1. Introduction
Oil pollution is defined as the emission of fluid hydrocarbon into the ocean. Humans release petroleum from tankers without refining and perform activities like drilling rigs, offshore platforms, and piping, which causes disastrous effects on marine life's eco and biological environment and leads to fatal repercussions. Thus, it is crucial to precisely predict the spread range of oil spills for an early stage countermeasure against a disaster to preserve the natural shoreline environmental system.
The area of spillage can be predicted by figuring the equation governing the flow field and related mass transport phenomenon. The diffusion(parabolic) equations are the most reasonable option. They allow statistics regarding the amount of oil that has outreached the ocean outlet, to be used as initial and boundary conditions for a mathematical model of oil diffusion and alteration in seas. As it involves the hyperbolic (advection/wave) component of the equation, the most reasonable choices are-diffusion and Allen-Cahn (AC) equations, which are difficult to solve numerically. In the past three decades, various researchers have studied the transport process of oil spills, premised on the trajectory method [1]. The methods have been applied to various water bodies, such as the river-lake systems [2], [3] and seas [4], [5], [6]
To model pollution in the seas by oil, the general diffusion equation with non-linearity will be considered, and the form of this equation will be [7]:
$\frac{\partial \psi }{\partial t}=D\frac{{{\partial }^{2}}\psi }{\partial {{x}^{2}}}+\beta \psi +\gamma {{\psi }^{m}}$
Where β and γ are from a set of real numbers, the diffusion coefficient is D, and ψ is concentration. Allen-Cahn(AC) equations is a particular case of diffusion equation obtained by substituting m=3, β=1 and γ=−1 in Eq. (1).
The AC equation is a parabolic nonlinear partial differential equation illustrating some natural physical phenomenon [8]. This equation has been extensively used to study numerous physical problems, such as quantum mechanics, fluid mechanics, propagation of shallow-water waves, chemical kinematics, optical fibers [9], and motion by mean curvature flows [10]. It has become a core model equation for the diffuse interface approach developed to investigate interfacial dynamics and phase transitions in material science [11]. It is also used to study phase separation in binary alloys that can be expressed as convection-diffusion equation in the case of fluid dynamics or as reaction-diffusion equation in the case of material sciences[12].
The Allen-Cahn (AC) equation is solved by using various techniques and appears in many studies; Hariharan [13] used Legendre wavelet-based approximation method to obtain an approximate solution of a few newell-whitehead (NW) and AC equations. Gui and Zhao [14] obtained travelling wave solutions of AC equation by using fractional laplacian. Yokus and Bulut [15] used the FDM to find a numerical solution for the AC equation. Javeed et al. [16] implemented the first integral method to get an analytical solution for space-time fractional AC equation and coupled space-time fractional Drinfeld's Sokolov-Wilson system. Chen et al. [12] proposed a time-space adaptive finite element method with the basis as 2nd-order, to solve the AC equation numerically. Hussain et al. [11] used the homotopy analysis and perturbation approaches to find the approximate solution of the bistable AC equation. Yin [17] proposed an algorithm for finite element (FE) method to get numerical solution of space fractional AC equation with continuous differentiable solutions. Li [18] investigated a reduced-order finite difference (ROFD) iterative approach based on, proper orthogonal decomposition method for one dimensional nonlinear AC equation with a modest parameter perturbation, that meets two intrinsic properties of the AC equation: discrete maximum-bound-principle(DMBP)-preserving and discrete energy-stability(DES)-preserving. Li et al. [19] investigated a ROFD technique for the AC equation with a minor parameter perturbation and a nonlinear factor concerned with the energy function. Khalid et al. [20] proposed a numerical solution of time-fractional AC equation with the help of redefined cubic B spline functions and finite difference method. Olshanskii et al. [21] investigated an equation of type AC defined on a surface that depends on time as a phase separation model in thin material along with the order-disorder transition. Ahmad et al. [7] discussed diffusion equation and Cahn-Allen equation arising in oil pollution with integer-order and solved by using modified versions of variational iteration algorithms (MVIM-I).
The real-world phenomenon has been governed by PDEs of integer order which can’t be described properly. Additionally, no method gives an exact solution for the fractional-order differential equation. Hence, non-linear PDEs of fractional order make the research more significant. Therefore, in this paper, we have applied the latest technique, FRDTM-an efficient and powerful method to find the solutions of the time-fractional diffusion equation and Cahn-Allen equation arising in oil pollution. The novelty of our work is that it provides an accurate prediction of the behaviour of oil and is vitally important to preserve the natural shoreline environmental system. Moreover, this method can also be applied to derive a variety of travelling wave solutions with distinct physical structures for nonlinear fractional equations arising in ocean engineering for examining nonlinear behaviour due to water waves.
2. A brief of FRDTM
In this section, RD represents operator of the reduced differential transform and $R_{D}^{-1}$ represents operator of the inverse reduced differential transform.
${{R}_{D}}\left[ \psi \left( x,t \right) \right]\approx {{\Psi}_{k}}\left( x \right)=\frac{1}{\Gamma \left( \alpha k+1 \right)}{{\left[ \frac{{{\partial }^{\alpha k}}}{\partial {{t}^{\alpha k}}}\psi \left( x,t \right) \right]}_{t=0}},$
where α represents the time-fractional order derivative.
$R_{D}^{-1}\left[ {{\Psi }_{k}}\left( x \right) \right]\approx \psi \left( x,t \right)=\underset{k=0}{\overset{\infty }{\mathop \sum }}\,{{\Psi}_{k}}\left( x \right){{\left( t \right)}^{\alpha k}},$
By inserting Eqs. (2) in (3), we obtain
${{\Psi}_{k}}\left( x \right)=\underset{k=0}{\overset{\infty }{\mathop \sum }}\,\frac{1}{\Gamma\left( \alpha k+1 \right)}{{\left[ \frac{{{\partial }^{\alpha k}}}{\partial {{t}^{\alpha k}}}\psi \left( x,t \right) \right]}_{t=0}}{{\left( t \right)}^{\alpha k}},$
Theorem 2.1: Let $\psi \left( x,t \right)={{\Psi}_{k}}\left( x \right)\cdot {{\left( t \right)}^{\alpha k}}$, then the series solution, stated in Eq. (3), $\forall k\in N\cup \left\{ 0 \right\}$,
i is convergent, if there exist 0<η<1 such that $\frac{\parallel {{\Psi}_{k+1}}\parallel }{\parallel {{\Psi}_{k}}\parallel }\le \eta $,
ii is divergent η>1, if there exist such that $\frac{\parallel {{\Psi}_{k+1}}\parallel }{\parallel {{\Psi}_{k}}\parallel }\ge \eta $.
Theorem 2.1 is a specific case of Banach's fixed point theorem and the proof is given in [25].
The ${{\eta }_{k}}$ can be obtained as ${{\eta }_{k}}=\left\{ \begin{matrix} \frac{\parallel {{\Psi}_{k+1}}\parallel }{\parallel {{\Psi}_{k}}\parallel }, & \parallel {{\Psi}_{k}}\parallel \ne 0, \\ 0, & \parallel {{\Psi}_{k}}\parallel =0, \\ \end{matrix} \right.$
Basic properties of the two-dimensional reduced differential transform are stated in [25].
3. Application of FRDTM
To understand the behavior of the diffusion equation and Allen-Cahn equation, we fractionalize the Eq. (1) into time-fractional partial differential equations and can write it as:
1. Time-fractional diffusion equation
$\frac{{{\partial }^{\alpha }}\psi }{\partial {{t}^{\alpha }}}=\frac{{{\partial }^{2}}\psi }{\partial {{x}^{2}}}+\cos x,$
2. Time-fractional Allen-Cahn equation
$\frac{{{\partial }^{\alpha }}\psi }{\partial {{t}^{\alpha }}}=\frac{{{\partial }^{2}}\psi }{\partial {{x}^{2}}}+\psi -{{\psi }^{3}},$
Example 1: Time-fractional diffusion equation
Consider the time-fractional diffusion equation,
$\frac{{{\partial }^{\alpha }}\psi }{\partial {{t}^{\alpha }}}=\frac{{{\partial }^{2}}\psi }{\partial {{x}^{2}}}+\cos x,0<\alpha 1,$
having the initial condition as [7],
$\psi \left( x,0 \right)=0,$
The Eq. (7) has an exact solution for α=1 is
$\psi \left( x,t \right)=\cos x\left( 1-{{e}^{-t}} \right),$
Using FRDT method on Eq. (7), we obtain:
$\frac{\Gamma\left( \alpha \left( k+1 \right)+1 \right)}{\Gamma\left( k\alpha +1 \right)}{{\Psi}_{k+1}}\left( x \right)=\left( \frac{{{\partial }^{2}}}{\partial {{x}^{2}}}{{\Psi}_{k}}\left( x \right)+\cos x\delta \left( k \right) \right),$
Or
${{\Psi}_{k+1}}\left( x \right)=\frac{\Gamma \left( k\alpha +1 \right)}{\Gamma\left( \alpha \left( k+1 \right)+1 \right)}\left( \frac{{{\partial }^{2}}}{\partial {{x}^{2}}}{{\Psi}_{k}}\left( x \right)+\cos x\delta \left( k \right) \right),$
From Eq. (8), initial condition can be written as
${{\Psi}_{0}}\left( x \right)=0,$
Substituting Eq. (12) into Eq. (11), we get
$\begin{matrix} & {{\Psi}_{1}}\left( x \right)=\frac{\cos x}{\Gamma\left( \alpha +1 \right)}; \\ & {{\Psi}_{2}}\left( x \right)=\frac{\cos x}{\Gamma\left( 2\alpha +1 \right)}; \\ & {{\Psi}_{3}}\left( x \right)=\frac{2}{3}\frac{\pi \cos \left( x \right){{27}^{-\alpha }}\sqrt{3}}{\Gamma\left( \alpha +1 \right)\Gamma\left( \alpha +1/3 \right)\Gamma\left( \alpha +2/3 \right)}; \\ & {{\Psi}_{4}}\left( x \right)=-\frac{\cos x}{\Gamma\left( 4\alpha +1 \right)}; \\ & {{\Psi}_{5}}\left( x \right)=\frac{4}{5}\frac{{{\pi }^{2}}{{3125}^{-\alpha }}\sqrt{5}\cos x}{\Gamma\left( \alpha +1 \right)\Gamma\left( \alpha +\frac{1}{5} \right)\Gamma\left( \alpha +\frac{2}{5} \right)\Gamma\left( \alpha +\frac{3}{5} \right)\Gamma\left( \alpha +\frac{4}{5} \right)}; \\ & \vdots \\ \end{matrix}$
Therefore, upto 5th term approximate analytical solution can be as follows
$\begin{align} & \psi \left( x,t \right) \\ & \text{=}\underset{k=0}{\overset{n}{\mathop \sum }}\,{{\Psi}_{k}}\left( x \right){{t}^{k\alpha }} \\ & \text{=}\frac{\cos x}{\Gamma\left( \alpha +1 \right)}{{t}^{\alpha }}-\frac{\cos x}{\Gamma\left( 2\alpha +1 \right)}{{t}^{2\alpha }} \\ & +\;\frac{2}{3}\frac{\pi \cos \left( x \right){{27}^{-\alpha }}\sqrt{3}}{\Gamma\left( \alpha +1 \right)\Gamma\left( \alpha +1/3 \right)\Gamma\left( \alpha +2/3 \right)}{{t}^{3\alpha }}-\frac{\cos x}{\Gamma\left( 4\alpha +1 \right)}{{t}^{4\alpha }} \\ & +\frac{4}{5}\frac{{{\pi }^{2}}{{3125}^{-\alpha }}\sqrt{5}\cos x}{\Gamma\left( \alpha +1 \right)\Gamma\left( \alpha +\frac{1}{5} \right)\Gamma\left( \alpha +\frac{2}{5} \right)\Gamma\left( \alpha +\frac{3}{5} \right)\Gamma\left( \alpha +\frac{4}{5} \right)}{{t}^{5\alpha }}+\ldots. \\ \end{align}$
Eq. (14) represents the approximate analytical solution of the time-fractional diffusion equation.
Hence, when $\alpha =1$, the series solution, $\underset{k=0}{\overset{n}{\mathop \sum }}\,{{\psi }_{k}}\left( x \right){{t}^{k\alpha }}$ converges efficiently to the exact solution $\psi \left( x,t \right)=\cos x\left( 1-{{e}^{-t}} \right)$ as $k\to \infty $. In addition, to check convergence analysis of example 1, computing ${{\eta }_{k}}$'s terms from corollary 2.1, we get
${{\eta }_{1}}=\frac{\parallel {{\psi }_{2}}\parallel }{\parallel {{\psi }_{1}}\parallel }=0.5<1$, ${{\eta }_{2}}=\frac{\parallel {{\psi }_{3}}\parallel }{\parallel {{\psi }_{2}}\parallel }=0.33<1$, ${{\eta }_{3}}=\frac{\parallel {{\psi }_{4}}\parallel }{\parallel {{\psi }_{3}}\parallel }=0.25<1$, ${{\eta }_{4}}=\frac{\parallel {{\psi }_{5}}\parallel }{\parallel {{\psi }_{4}}\parallel }=0.2<1$, …
This confirms that the FRDTM approach gives positive and bounded solution that converges to the exact solution, when k>1 and 0<α≤1.
Figures 1 and 2 depict the physical behavior of the solution of fractional order diffusion equation for different fractional order values α=0.25,0.5,0.75, integer order α=1, and different time levels t=0.5,1.0. From Figs. 1 and 2, it can be concluded that the FRDTM give accurate result as compared to exact solution and we see that the contours meet at x=0. To sum up, the series solution given in Eq. (14) works perfectly in dealing with time-fractional diffusion equations with less iterations. The three-dimensional plots given in Fig. 3 show absolute error obtained by FRDTM.
Fig. 1. Behavior of approximate solution for α=0.25,0.50,0.75 and for α=1 is compared with exact solution (circle) of Example 1 for t=0.5. |
Fig. 2. Behavior of approximate solution for α=0.25,0.50,0.75 and for α=1 is compared with exact solution (circle) of Example 1 for t=1.0. |
Fig. 3. Abs. error graph by FRDTM for Example 1. |
Comparison of absolute errors of FRDTM, MVIA-I, and VIA-I for different values of parameters is given in Table 1, shows that the error in FRDTM is less as compared to MVIT-I and VIA-I. Moreover, we considered only five iterations which are less than those in MVIT-I [7] and VIA-I [7]. This saves computational time and give better result.
Table 1. Numericaly comparison of abs. errors of Example 1. |
| x | t | Abs. error in FRDTM | Abs. error in MVIA - I | Abs. error in VIA - I |
|---|---|---|---|---|
| 1 | 0.5 | 1.000E-10 | 1.671E-10 | 6.344E-10 |
| 2 | 1.0 | 0.000E+00 | 3.563E-10 | 9.619E-09 |
| 3 | 1.5 | 9.000E-10 | 2.683E-09 | 1.905E-06 |
| 4 | 2.0 | 0.000E+00 | 1.871E-08 | 2.870E+05 |
| 5 | 2.5 | 3.000E-10 | 5.510E-08 | 1.399E-04 |
| 6 | 3.0 | 0.000E+00 | 5.522E-07 | 3.398E-03 |
| 7 | 3.5 | 4.000E-10 | 2.929E-06 | 1.406E-02 |
| 8 | 4.0 | 0.000E+00 | 8.666E-07 | 1.141E-02 |
| 9 | 4.5 | 2.100E-09 | 8.689E-06 | 2.528E-01 |
| 10 | 5.0 | 2.600E-09 | 1.682E-05 | 7.193E-01 |
Example 2: Time fractional Allen-Cahn equation
Consider the time fractional Allen-Cahn equation,
$\frac{{{\partial }^{\alpha }}\psi }{\partial {{t}^{\alpha }}}=\frac{{{\partial }^{2}}\psi }{\partial {{x}^{2}}}+\psi -{{\psi }^{3}},0<\alpha 1,$
having the initial condition as [7],
$\psi \left( x,0 \right)=\left( -0.5 \right)+\left( 0.5 \right)\tanh \left( 0.3536x \right),$
Eq. (15) has exact solution for α=1 as
$\psi \left( x,0 \right)=\left( -0.5 \right)+\left( 0.5 \right)\tanh \left( 0.3536x-0.75t \right),$
Using FRDTM on both side of Eq. (15), we obtain
$\begin{matrix} & \frac{\Gamma\left( \alpha \left( k+1 \right)+1 \right)}{\Gamma\left( k\alpha +1 \right)}{{\Psi}_{k+1}}\left( x \right) \\ & =\left( \frac{{{\partial }^{2}}}{\partial {{x}^{2}}}{{\Psi}_{k}}\left( x \right)+{{\Psi}_{k}}\left( x \right)-\underset{r=0}{\overset{k}{\mathop \sum }}\,\underset{s=0}{\overset{r}{\mathop \sum }}\,{{\Psi}_{s}}\left( x \right){{\Psi}_{r-s}}\left( x \right){{\Psi}_{k-r}}\left( x \right) \right), \\ \end{matrix}$
Or
$\begin{align} & {{\Psi }_{k+1}}(x)=\frac{\Gamma (k\alpha +1)}{\Gamma (\alpha (k+1)+1}\left( \frac{{{\partial }^{2}}}{\partial {{x}^{2}}} \right.{{\Psi }_{k}}(x)+{{\Psi }_{k}}(x) \\ & -\left. \sum\limits_{r=0}^{k}{\sum\limits_{s=0}^{r}{{{\Psi }_{s}}(x){{\Psi }_{r-s}}(x){{\Psi }_{k-r}}(x)}} \right), \\ \end{align}$
From Eq. (16), initial condition can be written as
${{\Psi}_{0}}\left( x \right)=\left( -0.5 \right)+\left( 0.5 \right)\tanh \left( 0.3536x \right),$
Substituting Eq. (20) into Eq. (19), we get
$\begin{matrix} \Psi_1(x) \quad=-3.2 \times 10^{-7}\left(\frac{103 \sinh (0.3536 x)+1171875 \cosh (0.3536 x)}{\Gamma(\alpha+1) \cosh ^3(0.3536 x)}\right) ; \\ \Psi_2(x)=-\frac{\left(3.5449 \times 10^{-14}\right)\left(\mathrm{e}^{-1.386294361 \alpha}\right)\left(\begin{array}{c} 2.812500022 \times 10^{13} \sinh (0.3536 x) \cosh ^2(0.3536 x) \\-1.236651817 \times 10^9 \sinh (0.3536 x) \\+4.944 \times 10^9 \cosh ^3(0.3536 x) \\-6.18 \times 10^9 \cosh (0.3536 x) \end{array}\right)}{\Gamma(\alpha+1) \Gamma(\alpha+0.5) \cosh ^5(0.3536 x)} ; \\ \vdots \\ \end{matrix}$
Therefore, up to 3rd term approximate analytical solution can be followed as
$\begin{matrix} & \psi (x,t)=\sum\nolimits_{k=0}^{n}{{{\Psi }_{k}}}(x){{t}^{k\alpha }} \\ & =(-0.5)+(0.5)\tanh (0.3536x)-3.2\times {{10}^{-7}}\left( \frac{\begin{matrix} & 103\sinh (0.3536x) \\ & +1171875\cosh (0.3536x) \\ \end{matrix}}{\Gamma (\alpha +1){{\cosh }^{3}}(0.3536x)} \right){{t}^{\alpha }} \\ & -\frac{(3.5449\times {{10}^{14}})({{e}^{-1.386294361\alpha }})\left(\begin{matrix} 2.812500022 \times 10^{13} \sinh (0.3536 x) \cosh ^2(0.3536 x) \\ -1.236651817 \times 10^9 \sinh (0.3536 x) \\ +4.944 \times 10^9 \cosh ^3(0.3536 x) \\ -6.18 \times 10^9 \cosh (0.3536 x)\end{matrix}\right)}{\Gamma (\alpha +1)\Gamma (\alpha +0.5){{\cosh }^{5}}(0.3536x)}{{t}^{2\alpha }}+\cdots \\ \end{matrix}$
Eq. (22) represents the approximate analytical solution of the time fractional order of Eq. (15), which converges efficiently to the exact solution $\psi \left( x,0 \right)=\left( -0.5 \right)+\left( 0.5 \right)\tanh \left( 0.3536x-0.75t \right)$ as $k\to \infty $. Figs. 4 and 5 show the effects of fractional order α on the solution of the AC equation with initial condition $\psi \left( x,0 \right)=\left( -0.5 \right)+\left( 0.5 \right)\tanh \left( 0.3536x \right)$ and distinct time t=0.5,1.0 using FRDTM. The results have been compared with the available exact solution for integer order α=1. It can be noted from Figs. 4 and 5 that for smaller values of α, the curved nonlinearity is observed; whereas when α is closer to 1 fewer nonlinear movements are found. The three-dimensional plots is given in Fig. 6 shows absolute error obtained by FRDTM. Comparison of absolute errors of FRDTM, MVIA-I, LLWM [13], ADM [14], and MQ [14] for different values of parameters is given in Table 2. It shows that the absolute errors in FRDTM are less compared to available methods.
Fig. 4. Behavior of approximate solution for α=0.25,0.50,0.75 and for α=1 is compared with exact solution (circle) of Example 2 for t=0.5. |
Fig. 5. Behavior of approximate solution for α=0.25,0.50,0.75 and for α=1 is compared with exact solution (circle) of Example 2 for t=1.0. |
Fig. 6. Abs. error graph by FRDTM for Example 2. |
Table 2. Numericaly comparison of abs. errors of Example 2. |
| x | t=0.1 | t=0.5 | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| FRDTM | MVIA-I [7] | LLWM [13] | ADM [14] | MQ [14] | FRDTM | MVIA-I [7] | LLWM [13] | ADM [14] | MQ [14] | |
| -25 | 1.00000E-10 | 1.17373E10 | 1.18943E11 | 1.1644E11 | 2.9200E19 | 1.00000E-10 | 1.25342E10 | 6.4747E12 | 1.45683E06 | 5.42902E09 |
| -15 | 2.00000E-10 | 1.38319E07 | 2.36636E09 | 1.37206E08 | 3.43787E06 | 7.00000E-10 | 1.47721E07 | 1.35653E10 | 1.23638E09 | 6.38144E06 |
| 25 | 5.42769E-12 | 1.57665E10 | 9.84744E10 | 4.8392E10 | 3.38887E09 | 3.04682E-11 | 1.68402E09 | 7.49924E10 | 1.35958E08 | 7.3109E10 |
| 30 | 1.11742E-12 | 4.59300 E12 | 3.57575E11 | 1.4096E11 | 9.7252E11 | 1.74795E-12 | 4.90550E12 | 2.44443E10 | 3.9604E10 | 2.0564E10 |
Example 3: Time fractional Allen-Cahn equation
Consider the time-fractional Allen-Cahn Eq. (6),
$\frac{{{\partial }^{\alpha }}\psi }{\partial {{t}^{\alpha }}}=\frac{{{\partial }^{2}}\psi }{\partial {{x}^{2}}}+\psi -{{\psi }^{3}},0<\alpha 1,$
having the initial condition as [7],
$\psi \left( x,0 \right)=-\frac{12\left[ -1+\tanh \left\{ 0.416667\left( 0.3+0.848528x \right) \right\} \right]}{24+30\left[ 1+\tanh \left\{ 0.416667\left( 0.3+0.848528x \right) \right\} \right]},$
Eq. (23) has exact solution for α=1 which is
$\psi \left( x,t \right)=-\frac{12\left[ -1+\tanh \left\{ 0.416667\left( 0.3-1.8t+0.848528x \right) \right\} \right]}{24+30\left[ 1+\tanh \left\{ 0.416667\left( 0.3-1.8t+0.848528x \right) \right\} \right]},$
Applying FRDTM on Eq. (23), we get:
$\frac{\Gamma\left( \alpha \left( k+1 \right)+1 \right)}{\Gamma\left( k\alpha +1 \right)}{{\Psi}_{k+1}}\left( x \right)=\left( \frac{{{\partial }^{2}}}{\partial {{x}^{2}}}{{\Psi}_{k}}\left( x \right)+{{\Psi}_{k}}\left( x \right)-\underset{r=0}{\overset{k}{\mathop \sum }}\,\underset{s=0}{\overset{r}{\mathop \sum }}\,{{\Psi}_{s}}\left( x \right){{\Psi}_{r-s}}\left( x \right){{\Psi}_{k-r}}\left( x \right) \right),$
Or
${{\Psi}_{k+1}}\left( x \right)=\frac{\Gamma\left( k\alpha +1 \right)}{\Gamma\left( \alpha \left( k+1 \right)+1 \right)}\left( \frac{{{\partial }^{2}}}{\partial {{x}^{2}}}{{\Psi}_{k}}\left( x \right)+{{\Psi}_{k}}\left( x \right)-\underset{r=0}{\overset{k}{\mathop \sum }}\,\underset{s=0}{\overset{r}{\mathop \sum }}\,{{\Psi}_{s}}\left( x \right){{\Psi}_{r-s}}\left( x \right){{\Psi}_{k-r}}\left( x \right) \right),$
From Eq. (24), initial condition can be written as
${{\Psi}_{0}}\left( x \right)=-\frac{12\left[ -1+\tanh \left\{ 0.416667\left( 0.3+0.848528x \right) \right\} \right]}{24+30\left[ 1+\tanh \left\{ 0.416667\left( 0.3+0.848528x \right) \right\} \right]},$
Substituting Eq. (28) into Eq. (27), we get
${{\Psi }_{1}}(x)=\frac{2\times {{10}^{34}}\left( \begin{matrix} \begin{matrix} & 24{{\cosh }^{4}}(0.3535x)\sinh (0.3535x) \\ & +6.244349535\times {{10}^{10}}{{\cosh }^{2}}(0.3535x) \\ \end{matrix} \\ -6.86518226\times {{10}^{8}}\sinh (0.3535x)+75{{\cosh }^{5}}(0.3535x) \\ +8.1928132150\times {{10}^{10}}{{\cosh }^{3}}(0.3535x) \\ -1.2120910680\times {{10}^{10}}\cosh (0.3535x) \\ \end{matrix} \right)}{\Gamma (\alpha +1){{\left( \begin{matrix} 2.909110077\times {{10}^{9}}\cosh (0.3535x) \\ +1.850113900\times {{10}^{9}}\sinh (0.3535x) \\ \end{matrix} \right)}^{3}}{{\left( \begin{matrix} 5.03911345\times {{10}^{8}}\cosh (0.3535x) \\ +6.2662938\times {{10}^{7}}\sinh (0.3535x) \\ \end{matrix} \right)}^{2}}};$
$\begin{matrix}\Psi_{2}(x)=\frac{\left(3.5449 \times 10^{82}\right)\left(e^{-1.362294361 \alpha}\right)\left(\begin{array}{c}-4.720014982 \times 10^{16} \cosh ^{2}(0.3535 x) \sinh (0.3535 x) \\+2.912118225 \times 10^{15} \cosh (0.3535 x) \\+7.4493939 \times 10^{13} \sinh (0.3535 x) \\+7.06068372 \times 10^{18} \cosh ^{5}(0.3535 x) \\-4.137746517 \times 10^{17} \cosh ^{3}(0.3535 x) \\+2.131297588 \times 10^{18} \cosh ^{4}(0.3535 x) \sinh (0.3535 x) \\+8.139498607 \times 10^{9} \cosh ^{11}(0.3535 x) \\+1.782848459 \times 10^{19} \cosh ^{9}(0.3535 x) \\+8.164475812 \times 10^{9} \cosh ^{10}(0.3535 x) \sinh (0.3535 x) \\-1.433217888 \times 10^{19} \cosh ^{6}(0.3535 x) \sinh (0.3535 x) \\+1.783564156 \times 10^{19} \cosh ^{8}(0.3535 x) \sinh (0.3535 x) \\-2.326720917 \times 10^{19} \cosh ^{7}(0.3535 x) \end{array}\right)}{\Gamma(\alpha+1) \Gamma(\alpha+0.5)\binom{5.03911345 \times 10^{8} \cosh (0.3535 x)}{+6.2662938 \times 10^{7} \sinh (0.3535 x)}^{4}\binom{2.909110077 \times 10^{9} \cosh (0.3535 x)}{+1.8501139 \times 10^{9} \sinh (0.3535 x)}^{7}}; \\ \vdots \\ \end{matrix}$
Therefore, upto 3rd approximate analytical solution can be followed as
$\begin{matrix} & \psi(x, t)= \sum_{k=0}^{n} \Psi_{k}(x) t^{k \alpha} \\ & -\frac{12[-1+\tanh \{0.416667(0.3+0.848528 x)\}]}{24+30[1+\tanh \{0.416667(0.3+0.848528 x)\}]} \\ & \frac{2 \times 10^{34}\left(\begin{array}{c} 24 \cosh ^{4}(0.3535 x) \sinh (0.3535 x) \\ +6.244349535 \times 10^{10} \cosh ^{2}(0.3535 x) \sinh (0.3535 x) \\ -6.86518226 \times 10^{8} \sinh (0.3535 x)+75 \cosh (0.3535 x) \\ +8.1928132150 \times 10^{10} \cosh ^{3}(0.3535 x) \\ -1.2120910680 \times 10^{10} \cosh (0.3535 x) \end{array}\right)}{\Gamma(\alpha+1)\binom{2.909110077 \times 10^{9} \cosh (0.3535 x)}{+1.850113900 \times 10^{9} \sinh (0.3535 x)}^{3}\binom{5.03911345 \times 10^{8} \cosh (0.3535 x)}{+6.2662938 \times 10^{7} \sinh (0.3535 x)}^{2}} {{t}^{\alpha }} \\ & \frac{\left(3.5449 \times 10^{82}\right)\left(e^{-1.3862243561 \alpha}\right)\left(\begin{array}{c} -4.720014982 \times 10^{16} \cosh ^{2}(0.3535 x) \sinh (0.3535 x) \\ +2.912118225 \times 10^{15} \cosh ^{(0.3535 x)} \\ \left.+7.4493939 \times 10^{13} \sinh ^{(0.3535 x}\right) \\ +7.06068372 \times 10^{18} \cosh ^{5}(0.3535 x) \\ -4.137746517 \times 10^{17} \cosh ^{3}(0.3535 x) \\ +2.131297588 \times 10^{18} \cosh ^{4}(0.3535 x) \sinh (0.3535 x) \\ +8.139498607 \times 10^{9} \cosh ^{11}(0.3535 x) \\ +1.782848459 \times 10^{11} \cosh ^{9}(0.3535 x) \\ +8.164475812 \times 10^{9} \cosh ^{10}(0.3535 x) \sinh (0.3535 x) \\ -1.433217888 \times 10^{19} \cosh ^{6}(0.3535 x) \sinh (0.3535 x) \\ +1.783564156 \times 10^{19} \cosh ^{8}(0.3535 x) \sinh (0.3535 x) \\ -2.326720917 \times 10^{19} \cosh ^{7}(0.3535 x) \end{array}\right)}{ \Gamma(\alpha+1) \Gamma(\alpha+0.5)\binom{5.03911345 \times 10^{8} \cosh (0.3535 x)}{+6.2662938 \times 10^{7} \sinh (0.3535 x)}^{4}\binom{2.909110077 \times 10^{9} \cosh (0.3535 x)}{+1.8501139 \times 10^{9} \sinh (0.3535 x)}} {{t}^{2\alpha }}+\cdots \\ \end{matrix}$
Eq. (31) represents the approximate analytical solution of the time fractional order of Eq. (23), which converges efficiently to the exact solution given in Eq. (25) as $k\to \infty $.
Figures 7 and 8 show the physical behavior of the AC equation with initial condition by [7] for different fractional values α=0.25,0.5,0.75, integer order α=1, and fixed time t=0.5,1.0. From Figs. 7 and 8 it can be observed that the curved nonlinearity is noticed for smaller values of α, whereas less nonlinear movements are found for α which is closer to 1. The 3D plots given in Fig. 9 shows absolute error obtained by FRDTM.
Fig. 7. Behavior of approximate solution for α=0.25,0.50,0.75 and for α=1 is compared with exact solution (circle) of Example 3 for t=0.5. |
Fig. 8. Behavior of approximate solution for α=0.25,0.50,0.75 and for α=1 is compared with exact solution (circle) of Example 3 for t=1.0. |
Fig. 9. Abs. error graph by FRDTM for Example 3. |
Table 3. Numericaly comparison of abs. errors of Example 3. |
| x | t | Exact | Solutions | Abs.errors | ||||
|---|---|---|---|---|---|---|---|---|
| FRDTM | MVIA-I [7] | FDM [15] | FRDTM | MVIA-I [7] | FDM [15] | |||
| 0.000 | 0.01 | 0.184257703 | 0.184257702 | 0.184257684 | 0.184247 | 1.20E-09 | 1.908E08 | 1.06626E05 |
| 0.001 | 0.01 | 0.183197245 | 0.183197243 | 0.183197225 | 0.183187 | 1.20E-09 | 1.924E08 | 1.06500E05 |
| 0.002 | 0.01 | 0.182141527 | 0.182141526 | 0.182141507 | 0.182131 | 1.30E-09 | 1.939E08 | 1.06370E05 |
| 0.003 | 0.01 | 0.181090544 | 0.181090543 | 0.181090525 | 0.18108 | 1.10E-09 | 1.954E08 | 1.06236E05 |
| 0.004 | 0.01 | 0.180044291 | 0.180044290 | 0.180044271 | 0.180034 | 1.10E+09 | 1.969E08 | 1.06098E05 |
| 0.005 | 0.01 | 0.179002761 | 0.179002760 | 0.179002741 | 0.178992 | 1.20E-09 | 1.984E08 | 1.05956E05 |
| 0.006 | 0.01 | 0.177965948 | 0.177965947 | 0.177965928 | 0.177955 | 1.10E+09 | 1.999E08 | 1.05810 E05 |
4. Conclusion
In this study, the FRDTM is an efficient mathematical tool that has been applied to solve fractional-order diffusion equations arising in oil pollution for three fractional orders α=0.25,0.5,0.75 and integer-order α=1.0. Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9 and Table 1, Table 2, Table 3 convey that the FRDTM can tackle the problems perfectly, and it can be deployed in diffusion equations for analyzing oil pollution in the sea with linear and non-linear nature. Also, absolute errors are less compared to MVIA-I [7] and other available results in the literature, which conclude that FRDTM provides highly accurate numerical results without using Adomian polynomials, discretization, transformation, shape parameters, restrictive assumptions, or linearization for nonlinear time-fractional differential equations. Thus, the FRDTM methodology displays a faster convergence of the solutions.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
