{F}\left[\frac{\partial^n u_0}{\partial r^n}\right]-p \mathcal{F}\left[\begin{array}{c}\frac{\partial^n}{\partial r^n}\left(u_0\right)+ \\ {[R(U)+N(U)-q(r)]}\end{array}\right]$                  (12)

where, $\mathcal{F}[g(t)]=\mathcal{F}(\omega)=\int_{-\infty}^{\infty} g(t) e^{-i \omega t} d t$.

Using the differentiation property of the Fourier transform, we obtain:

$(i \omega)^n \mathcal{F}[U]=\mathcal{F}\left[\frac{\partial^n u_0}{\partial r^n}-p\left(\begin{array}{c}\frac{\partial^n}{\partial r^n}\left(u_0\right)+ \\ {[R(U)+N(U)-q(r)]}\end{array}\right)\right]$                  (13)

The rearrangement of Eq. (13) leads to:

$\mathcal{F}[U]=\frac{1}{(i \omega)^n} \mathcal{F}\left[\frac{\partial^n u_0}{\partial r^n}-p\left(\begin{array}{c}\frac{\partial^n}{\partial r^n}\left(u_0\right)+ \\ {[R(U)+N(U)-q(r)]}\end{array}\right)\right]$                  (14)

according to the properties of the Fourier transform, we get:

$\mathcal{F}[U]=\left\{\begin{array}{c}\frac{1}{2(n-1) !} \mathcal{F}\left[r^{n-1} \operatorname{sgn}(r)\right] \times \\ \mathcal{F}\left[\begin{array}{c}\frac{\partial^n}{\partial r^n}\left(u_0\right)- \\ p\left(\frac{\partial^n}{\partial r^n}\left(u_0\right)+[R(U)+N(U)-q(r)]\right)\end{array}\right]\end{array}\right\}$                  (15)

where, $\mathcal{F}\left(t^n \operatorname{sgn}(t)\right)=(-i)^{n+1} \frac{2(n !)}{\omega^{n+1}}$ .

Applying the idea of convolution theory on the right-hand side of the Eq. (15) to get the following result:

$\mathcal{F}[U]=\mathcal{F}\left\{\left[\begin{array}{c}\frac{1}{2(n-1) !}\left[r^{n-1} \operatorname{sgn}(r)\right] * \\ \frac{\partial^n}{\partial r^n}\left(u_0\right)- \\ p\left(\frac{\partial^n}{\partial r^n}\left(u_0\right)+[R(U)+N(U)-q(r)]\right)\end{array}\right]\right\}$                  (16)

where, the operation * is given by:

$\begin{gathered}f(t) * g(t)=\int_{-\infty}^{\infty} f(t-\eta) g(\eta) d \eta \equiv \int_{-\infty}^{\infty} f(\eta) g(t-\eta) d \eta \\ \text { and } \mathcal{F}[f(t) * g(t)]=\mathcal{F}[f(t)] * \mathcal{F}[g(t)].\end{gathered}$

By Taking the inverse Fourier transform for both sides of Eq. (16), we obtain:

$U=\left\{\frac{1}{2(n-1) !}\left[r^{n-1} \operatorname{sgn}(r)\right] *\left[\frac{\partial^n}{\partial r^n}\left(u_0\right)-p\left(\frac{\partial^n}{\partial r^n}\left(u_0\right)+[R(U)+N(U)-q(r)]\right)\right]\right\}$                  (17)

Eq. (17) can be described as follows:

$U=\int_{-\infty}^{\infty}\left\{\begin{array}{c}{\left[\frac{(r-\eta)^{n-1} \operatorname{sgn}(r-\eta)}{2(n-1) !}\right] \times} \\ {\left.\left[\begin{array}{c}\frac{\partial^n}{\partial r^n}\left(u_0\right)-p \frac{\partial^n}{\partial r^n}\left(u_0\right)- \\ p\{R(U)+N(U)-q(r)\}\end{array}\right]\right|_{r=n}}\end{array}\right\} d \eta$           (18)

From the assumption of the HPM, we have:

$U=\sum_{m=0}^{\infty} p^j U_j$              (19)

and the non-linear terms can be decomposed as:

$N(U)=\sum_{j=0}^{\infty} p^j H_j$              (20)

where, H_j(U) represents the He’s polynomials [3] that are given by:

$\begin{gathered}H_j\left(U_0, U_1, U_2, \ldots, U_j\right)=\frac{1}{j !} \frac{\partial^j}{\partial p^j}\left[N\left(\sum_{i=0}^{\infty} p^i U_i\right)\right]_{p=0} \\ j=0,1,2,3, \ldots\end{gathered}$              (21)

Putting Eqns. (19) and (20) in to Eq. (18), we obtain:

$\sum_{m=0}^{\infty} p^j U_j=\int_{-\infty}^{\infty}\left\{\begin{array}{c}{\left[\frac{(r-\eta)^{n-1} \operatorname{sgn}(r-\eta)}{2(n-1) !}\right] \times} \\ {\left.\left[\frac{\partial^n u_0}{\partial r^n}-p\left(\frac{\partial^n u_0}{\partial r^n}+\left[\begin{array}{c}\left.R\left(\sum_{m=0}^{\infty} p^j U_j\right)+\right] \\ \sum_{j=0}^{\infty} p^j H_j-q(r)\end{array}\right]\right)\right]\right|_{r=\eta}}\end{array}\right\} d \eta$              (22)

By equating the coefficients of the same powers of p, we deduce:

$p^0: U_0=\int_{-\infty}^{\infty}\left\{\left.\left[\frac{(r-\eta)^{n-1} \operatorname{sgn}(r-\eta)}{2(n-1) !}\right]\left[\frac{\partial^n u_0}{\partial r^n}\right]\right|_{r=\eta}\right\} d \eta$              (23)

$p^1: U_1=-\int_{-\infty}^{\infty}\left\{\begin{array}{c}{\left[\frac{(r-\eta)^{n-1} \operatorname{sgn}(r-\eta)}{2(n-1) !}\right] \times} \\ {\left.\left[\frac{\partial^n u_0}{\partial r^n}+R\left(U_0\right)+H_0-q(r)\right]\right|_{r=\eta}}\end{array}\right\} d \eta$              (24)

$p^2: U_2=-\int_{-\infty}^{\infty}\left\{\begin{array}{c}{\left[\frac{(r-\eta)^{n-1} \operatorname{sgn}(r-\eta)}{2(n-1) !}\right] \times} \\ {\left.\left[R\left(U_1\right)+H_1\right]\right|_{r=\eta}}\end{array}\right\} d \eta$              (25)

$p^j: U_j=-\int_{-\infty}^{\infty}\left\{\begin{array}{c}{\left[\frac{(r-\eta)^{n-1} \operatorname{sgn}(r-\eta)}{2(n-1) !}\right] \times} \\ {\left.\left[R\left(U_{j-1}\right)+H_{j-1}\right]\right|_{r=\eta}}\end{array}\right\} d \eta$              (26)

Taking p=1, then the analytical approximate solution u can be given by:

$u=\lim _{p \rightarrow 1} U=\sum_{j=0}^{\infty} U_j$              (27)

Now, to apply the algorithm of FT-HPM, first, we simplify Eqns. (28) and (29) to get the following system:

$\begin{aligned} & \frac{\partial^4 \psi}{\partial r^4}+\frac{1}{r} \frac{\partial^3 \psi}{\partial r^3}+\frac{1}{r^2} \frac{\partial^4 \psi}{\partial \theta^2 \partial r^2}-\frac{1}{P r} \frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial \nabla^2 \psi}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial \nabla^2 \psi}{\partial \theta}\right) \\ & +\nabla^2\left(\frac{1}{r} \frac{\partial \psi}{\partial r}+\frac{1}{r^2} \frac{\partial^2 \psi}{\partial \theta^2}\right)-R a\left(\frac{\cos \theta}{r} \frac{\partial T}{\partial \theta}+\sin \theta \frac{\partial T}{\partial r}\right)=0\end{aligned}$     (31)

$\frac{\partial^2 T}{\partial r^2}+\frac{1}{r} \frac{\partial T}{\partial r}+\frac{1}{r^2} \frac{\partial^2 T}{\partial \theta^2}-\frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial T}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial T}{\partial \theta}\right)=0$    (32)

Then, the basic steps of the new technique for Eq. (31) and Eq. (32) are illustrated as follows.

By using HPM of Eq. (31) and Eq. (32), we obtain:

$\begin{aligned} & (1-p)\left[\frac{\partial^4 \psi}{\partial r^4}-\frac{\partial^4 \psi_0^*}{\partial r}\right]+ \\ & p\left[\begin{array}{l}\frac{\partial^4 \psi}{\partial r^4}+\frac{1}{r} \frac{\partial^3 \psi}{\partial r^3}+\frac{1}{r^2} \frac{\partial^4 \psi}{\partial \theta^2 \partial r^2}+\nabla^2\left(\frac{1}{r} \frac{\partial \psi}{\partial r}+\frac{1}{r^2} \frac{\partial^2 \psi}{\partial \theta^2}\right)- \\ \frac{1}{P r} \frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial \nabla^2 \psi}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial \nabla^2 \psi}{\partial \theta}\right)-R a\left(\frac{\cos \theta}{r} \frac{\partial T}{\partial \theta}+\sin \theta \frac{\partial T}{\partial r}\right)\end{array}\right]=0 \\ & (1-p)\left[\frac{\partial^2 T}{\partial r^2}-\frac{\partial^2 T_0^*}{\partial r^2}\right]+p\left[\begin{array}{l}\frac{\partial^2 T}{\partial r^2}+\frac{1}{r} \frac{\partial T}{\partial r}+\frac{1}{r^2} \frac{\partial^2 T}{\partial \theta^2} \\ -\frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial T}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial T}{\partial \theta}\right)\end{array}\right]=0 \\ & \end{aligned}$

where, $\psi_0^*$ and $T_0^*$ are the initial conditions. The rearrangement of the above equations yields:

$\frac{\partial^4 \psi}{\partial r^4}-\frac{\partial^4 \psi_0^*}{\partial r}+p \frac{\partial^4 \psi_0^*}{\partial r}+p\left[\begin{array}{c}\frac{1}{r} \frac{\partial^3 \psi}{\partial r^3}+\frac{1}{r^2} \frac{\partial^4 \psi}{\partial \theta^2 \partial r^2}-\frac{1}{P r} \frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial \nabla^2 \psi}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial \nabla^2 \psi}{\partial \theta}\right)+ \\ \nabla^2\left(\frac{1}{r} \frac{\partial \psi}{\partial r}+\frac{1}{r^2} \frac{\partial^2 \psi}{\partial \theta^2}\right)-R a\left(\frac{\cos \theta}{r} \frac{\partial T}{\partial \theta}+\sin \theta \frac{\partial T}{\partial r}\right)\end{array}\right]=0$    (33)

$\begin{gathered}\frac{\partial^2 T}{\partial r^2}-\frac{\partial^2 T_0^*}{\partial r^2}+p\left[\frac{\partial^2 T_0^*}{\partial r^2}+\frac{1}{r} \frac{\partial T}{\partial r}+\frac{1}{r^2} \frac{\partial^2 T}{\partial \theta^2}-\frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial T}{\partial r}-\right.\right. \left.\left.\frac{\partial \psi}{\partial r} \frac{\partial T}{\partial \theta}\right)\right]=0\end{gathered}$    (34)

From the boundary conditions, the second and third terms in Eqns. (33) and (34) are equal to zero, therefore,

$\begin{aligned} & \frac{\partial^4 \psi}{\partial r^4}= \\ & -p\left[\begin{array}{c}\frac{1}{r} \frac{\partial^3 \psi}{\partial r^3}+\frac{1}{r^2} \frac{\partial^4 \psi}{\partial \theta^2 \partial r^2}-\frac{1}{\operatorname{Pr}} \frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial \nabla^2 \psi}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial \nabla^2 \psi}{\partial \theta}\right)+ \\ \nabla^2\left(\frac{1}{r} \frac{\partial \psi}{\partial r}+\frac{1}{r^2} \frac{\partial^2 \psi}{\partial \theta^2}\right)-R a\left(\frac{\cos \theta}{r} \frac{\partial T}{\partial \theta}+\sin \theta \frac{\partial T}{\partial r}\right)\end{array}\right] \\ & \end{aligned}$    (35)

$\frac{\partial^2 T}{\partial r^2}=-p\left[\frac{1}{r} \frac{\partial T}{\partial r}+\frac{1}{r^2} \frac{\partial^2 T}{\partial \theta^2}-\frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial T}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial T}{\partial \theta}\right)\right]$       (36)

Applying the Fourier transform with respect to r on both sides of Eq. (35) and Eq. (36), we have:

$\begin{gathered}\mathcal{F}\left[\frac{\partial^4 \psi}{\partial r^4}\right]= \\ -p \mathcal{F}\left[\begin{array}{c}\frac{1}{r} \frac{\partial^3 \psi}{\partial r^3}+\frac{1}{r^2} \frac{\partial^4 \psi}{\partial \theta^2 \partial r^2}-\frac{1}{P r} \frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial \nabla^2 \psi}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial \nabla^2 \psi}{\partial \theta}\right)+ \\ \nabla^2\left(\frac{1}{r} \frac{\partial \psi}{\partial r}+\frac{1}{r^2} \frac{\partial^2 \psi}{\partial \theta^2}\right)-R a\left(\frac{\cos \theta}{r} \frac{\partial T}{\partial \theta}+\sin \theta \frac{\partial T}{\partial r}\right)\end{array}\right]\end{gathered}$    (37)

$\mathcal{F}\left[\frac{\partial^2 T}{\partial r^2}\right]=-p \mathcal{F}\left[\frac{1}{r} \frac{\partial T}{\partial r}+\frac{1}{r^2} \frac{\partial^2 T}{\partial \theta^2}-\frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial T}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial T}{\partial \theta}\right)\right]$      (38)

Using the differentiation property of the Fourier transform, we deduce:

$(i \omega)^4 \mathcal{F}[\psi]=-p \mathcal{F}\left[\begin{array}{c}\frac{1}{r} \frac{\partial^3 \psi}{\partial r^3}+\frac{1}{r^2} \frac{\partial^4 \psi}{\partial \theta^2 \partial r^2}-\frac{1}{P r} \frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial \nabla^2 \psi}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial \nabla^2 \psi}{\partial \theta}\right)+ \\ \nabla^2\left(\frac{1}{r} \frac{\partial \psi}{\partial r}+\frac{1}{r^2} \frac{\partial^2 \psi}{\partial \theta^2}\right)-R a\left(\frac{\cos \theta}{r} \frac{\partial T}{\partial \theta}+\sin \theta \frac{\partial T}{\partial r}\right)\end{array}\right]$   (39)

$(i \omega)^2 \mathcal{F}[T]=-p \mathcal{F}\left[\frac{1}{r} \frac{\partial T}{\partial r}+\frac{1}{r^2} \frac{\partial^2 T}{\partial \theta^2}-\frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial T}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial T}{\partial \theta}\right)\right]$          (40)

The rearrangement of the above system leads to:

$\mathcal{F}[\psi]=\frac{1}{\omega^4} p \mathcal{F}\left[\begin{array}{c}\frac{1}{P r} \frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial \nabla^2 \psi}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial \nabla^2 \psi}{\partial \theta}\right)-\frac{1}{r} \frac{\partial^3 \psi}{\partial r^3}-\frac{1}{r^2} \frac{\partial^4 \psi}{\partial \theta^2 \partial r^2}- \\ \nabla^2\left(\frac{1}{r} \frac{\partial \psi}{\partial r}+\frac{1}{r^2} \frac{\partial^2 \psi}{\partial \theta^2}\right)+R a\left(\frac{\cos \theta}{r} \frac{\partial T}{\partial \theta}+\sin \theta \frac{\partial T}{\partial r}\right)\end{array}\right]$     (41)

$\mathcal{F}[T]=\frac{1}{\omega^2} p \mathcal{F}\left[\frac{1}{r} \frac{\partial T}{\partial r}+\frac{1}{r^2} \frac{\partial^2 T}{\partial \theta^2}-\frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial T}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial T}{\partial \theta}\right)\right]$     (42)

By using the properties of the Fourier transform, we get:

$\begin{gathered}\mathcal{F}[\psi]=p\left(\frac{1}{2(3 !)}\right) \mathcal{F}\left[r^3 \operatorname{sgn}(r)\right] \times \\ \mathcal{F}\left[\begin{array}{c}\frac{1}{\operatorname{Pr}} \frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial \nabla^2 \psi}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial \nabla^2 \psi}{\partial \theta}\right)-\frac{1}{r} \frac{\partial^3 \psi}{\partial r^3}-\frac{1}{r^2} \frac{\partial^4 \psi}{\partial \theta^2 \partial r^2}- \\ \nabla^2\left(\frac{1}{r} \frac{\partial \psi}{\partial r}+\frac{1}{r^2} \frac{\partial^2 \psi}{\partial \theta^2}\right)+R a\left(\frac{\cos \theta}{r} \frac{\partial T}{\partial \theta}+\sin \theta \frac{\partial T}{\partial r}\right)\end{array}\right]\end{gathered}$     (43)

$\begin{gathered}\mathcal{F}[T]=p\left(\frac{-1}{2(1 !)}\right) \mathcal{F}[r \operatorname{sgn}(r)] \mathcal{F}\left[\frac{1}{r} \frac{\partial T}{\partial r}+\frac{1}{r^2} \frac{\partial^2 T}{\partial \theta^2}-\right. \\ \left.\frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial T}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial T}{\partial \theta}\right)\right]\end{gathered}$     (44)

Applying the idea of convolution theory on the right-hand side of the above system to get the following result:

$\mathcal{F}[\psi]=\frac{p}{12} \mathcal{F}\left\{\left[\begin{array}{c}{\left[r^3 \operatorname{sgn}(r)\right] *} \\ \left.\frac{1}{P r} \frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial \nabla^2 \psi}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial \nabla^2 \psi}{\partial \theta}\right)-\frac{1}{r} \frac{\partial^3 \psi}{\partial r^3}-\frac{1}{r^2} \frac{\partial^4 \psi}{\partial \theta^2 \partial r^2}-\right] \\ \nabla^2\left(\frac{1}{r} \frac{\partial \psi}{\partial r}+\frac{1}{r^2} \frac{\partial^2 \psi}{\partial \theta^2}\right)+R a\left(\frac{\cos \theta}{r} \frac{\partial T}{\partial \theta}+\sin \theta \frac{\partial T}{\partial r}\right)\end{array}\right]\right\}$     (45)

$\mathcal{F}[T]=\frac{p}{2} \mathcal{F}\left\{[r s g n(r)] *\left[\begin{array}{c}\frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial T}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial T}{\partial \theta}\right) \\ -\frac{1}{r} \frac{\partial T}{\partial r}-\frac{1}{r^2} \frac{\partial^2 T}{\partial \theta^2}\end{array}\right]\right\}$     (46)

Taking the Fourier inverse for both sides of Eqns. (45) and (46) we get:

$\psi=\frac{1}{12} p\left\{\left[\begin{array}{c}{\left[r^3 \operatorname{sgn}(r)\right] *} \\ \frac{1}{P r} \frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial \nabla^2 \psi}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial \nabla^2 \psi}{\partial \theta}\right)-\frac{1}{r} \frac{\partial^3 \psi}{\partial r^3}-\frac{1}{r^2} \frac{\partial^4 \psi}{\partial \theta^2 \partial r^2}- \\ \nabla^2\left(\frac{1}{r} \frac{\partial \psi}{\partial r}+\frac{1}{r^2} \frac{\partial^2 \psi}{\partial \theta^2}\right)+R a\left(\frac{\cos \theta}{r} \frac{\partial T}{\partial \theta}+\sin \theta \frac{\partial T}{\partial r}\right)\end{array}\right]\right\}$     (47)

$T=\frac{1}{2} p\left\{[r \operatorname{sgn}(r)] *\left[\begin{array}{c}\frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial T}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial T}{\partial \theta}\right) \\ -\frac{1}{r} \frac{\partial T}{\partial r}-\frac{1}{r^2} \frac{\partial^2 T}{\partial \theta^2}\end{array}\right]\right\}$     (48)

Eq. (47) and Eq. (48) can be described as follows:

$\psi=\int_{-\infty}^{\infty}\left\{\left[\begin{array}{c}\frac{1}{12} p\left[(r-\eta)^3 \operatorname{sgn}(r-\eta)\right] \times \\ {\left.\left[\begin{array}{c}\frac{1}{P r} \frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial \nabla^2 \psi}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial \nabla^2 \psi}{\partial \theta}\right)-\frac{1}{r} \frac{\partial^3 \psi}{\partial r^3}-\frac{1}{r^2} \frac{\partial^4 \psi}{\partial \theta^2 \partial r^2}- \\ \nabla^2\left(\frac{1}{r} \frac{\partial \psi}{\partial r}+\frac{1}{r^2} \frac{\partial^2 \psi}{\partial \theta^2}\right)+R a\left(\frac{\cos \theta}{r} \frac{\partial T}{\partial \theta}+\sin \theta \frac{\partial T}{\partial r}\right)\end{array}\right]\right|_{r=\eta}}\end{array}\right\} d \eta\right.$     (49)

$T=\frac{1}{2} p \int_{-\infty}^{\infty}\left\{\begin{array}{c}{[(r-\eta) \operatorname{sgn}(r-\eta)] \times} \\ {\left.\left[\frac{1}{r}\left(\frac{\partial \psi}{\partial \theta} \frac{\partial T}{\partial r}-\frac{\partial \psi}{\partial r} \frac{\partial T}{\partial \theta}\right)-\frac{1}{r} \frac{\partial T}{\partial r}-\frac{1}{r^2} \frac{\partial^2 T}{\partial \theta^2}\right]\right|_{r=\eta}}\end{array}\right\} d \eta$     (50)

From the assumption of the HPM, we have:

$\psi=\sum_{m=0}^{\infty} p^j \psi_j$ and $T=\sum_{m=0}^{\infty} p^j T_j$     (51)

and the nonlinear terms can be represented as:

$\left.\begin{array}{l}\frac{\partial \psi}{\partial \theta} \frac{\partial \nabla^2 \psi}{\partial r}=\sum_{j=0}^{\infty} p^j H_j, \frac{\partial \psi}{\partial r} \frac{\partial \nabla^2 \psi}{\partial \theta}=\sum_{j=0}^{\infty} p^j H_j^* \\ \frac{\partial \psi}{\partial \theta} \frac{\partial T}{\partial r}=\sum_{j=0}^{\infty} p^j G_j \text { and } \frac{\partial \psi}{\partial r} \frac{\partial T}{\partial \theta}=\sum_{j=0}^{\infty} p^j G_j^*\end{array}\right\}$     (52)

Substituting Eqns. (51) and (52) in Eqns. (49) and (50), we get:

$\begin{gathered}\sum_{j=0}^{\infty} p^j \psi_j=\int_{-\infty}^{\infty}\left\{p\left[\frac{(r-\eta)^3 \operatorname{sgn}(r-\eta)}{12}\right] \times\left[\frac{1}{P r} \frac{1}{r}\left(\sum_{j=0}^{\infty} p^j H_j-\sum_{j=0}^{\infty} p^j H_j^*\right)-\frac{1}{r^2} \frac{\partial^4}{\partial \theta^2 \partial r^2} \sum_{j=0}^{\infty} p^j \psi_j-\frac{1}{r} \frac{\partial^3}{\partial r^3} \sum_{j=0}^{\infty} p^j \psi_j-\nabla^2\left(\frac{1}{r} \frac{\partial}{\partial r} \sum_{j=0}^{\infty} p^j \psi_j+\right.\right.\right. \\ \left.\left.\left.\frac{1}{r^2} \frac{\partial^2}{\partial \theta^2} \sum_{j=0}^{\infty} p^j \psi_j\right)+R a\left(\frac{\cos \theta}{r} \frac{\partial}{\partial \theta} \sum_{j=0}^{\infty} p^j T_j+\sin \theta \frac{\partial}{\partial r} \sum_{j=0}^{\infty} p^j T_j\right)\right]\left.\right|_{r=\eta}\right\} d \eta\end{gathered}$    (53)

$\sum_{j=0}^{\infty} p^j T_j=\int_{-\infty}^{\infty}\left\{p\left[\frac{(r-\eta) \operatorname{sgn}(r-\eta)}{2}\right] \times\left.\left[\begin{array}{c}\frac{1}{r}\left(\sum_{j=0}^{\infty} p^j G_j-\sum_{j=0}^{\infty} p^j G_j^*\right)- \\ \frac{1}{r} \frac{\partial}{\partial r} \sum_{j=0}^{\infty} p^j T_j-\frac{1}{r^2} \frac{\partial^2}{\partial \theta^2} \sum_{j=0}^{\infty} p^j T_j\end{array}\right]\right|_{r=\eta}\right\} d \eta$     (54)

Comparing the coefficients of the same powers of p, we obtain:

$p^0:\left\{\begin{array}{c}\psi_0=\int_{-\infty}^{\infty}\left\{\left.\left[\frac{(r-\eta)^3 \operatorname{sgn}(r-\eta)}{12}\right][0]\right|_{r=\eta}\right\} d \eta \Rightarrow \\ (i \omega)^4 \mathcal{F}[\psi]=0 \Rightarrow \frac{\partial^4 \psi_0}{\partial r^4}=0 \\ T_0=\int_{-\infty}^{\infty}\left\{\left.\left[\frac{(r-\eta) \operatorname{sgn}(r-\eta)}{2}\right][0]\right|_{r=\eta}\right\} d \eta \Rightarrow \\ (i \omega)^2 \mathcal{F}[T]=0 \Rightarrow \frac{\partial^2 T_0}{\partial r^2}=0\end{array}\right.$     (55)

$p^1:\left\{\begin{array}{l}\psi_1=\int_{-\infty}^{\infty}\left\{\left[\frac{(r-\eta)^3 \operatorname{sgn}(r-\eta)}{12}\right] \times\left.\left[\frac{1}{P r} \frac{1}{r}\left(H_0-H_0^*\right)-\frac{1}{r^2} \frac{\partial^4 \psi_0}{\partial \theta^2 \partial r^2}+R a\left(\frac{\cos \theta}{r} \frac{\partial T_0}{\partial \theta}+\sin \theta \frac{\partial T_0}{\partial r}\right)-\frac{1}{r} \frac{\partial^3 \psi_0}{\partial r^3}-\nabla^2\left(\frac{1}{r} \frac{\partial \psi_0}{\partial r}+\frac{1}{r^2} \frac{\partial^2 \psi_0}{\partial \theta^2}\right)\right]\right|_{r=\eta}\right\} d \eta \\ T_1=\int_{-\infty}^{\infty}\left\{\left[\frac{(r-\eta) \operatorname{sgn}(r-\eta)}{2}\right] \times\left.\left[\frac{1}{r}\left(G_0-G_0^*\right)-\frac{1}{r} \frac{\partial T_0}{\partial r}-\frac{1}{r^2} \frac{\partial^2 T_0}{\partial \theta^2}\right]\right|_{r=\eta}\right\} d \eta\end{array}\right.$     (56)

$p^2:\left\{\begin{array}{c}\psi_2=\int_{-\infty}^{\infty}\left\{\left[\frac{(r-\eta)^3 \operatorname{sgn}(r-\eta)}{12}\right] \times\left.\left[\frac{1}{\operatorname{Pr}} \frac{1}{r}\left(H_1-H_1^*\right)-\frac{1}{r^2} \frac{\partial^4 \psi_1}{\partial \theta^2 \partial r^2}+R a\left(\frac{\cos \theta}{r} \frac{\partial T_1}{\partial \theta}+\sin \theta \frac{\partial T_1}{\partial r}\right)-\frac{1}{r} \frac{\partial^3 \psi_1}{\partial r^3}-\nabla^2\left(\frac{1}{r} \frac{\partial \psi_1}{\partial r}+\frac{1}{r^2} \frac{\partial^2 \psi_1}{\partial \theta^2}\right)\right]\right|_{r=\eta}\right\} d \eta \\ T_2=\int_{-\infty}^{\infty}\left\{\left[\frac{(r-\eta) \operatorname{sgn}(r-\eta)}{2}\right] \times\left.\left[\frac{1}{r}\left(G_1-G_1^*\right)-\frac{1}{r} \frac{\partial T_1}{\partial r}-\frac{1}{r^2} \frac{\partial^2 T_1}{\partial \theta^2}\right]\right|_{r=\eta}\right\} d \eta\end{array}\right.$    (57)

$p^j:\left\{\begin{array}{c}\psi_j=\int_{-\infty}^{\infty}\left\{\left[\frac{(r-\eta)^3 \operatorname{sgn}(r-\eta)}{12}\right] \times\left.\left[\frac{1}{P r} \frac{1}{r}\left(H_{j-1}-H_{j-1}^*\right)-\frac{1}{r^2} \frac{\partial^4 \psi_{j-1}}{\partial \theta^2 \partial r^2}+R a\left(\frac{\cos \theta}{r} \frac{\partial T_{j-1}}{\partial \theta}+\sin \theta \frac{\partial T_{j-1}}{\partial r}\right)-\frac{1}{r} \frac{\partial^3 \psi_{j-1}}{\partial r^3}-\nabla^2\left(\frac{1}{r} \frac{\partial \psi_{j-1}}{\partial r}+\frac{1}{r^2} \frac{\partial^2 \psi_{j-1}}{\partial \theta^2}\right)\right]\right|_{r=\eta}\right\} d \eta \\ T_j=\int_{-\infty}^{\infty}\left\{\left[\frac{(r-\eta) \operatorname{sgn}(r-\eta)}{2}\right] \times\left.\frac{1}{r}\left[\left(G_{j-1}-G_{j-1}^*\right)-\frac{\partial T_{j-1}}{\partial r}-\frac{1}{r} \frac{\partial^2 T_{j-1}}{\partial \theta^2}\right]\right|_{r=\eta}\right\} d \eta\end{array}\right.$  (58)

where,

$\left.\begin{array}{c}H_0=\frac{\partial \psi_0}{\partial \theta} \frac{\partial \nabla^2 \psi_0}{\partial r}, H_1=\frac{\partial \psi_0}{\partial \theta} \frac{\partial \nabla^2 \psi_1}{\partial r}+\frac{\partial \psi_1}{\partial \theta} \frac{\partial \nabla^2 \psi_0}{\partial r} \\ H_0^*=\frac{\partial \psi_0}{\partial r} \frac{\partial \nabla^2 \psi_0}{\partial \theta}, H_1^*=\frac{\partial \psi_0}{\partial r} \frac{\partial \nabla^2 \psi_1}{\partial \theta}+\frac{\partial \psi_1}{\partial r} \frac{\partial \nabla^2 \psi_0}{\partial \theta} \\ G_0=\frac{\partial \psi_0}{\partial \theta} \frac{\partial T_0}{\partial r}, G_1=\frac{\partial \psi_0}{\partial \theta} \frac{\partial T_1}{\partial r}+\frac{\partial \psi_1}{\partial \theta} \frac{\partial T_0}{\partial r} \\ G_0^*=\frac{\partial \psi_0}{\partial r} \frac{\partial T_0}{\partial \theta} \text { and } G_1^*=\frac{\partial \psi_0}{\partial r} \frac{\partial T_1}{\partial \theta}+\frac{\partial \psi_1}{\partial r} \frac{\partial T_0}{\partial \theta}\end{array}\right\}$     (59)

To solve Eq. (55), we use the integration with respect to r subject to the boundary conditions in Eq. (30), to get:

$\psi_0=\phi(\theta)\left(\begin{array}{c}\frac{1}{6} r^3-\frac{R+1}{4} r^2+\frac{R}{2} r+ \\ \frac{1}{24}\left(R^3-3 R^2-3 R+1\right)\end{array}\right), T_0=\frac{(R-r)}{R-1}$     (60)

$\psi_1=\frac{1}{48} \frac{1}{\left(R^2-R\right)}\left[\begin{array}{c}-R a R\left\{\begin{array}{c}2 r^4-4(R+1) r^3+6\left(R^2+1\right) r^2 \\ -4\left(R^3+1\right) r+R^4+1\end{array}\right\} \sin \theta \\ -\frac{1}{M}\left\{\begin{array}{c}2 R a R(R-1)\left(R^3-3 R^2+3 R-1\right) \\ \times\left[\left(-3 R r^3-3 R^2 r\right) \ln (R)+\ldots\right]\end{array}\right\} \sin \theta\end{array}\right]$     (61)

$T_1=\frac{1}{144} \frac{1}{(R-1)}\left[\begin{array}{c}\frac{3}{M} R a R(R-1)^3 \times \\ \left\{\begin{array}{c}r(R+1)\left(R^2-4 R+1\right)\left(\begin{array}{c}\ln (R)- \\ 2 \ln (r)\end{array}\right) \\ +\left(2 r^3-12 r^2+3 r-2\right) R-\frac{2}{3} r^4+ \\ 2 r^3+(2 r+6) R^2+6\left(R^2+1\right) r^2 \\ +\frac{1}{3} r-\left(\frac{1}{3} r-2\right) R^3\end{array}\right\} \cos (\theta) \\ -\frac{144}{2}((2 \ln (r)-\ln (R)-2) r+R+1)\end{array}\right]$    (62)

$\psi_2=\frac{1}{144} \frac{\operatorname{Raln}(R)}{(R-1) M}\left[\begin{array}{c}\left(72 r^4+390 r^3-54 r^2-9 r+6\right) R \\ +72 R \ln (R)(R+1)^2 r^3-15 r^3- \\ 36 R r^3\left(\begin{array}{l}-4 R^3+10 R^2 \\ +R^4+4 R+5\end{array}\right) \ln (r)+\cdots+ \\ \left(\begin{array}{c}-\frac{9671}{735}-\frac{24067}{210} r-\frac{1766}{7} r^2+\frac{26525}{42} r^3 \\ +\frac{2200}{3} r^4-\frac{9873}{35} r^5+\frac{3064}{35} r^6-\frac{2048}{147} r^7\end{array}\right) \\ \quad \times\left(\frac{7(R-1) \cos (\theta) R a}{160 P r \ln (R) M}\right) R^2+\cdots\end{array}\right] \sin \theta$   (63)

$T_2=\frac{31(R-1)^3 R a}{161280 M^2}$$\left[\begin{array}{c}\frac{140 R a}{31}\left(R^3-4 R^2+R\right)(R+1) r \times \\ \left(\begin{array}{c}\left(R^3-3 R^2+3 R+1\right) \ln (r) \ln (R) \\ +\cdots\end{array}\right) \\ +\left(\begin{array}{c}\frac{35}{31}-\frac{280}{31} r-\frac{560}{31} r^2-\frac{840}{31} r^3 \\ +\frac{2870}{31} r^4-\frac{1400}{31} r^5+\frac{560}{93} r^6\end{array}\right) R^2+\cdots\end{array}\right]$$\cos \theta$    (64)

where

$\begin{gathered}\phi(\theta)=\frac{\left(R^3-3\left(R^2-R\right)+1\right)}{M} \sin \theta, \\ M=\left[6\left(R^2+R\right) \ln (R)-2 R^4+7 R^3-21 R^2+17 R-1\right] .\end{gathered}$

The analytical approximate solutions ψ, T are given by putting p=1 as:

$\psi=\lim _{N \rightarrow \infty} \sum_{j=0}^N \psi_j$     (65)

$T=\lim _{N \rightarrow \infty} \sum_{j=0}^N T_j$     (66)

In this section, we will present some important definitions and theories through which we can study convergence analysis with finding the necessary condition for convergence of the approximate analytic solutions (65, 66) that were found depending on the new algorithm (FT-HPM). Moreover, we must begin with the following definition:

Definition 7.1 Let $\mathcal{N}: \mathcal{H} \rightarrow \mathbb{R}$ be a non-linear mapping, where $\mathcal{H}, \mathbb{R}$ represents the Banach space, the set of real numbers, respectively. Then, the sequence of the solutions can be written as:

$E_{n+1}=\mathcal{N}\left(E_n\right), E_n=\sum_{j=0}^n h_i, j=0,1,2,3, \ldots$             (71)

where, $\mathcal{N}$ satisfies the Lipschitz condition, such that for $\gamma \in$ $\mathbb{R}$, we have:

$\left\|\mathcal{N}\left(E_n\right)-\mathcal{N}\left(E_{n-1}\right)\right\| \leq \gamma\left\|E_n-E_{n-1}\right\|, 0<\gamma<1$             (72)

Theorem 7.1 The series of the analytical-approximate solution $\psi(r, \theta)=\sum_{j=0}^{\infty} \psi_j(r, \theta)$ that is generated from applying the new scheme (FT-HPM) converges if it satisfies the following condition:

$\left\|E_{n+1}-E_n\right\| \rightarrow 0$ as $n \rightarrow \infty$ for $0<\gamma<1$             (73)

Proof:

$\begin{gathered}\left\|E_{n+1}-E_n\right\|=\left\|\sum_{j=0}^{n+1} \psi_j-\sum_{j=0}^n \psi_j\right\|= \\ \left\|\begin{array}{l}\psi_0+\sum_{j=1}^{n+1} \psi_j-\| \\ {\left[\psi_0+\sum_{j=1}^n \psi_j\right]}\end{array}\right\| \\ =\left\|\begin{array}{l}\psi_0+\sum_{j=1}^{n+1} L_1^{-1}\left[\mathcal{W}_{j-1}\right]-\| \\ \left\{\psi_0+\sum_{j=1}^n L_1^{-1}\left[\mathcal{W}_{j-1}\right]\right\}\end{array}\right\| \\ =\left\|\begin{array}{l}\psi_0+L_1^{-1} \sum_{j=1}^{n+1}\left[\mathcal{W}_{j-1}\right]-\| \\ \left\{\psi_0+L_1^{-1} \sum_{j=1}^n\left[\mathcal{W}_{j-1}\right]\right\}\end{array}\right\|\end{gathered}$

Since, $E_{n+1}=\mathcal{N}\left(E_n\right)$, then

$\begin{gathered}\left\|E_{n+1}-E_n\right\|=\left\|\begin{array}{c}L_1^{-1} \mathcal{N} \sum_{j=0}^n\left[\mathcal{W}_{j-1}\right]- \\ L_1^{-1} \mathcal{N} \sum_{j=0}^{n-1}\left[\mathcal{W}_{j-1}\right]\end{array}\right\| \\ \quad=\left\|L_1^{-1} \mathcal{N}\left[\sum_{j=0}^n \psi_j\right]-L_1^{-1} N\left[\sum_{j=0}^{n-1} \psi_j\right]\right\| \\ \quad \leq\left|L_1^{-1}\right|\left\|\mathcal{N}\left[\sum_{j=0}^n \psi_j\right]-N\left[\sum_{j=0}^{n-1} \psi_j\right]\right\| \\ \leq \gamma\left\|\sum_{j=0}^n L_1^{-1}\left[\mathcal{W}_{j-1}\right]-\sum_{j=0}^{n-1} L_1^{-1}\left[\mathcal{W}_{j-1}\right]\right\| \\ \leq \gamma^2\left\|\sum_{j=0}^{n-1} L_1^{-1}\left[\mathcal{W}_{j-1}\right]-\sum_{j=0}^{n-2} L_1^{-1}\left[\mathcal{W}_{j-1}\right]\right\| \\ \quad \vdots \\ \leq \gamma^n\left\|\sum_{j=0}^1 L_1^{-1}\left[\mathcal{W}_{j-1}\right]-\sum_{j=0}^0 L_1^{-1}\left[\mathcal{W}_{j-1}\right]\right\| \\ =\gamma^n\left\|E_1-E_0\right\| \rightarrow 0 \text { as } n \rightarrow \infty \text { for } 0<\gamma<1 .\end{gathered}$

where,

$L_1^{-1}(\cdot)=\int_{-\infty}^{\infty}\left[\frac{(r-\eta)^3 \operatorname{sgn}(r-\eta)}{12}\right](\cdot)_{r=\eta} d \eta,W_{j-1}=\left[\begin{array}{c}\frac{1}{P r} \frac{1}{r}\left(H_{j-1}-H_{j-1}^*\right)+R a\left(\frac{\cos \theta}{r} \frac{\partial T_{j-1}}{\partial \theta}+\sin \theta \frac{\partial T_{j-1}}{\partial r}\right) \\ -\frac{1}{r^2} \frac{\partial^4 \psi_{j-1}}{\partial \theta^2 \partial r^2}-\frac{1}{r} \frac{\partial^2 \psi_{j-1}}{\partial r^3}-\nabla^2\left(\frac{1}{r} \frac{\partial \psi_{j-1}}{\partial r}+\frac{1}{r^2} \frac{\partial^2 \psi_{j-1}}{\partial \theta^2}\right)\end{array}\right]$.

Theorem 7.2 The necessary condition for the convergence of the series of the solutions $T(r, \theta)=\sum_{j=0}^{\infty} T_j(r, \theta)$ that is obtained by the based algorithm (FT-HPM), is achieving of the following property:

$\left\|E_{n+1}-E_n\right\| \rightarrow 0$ as $n \rightarrow \infty$ for $0<\gamma<1$             (74)

Proof:

$\left\|E_{n+1}-E_n\right\|=\left\|\sum_{j=0}^{n+1} T_j-\sum_{j=0}^n T_j\right\|=\left\|T_0+\sum_{j=1}^{n+1} T_j-\left[T_0+\sum_{j=1}^n T_j\right]\right\|=\left\|T_0+\sum_{j=1}^{n+1} L_2^{-1}\left[\dddot{\mathcal{W}}_{j-1}\right]-\left\{T_0+\sum_{j=1}^n L_2^{-1}\left[\dddot{\mathcal{W}}_{j-1}\right]\right\}\right\|=\left\|T_0+L_2^{-1} \sum_{j=1}^{n+1}\left[\dddot{\mathcal{W}}_{j-1}\right]-\left\{T_0+L_2^{-1} \sum_{j=1}^n\left[\dddot{\mathcal{W}}_{j-1}\right]\right\}\right\|$

Since, $E_{n+1}=\mathcal{N}\left(E_n\right)$, then:

$\begin{gathered}\left\|E_{n+1}-E_n\right\|=\left\|L_2^{-1} \mathcal{N} \sum_{j=0}^n\left[\dddot{\mathcal{W}}_{j-1}\right]-L_2^{-1} \mathcal{N} \sum_{j=0}^{n-1}\left[\dddot{\mathcal{W}}_{j-1}\right]\right\| \\ =\left\|L_2^{-1} \mathcal{N}\left[\sum_{j=0}^n T_j\right]-L_2^{-1} N\left[\sum_{j=0}^{n-1} T_j\right]\right\| \\ \leq\left|L_2^{-1}\right|\left\|\mathcal{N}\left[\sum_{j=0}^n T_j\right]-N\left[\sum_{j=0}^{n-1} T_j\right]\right\| \\ \leq \gamma\left\|\sum_{j=0}^n L_2^{-1}\left[\dddot{\mathcal{W}}_{j-1}\right]-\sum_{j=0}^{n-1} L_2^{-1}\left[\dddot{\mathcal{W}}_{j-1}\right]\right\| \\ \leq \gamma^2\left\|\sum_{j=0}^{n-1} L_2^{-1}\left[\dddot{\mathcal{W}}_{j-1}\right]-\sum_{j=0}^{n-2} L_2^{-1}\left[\dddot{\mathcal{W}}_{j-1}\right]\right\| \\ \vdots \\ \leq \gamma^n\left\|\sum_{j=0}^1 L_2^{-1}\left[\dddot{\mathcal{W}}_{j-1}\right]-\sum_{j=0}^0 L_2^{-1}\left[\dddot{\mathcal{W}}_{j-1}\right]\right\| \\ =\gamma^n\left\|E_1-E_0\right\| \rightarrow 0 \text { as } n \rightarrow \infty \text { for } 0<\gamma<1 .\end{gathered}$

where,

$\begin{gathered}L_2^{-1}(\cdot)=\int_{-\infty}^{\infty}\left[\frac{(r-\eta) \operatorname{sgn}(r-\eta)}{2}\right](\cdot)_{r=\eta} d \eta, \\ \dddot{W}_{j-1}=\frac{1}{r}\left(G_{j-1}-G_{j-1}^*\right)-\frac{1}{r} \frac{\partial T_{j-1}}{\partial r}-\frac{1}{r^2} \frac{\partial^2 T_{j-1}}{\partial \theta^2} .\end{gathered}$

The results of the above theorems )7.1, 7.2(, can be applied to calculate the values of the parameter γⁿ by formulating the following definition.

Definition 7.2 For n=1,2,3, …

$\gamma^n=\left\{\begin{array}{c}\frac{\left\|E_{n+1}-E_n\right\|}{\left\|E_1-E_0\right\|}=\frac{\left\|h_{n+1}\right\|}{\left\|h_1\right\|},\left\|h_1\right\| \neq 0, n=1,2,3, \ldots \\ 0,\left\|h_1\right\|=0\end{array}\right.$             (75)

The convergence of the analytical solutions to the problem under study, can be tested by using Eq. (75). Accordingly, the results of convergence of the solutions that were found using the two methods FT-HPM and HPM were compared, as shown in the table below:

Table 5a. Comparison of the values of the parameter γⁿ for ψ(r, θ) between FT-HPM and HPM at Pr=0.7

Table 5b. Comparison of the values of the parameter γⁿ for T(r, θ) between FT-HPM and HPM at Pr=0.7

Table 5a, b shows that, γⁿ→0 as n→∞ for 0<γ<1. In addition, through this tables, the difference in convergence between FT-HPM and HPM methods can be seen, which show that the powers of γ calculated using FT-HPM, approach zero faster than the powers of γ calculated based on HPM. Moreover, it can be said that FT-HPM represents an evolution of HPM with better convergence.