# Application of the Elzaki iterative method to fractional partial differential equations

## Abstract

In this article, we present an iterative transformation method for solving fractional partial differential equations that combines the Elzaki transform and iterative methods. By this iterative transformation method, numerical solutions in the form of series are obtained. When we apply this method to the fractional linear Klein–Gordon equation, we find that it yields the same results, just like the Homotopy perturbation method. The procedures and results of this method for solving the new generalized fractional Hirota–Satsuma coupled KdV equation are given in the paper.

## Introduction

In recent decades, fractional-order partial differential equations have been widely used and developed in physics, engineering, and fluid mechanics. Compared with integer-order partial differential equations, they are more suitable to portray complex phenomena and processes. Therefore, the method to solve fractional partial differential equations is also a relatively important problem. Now, there are methods to solve fractional-order partial differential equations. For example, in , the finite-difference methods, the Galerkin finite-element methods, and the spectral methods to solve fractional-order partial differential equations are mentioned; Gepreel uses the Homotopy perturbation method to obtain the solution of the fractional Klein–Gordon equation in ; Khalid used the Elzaki transform method to solve the equations ; in , Ziane used the fractional Elzaki variational iteration method to solve the equations; Jafari introduced the Iterative Laplace transform method in ; Tarig used the Sumudutrans form of the variational iteration method to solve linear homogeneous partial differential equations; Thabet  introduced a new analytic method to solve partial differential equations with fractional order, and El-Rashidy  used the method to obtain new traveling-wave solutions of the equations. Hosseini  used the modified Kudryashov method to obtain exact solutions of the coupled sine-Gordon equations. Mohammad Tamsir  employed a semianalytical approach to obtain the approximate analytical solution of the Klein–Gordon equations; The Klein–Gordon equation [12, 13] is a crucial equation in the study of relativistic quantum mechanics.

Many authors have solved the generalized Hirota–Satsuma coupled KdV equation utilizing various equations in , including the homotopy analysis approach . This is a significant class of equations in mathematics and physics. In this study, we employ the Elzaki transform [19, 20] in conjunction with an iterative approach  to generate approximations to partial differential equations with fractional order. The results demonstrate the method’s validity, further, it also may be applied to other fractional-order partial differential equations.

## Basic definition

### Definition 1

The fractional Riemann–Liouville of operator $$D^{p}$$ is as follows

$$D^{p}w(x)= \textstyle\begin{cases} \frac{\partial ^{m}w(x) }{ \partial x^{m}} ,&p=m, \\ \frac{\partial}{\Gamma (m-p)\partial x^{m}}\int _{0}^{x} \frac{w(x)}{(x-\xi )^{p-m+1}}\,d\xi ,&m-1< p< m, \end{cases}$$
(1)

where $$m\in Z^{+}$$, $$p\in R^{+}$$ when $$0< p\le 1$$,

$$D^{p}w(x)=\frac{1 }{\Gamma (p)} \int _{0}^{x} \frac{w(x)}{(x-\xi )^{1-p}}\,d\xi .$$

### Definition 2

The Riemann–Liouville integral operator with fractional order is defined as follows

$$I^{p}w(x)=\frac{1}{\Gamma (p)} \int _{0}^{x}(x-\xi )^{p-1}w( \xi )\,d \xi ,\quad p>0,\xi >0.$$
(2)

### Definition 3

The Caputo fractional derivative of $$w(x)$$ is defined as follows, $$m\in N$$

$$c_{D^{p}w(x)}= \textstyle\begin{cases} I^{m-p}[\frac{\partial ^{m}w(x)}{\partial x^{m}}] , &m-1< p< m, \\ \frac{\partial ^{m}w(x)}{\partial x^{m}} , &p=m. \end{cases}$$
(3)

### Definition 4

The Elzaki [22, 23] transform of $$w(x)$$ is defined as follows

$$E\bigl[w(x)\bigr]=T(v)=v \int _{0}^{\infty}w(x)e^{\frac{-x}{v}}\,dx,\quad x\ge 0, k_{1}\le v\le k_{2}.$$
(4)

### Definition 5

The fractional Caputo operator of the Elzaki transform is 

$$E\bigl[D_{x}^{\alpha}w(x)\bigr]=v^{-\alpha}E \bigl[w(x)\bigr]-\sum_{k=0}^{m-1}v^{2- \alpha +k}w^{(k)}(0),\quad \text{where } m-1< \alpha < m.$$
(5)

## Methodology of the Elzaki transform iterative method

To briefly describe this equation in detail, we consider the following fractional partial differential equations

$$\frac{\partial ^{\alpha} w(x,t)}{\partial t^{\alpha}}=M\bigl(w_{1}(x,t),w_{2}(x,t), \dots ,w_{n}(x,t)\bigr)+N\bigl(w_{1}(x,t),w_{2}(x,t), \dots ,w_{n}(x,t)\bigr),$$
(6)

where M and N are the nonlinear and linear operators from Banach space B → B, respectively, $$\alpha \in R^{+}$$, $$m-1<\alpha \leq m$$, $$m=0,1,\dots ,n$$,

subject to the initial condition

$$\frac{\partial ^{k} w(x,0)}{\partial t^{k}}=w_{k}(x,0),\quad k=0,1, \dots ,m-1,m \in N.$$
(7)

Then, the Ezaki transformation acts simultaneously on both sides of the equation, and we obtain

\begin{aligned}[b] E\biggl[\frac{\partial ^{k} w(x,0)}{\partial t^{k}}\biggr]={}&E\bigl[M \bigl(w_{1}(x,t),w_{2}(x,t), \dots ,w_{n}(x,t) \bigr) \\ &{}+N\bigl(w_{1}(x,t),w_{2}(x,t),\dots ,w_{n}(x,t)\bigr)\bigr], \end{aligned}
(8)

hence,

\begin{aligned}[b] v^{-k}E\bigl[w(x,t)\bigr]-\sum _{n=0}^{m-1}v^{2-k+n}w^{(n)}(x,0)={}&E \bigl[M\bigl(w_{1}(x,t),w_{2}(x,t), \dots ,w_{n}(x,t)\bigr) \\ &{}+N\bigl(w_{1}(x,t),w_{2}(x,t),\dots ,w_{n}(x,t)\bigr)\bigr]. \end{aligned}
(9)

Through the use of the inverse Elzaki transform, we obtain

\begin{aligned}[b]w(x,t)= {}&E^{-1}\Biggl[\sum _{n=0}^{m-1}v^{2+n}w^{(n)}(x,0) \Biggr]+E^{-1}\bigl[v^{k}E\bigl[M\bigl(w_{1}(x,t),w_{2}(x,t), \dots ,w_{n}(x,t)\bigr) \\ &{}+N\bigl(w_{1}(x,t),w_{2}(x,t),\dots ,w_{n}(x,t)\bigr)\bigr]\bigr]. \end{aligned}
(10)

The following iterative method is utilized

$$w(x,t)=\sum_{i=1}^{\infty}w_{i}(x,t).$$
(11)

The nonlinear operator M can be decomposed into

\begin{aligned}[b] &M\bigl(w_{1}(x,t),w_{2}(x,t), \dots ,w_{n}(x,t)\bigr)\\ &\quad=M\bigl(w_{10}(x,t),w_{20}(x,t), \dots ,w_{n0}(x,t)\bigr) \\ &\qquad{}+\sum_{m=0}^{\infty}\Biggl[M\Biggl(\sum _{i=0}^{m}w_{1i}(x,t),\sum _{i=0}^{m}w_{2i}(x,t), \dots ,\sum_{i=0}^{m}w_{ni}(x,t) \Biggr) \\ &\qquad{}-M\Biggl(\sum_{i=0}^{m-1}w_{1i}(x,t), \sum_{i=0}^{m-1}w_{2i}(x,t), \dots , \sum_{i=0}^{m-1}w_{ni}(x,t) \Biggr)\Biggr]. \end{aligned}
(12)

Then, we can obtain

\begin{aligned} \sum_{i=0}^{\infty}w_{i}(x,t)={}&E^{-1} \Biggl[\sum_{n=0}^{m-1}v^{2+n}w^{(n)}(x,0) \Biggr]+ E^{-1}v^{k}\bigl[E\bigl[M\bigl(w_{10}(x,t),w_{20}(x,t), \dots ,w_{n0}(x,t)\bigr) \\ & {}+ N\bigl(w_{10}(x,t),w_{20}(x,t),\dots ,w_{n0}(x,t)\bigr)\bigr]\bigr] \\ & {}+ E^{-1}\Biggl[v^{k}E\Biggl[\sum _{m=0}^{\infty}\Biggl[M\Biggl(\sum _{i=0}^{m}w_{1i}(x,t), \sum _{i=0}^{m}w_{2i}(x,t),\dots ,\sum _{i=0}^{m}w_{ni}(x,t)\Biggr) \\ & {}- M\Biggl(\sum_{i=0}^{m-1}w_{1i}(x,t), \sum_{i=0}^{m-1}w_{2i}(x,t), \dots , \sum_{i=0}^{m-1}w_{ni}(x,t) \Biggr)\Biggr]\Biggr]\Biggr] \\ & {}+ E^{-1}\Biggl[v^{k}E\Biggl[\sum _{m=0}^{\infty}\Biggl[N\Biggl(\sum _{i=0}^{m}w_{1i}(x,t), \sum _{i=0}^{m}w_{2i}(x,t),\dots ,\sum _{i=0}^{m}w_{ni}(x,t)\Biggr) \\ & {}- N\Biggl(\sum_{i=0}^{m-1}w_{1i}(x,t), \sum_{i=0}^{m-1}w_{2i}(x,t), \dots , \sum_{i=0}^{m-1}w_{ni}(x,t) \Biggr)\Biggr]\Biggr]\Biggr]. \end{aligned}
(13)

We make the following settings

$$\textstyle\begin{cases} w_{0}=E^{-1}[\sum_{n=0}^{m-1}v^{2+n}w^{(n)}(x,0)], \\ w_{1}=E^{-1}v^{k}[E[M(w_{10}(x,t),w_{20}(x,t),\dots ,w_{n0}(x,t))\\ \hphantom{w_{1}=}{}+N(w_{10}(x,t),w_{20}(x,t), \dots ,w_{n0}(x,t))]], \\ w_{i}=E^{-1}[v^{k}E[\sum_{i=0}^{\infty}[M(\sum_{i=0}^{m}w_{1i}(x,t), \sum_{i=0}^{m}w_{2i}(x,t),\dots ,\sum_{i=0}^{m}w_{ni}(x,t)) \\ \hphantom{w_{i}=}{}-M(\sum_{i=0}^{m-1}w_{1i}(x,t),\sum_{i=0}^{m-1}w_{2i}(x,t),\dots , \sum_{i=0}^{m-1}w_{ni}(x,t))]]] \\ \hphantom{w_{i}=}{}+E^{-1}[v^{k}E[\sum_{i=0}^{\infty}[N(\sum_{i=0}^{m}w_{1i}(x,t), \sum_{i=0}^{m}w_{2i}(x,t),\dots ,\sum_{i=0}^{m}w_{ni}(x,t)) \\ \hphantom{w_{i}=}{}-N(\sum_{i=0}^{m-1}w_{1i}(x,t),\sum_{i=0}^{m-1}w_{2i}(x,t),\dots , \sum_{i=0}^{m-1}w_{ni}(x,t))]]]. \end{cases}$$
(14)

Finally, we obtain the approximate solution of the fractional-order partial differential equation

$$w(x,t)\cong w_{0}(x,t)+ w_{1}(x,t)+\cdots +w_{m}(x,t), \quad m=1,2,\dots .$$
(15)

### Theorem

B is the Banach space, if there exists $$0< K<1$$, $$\|w_{n}\| \le K\|w_{n-1}\|$$, for $$\forall x\in N$$, then the approximate solution $$w(x,t)$$ converges to A.

### Proof

Define the sequence $$A_{i}$$, $$i=0,1,\dots ,n$$

$$\textstyle\begin{cases} A_{0}=w_{0}, \\ A_{1}=w_{0}+w_{1}, \\ A_{2}=w_{0}+w_{1}+w_{2}, \\ \dots, \\ A_{n}=w_{0}+w_{1}+\cdots +w_{n} \end{cases}$$
(16)

and prove that $$(A_{i})_{i\ge 0}$$ is a Cauchy sequence, and we consider

$$\Vert A_{n}-A_{n-1} \Vert \le \Vert w_{n} \Vert \le K^{n}w_{0}$$
(17)

for $$m>n>0 \in N$$, we have

$$\textstyle\begin{cases} \Vert A_{n}-A_{m} \Vert = \Vert A_{n}-A_{n-1}+A_{n-1}-A_{n-2}+\cdots +A_{m+1}-A_{m} \Vert \\ \quad \le \Vert A_{n}-A_{n-1} \Vert + \Vert A_{n-1}-A_{n-2} \Vert +\cdots + \Vert A_{m+1}-A{m} \Vert \\ \quad \le (K^{n}+K^{n-1}+\cdots +K^{m+1})A_{0} \\ \quad \le \Vert \frac{K^{m+1}(1-K^{n-m})}{1-K} \Vert A_{0}, \end{cases}$$
(18)

where $$w_{0}$$ is bounded, and we have

$$\lim_{n,m\to \infty} \Vert A_{n}-A_{m} \Vert =0.$$
(19)

Therefore, the sequence $$(A_{i})_{i\ge 0}$$ is a Cauchy sequence in B, so the solution of Eq. (6) is convergent.

The error estimates are as follows:

$$\sup \Biggl\vert w(x,t)-\sum_{i=0}^{m}w_{i}(x,t) \Biggr\vert \le \frac{K^{m+1}}{1-K}\sup \bigl\vert w_{0}(x,t) \bigr\vert .$$
(20)

□

### Remark

Similar proofs can be found in .

## Test example

### Example 1

Consider the linear fractional Klein–Gordon equation 

$$\frac{\partial ^{\alpha}u}{\partial t^{\alpha}}- \frac{\partial ^{2}u}{\partial x^{2}}-u=0 ,\quad 0< \alpha \le 1$$
(21)

subject to the initial condition:

$$u(x,0)=1+\sin x.$$
(22)

The Elzaki transform of the linear fractional Klein–Gordon equation  is

$$v^{-\alpha}E \bigl[u(x,t) \bigr]=v^{2-\alpha}u(x,0)+E \biggl[ \frac{\partial ^{2}u(x,t)}{\partial x^{2}}+u(x,t) \biggr].$$
(23)

Using the inverse Elzaki transform of the above equation, we obtain

$$u(x,t)=E^{-1} \bigl[v^{2}u(x,0) \bigr]+E^{-1} \biggl[v^{ \alpha} E \biggl[\frac{\partial ^{2}u(x,t)}{\partial x^{2}}+u(x,t) \biggr] \biggr],$$
(24)

then, we use the iterative method above, and we have

\begin{aligned}& u_{0}(x,t)=E^{-1} \bigl[v^{2}(1+\sin x) \bigr], \end{aligned}
(25)
\begin{aligned}& u_{0}(x,t)=1+\sin x, \end{aligned}
(26)
\begin{aligned}& u_{1}(x,t)=E^{-1} \biggl[v^{\alpha}E \biggl[ \frac{\partial ^{2}u_{0}(x,t)}{\partial x^{2}}+u_{0}(x,t) \biggr] \biggr], \end{aligned}
(27)
\begin{aligned}& u_{1}(x,t)=\frac{t^{\alpha}}{\Gamma (1+\alpha )}, \end{aligned}
(28)
\begin{aligned}& \begin{aligned}[b]u_{2}(x,t)={}&v^{\alpha}E \biggl[ \frac{\partial ^{2}u_{1}(x,t)+u_{0}(x,t)}{\partial x^{2}}+u_{1}(x,t)+u_{0}(x,t) \biggr]\\ &{}-v^{\alpha}E \biggl[ \frac{\partial ^{2}u_{0}(x,t)}{\partial x^{2}}+u_{0}(x,t) \biggr], \end{aligned} \end{aligned}
(29)
\begin{aligned}& u_{2}(x,t)=\frac{t^{2\alpha}}{\Gamma (2\alpha +1)}, \end{aligned}
(30)
\begin{aligned}& \begin{aligned}[b]u_{3}(x,t)={}&v^{\alpha}E \biggl[ \frac{\partial ^{2} (u_{2}(x,t)+u_{1}(x,t)+u_{0}(x,t) )}{\partial x^{2}}+u_{2}(x,t)+u_{1}(x,t)+u_{0}(x,t) \biggr] \\ &{}-v^{\alpha}E \biggl[ \frac{\partial ^{2} (u_{1}(x,t)+u_{0}(x,t) )}{\partial x^{2}}+u_{1}(x,t)+u_{0}(x,t) \biggr] \end{aligned} \end{aligned}
(31)
\begin{aligned}& u_{3}(x,t)=\frac{t^{3\alpha}}{\Gamma (3\alpha +1)}, \end{aligned}
(32)
\begin{aligned}& \dots \\& u_{n}(x,t)=\frac{t^{n\alpha}}{\Gamma (n\alpha +1)}. \end{aligned}
(33)

The result is

$$u(x,t)=1+\sin x+\frac{t^{\alpha}}{\Gamma (1+\alpha )}+ \frac{t^{2\alpha}}{\Gamma (1+2\alpha )}+ \frac{t^{3\alpha}}{\Gamma (1+3\alpha )}+\cdots + \frac{t^{n\alpha}}{\Gamma (1+n\alpha )}+\cdots .$$
(34)

When $$\alpha =1$$, the exact solution of the linear fractional Klein–Gordon equation is as follows:

$$u(x,t)=e^{t}+\sin x.$$
(35)

In Fig. 1, the approximate solution of u is depicted for the case where the value of α is 0.01, 0.002, and 0.1.

### Example 2

Consider the new generalized fractional Hirota–Satsuma coupled KdV equation

\begin{aligned} &\frac{\partial ^{\alpha} u}{\partial t^{\alpha}}= \frac{1}{2}u_{xxx}-3uu_{x}+3(vw)_{x}, \\ &\frac{\partial ^{\alpha} v}{\partial t^{\alpha}}=-v_{xxx}+3uv_{x}, \quad 0< \alpha \le 1, \\ &\frac{\partial ^{\alpha}w}{\partial t^{\alpha}}=-w_{xxx}+3uw_{x} \end{aligned}
(36)

subject to the initial condition

\begin{aligned} &u(x,0)=\frac{1}{3}\bigl(\beta -2k^{2}\bigr)+2k^{2}\tanh^{2}(kx), \\ &v(x,0)=-\frac{4k^{2}c_{0}(\beta +k^{2})}{3c_{1}^{2}}+ \frac{4k^{2}(\beta +k^{2})}{3c_{1}}\tanh(kx), \\ &w(x,0)=c_{0}+c_{1}\tanh(kx). \end{aligned}
(37)

When $$\alpha =1$$, the exact results of the new generalized Hirota–Satsuma coupled KdV equation is as follows:

\begin{aligned} &u(x,t)=\frac{1}{3}\bigl(\beta -2k^{2}\bigr)+2k^{2}\tanh^{2}\bigl(k(x+ \beta t)\bigr), \\ &v(x,t)=-\frac{4k^{2}c_{0}(\beta +k^{2})}{3c_{1}^{2}}+ \frac{4k^{2}(\beta +k^{2})}{3c_{1}}\tanh\bigl(k(x+\beta t) \bigr), \\ &w(x,t)=c_{0}+c_{1}\tanh\bigl(k(x+\beta t)\bigr). \end{aligned}
(38)

The Elzaki transform of the new generalized fractional Hirota–Satsuma coupled KdV equation is

\begin{aligned} &v^{-\alpha}E\bigl[u(x,t)\bigr]=v^{2-\alpha}u(x,0)+E \biggl[\frac{1}{2}u_{xxx}-3uu_{x}+3(vw)_{x} \biggr], \\ &v^{-\alpha}E\bigl[v(x,t)\bigr]=v^{2-\alpha}v(x,0)+E[-v_{xxx}+3uv_{x}], \\ &v^{-\alpha}E\bigl[w(x,t)\bigr]=v^{2-\alpha}w(x,0)+E[-w_{xxx}+3uw_{x}]. \end{aligned}
(39)

Using the inverse Elzaki transform, we obtain

\begin{aligned} &u(x,t)=E^{-1}\bigl[v^{2}u(x,0) \bigr]+E^{-1}\biggl[v^{\alpha}E\biggl[ \frac{1}{2}u_{xxx}-3uu_{x}+3(vw)_{x} \biggr]\biggr], \\ &v(x,t)=E^{-1}\bigl[v^{2}v(x,0)\bigr]+E^{-1} \bigl[v^{\alpha}E[-v_{xxx}+3uv_{x}]\bigr], \\ &w(x,t)=E^{-1}\bigl[v^{2}w(x,0)\bigr]+E^{-1} \bigl[v^{\alpha}E[-w_{xxx}+3uw_{x}]\bigr]. \end{aligned}
(40)

Next, in terms of the iterative method above, we have

\begin{aligned}& u_{0}(x,t)=E^{-1} \bigl[v^{2}u(x,0)\bigr], \\& v_{0}(x,t)=E^{-1}\bigl[v^{2}u(x,0)\bigr], \end{aligned}
(41)
\begin{aligned}& w_{0}(x,t)=E^{-1}\bigl[v^{2}u(x,0)\bigr], \\& u_{0}(x,t)=\frac{1}{3}\bigl( \beta -2k^{2}\bigr)+2k^{2}\tanh^{2}(kx), \\& v_{0}(x,t)=-\frac{4k^{2}c_{0}(\beta +k^{2})}{3c_{1}^{2}}+ \frac{4k^{2}(\beta +k^{2})}{3c_{1}}\tanh(kx), \end{aligned}
(42)
\begin{aligned}& w_{0}(x,t)=c_{0}+c_{1}\tanh(kx), \\& u_{1}(x,t)=E^{-1} \biggl[v^{\alpha}E\biggl[\frac{1}{2}{u_{0}}_{xxx}-3u_{0}{u_{0}}_{x}+3(v_{0}w_{0})_{x} \biggr]\biggr], \\& v_{1}(x,t)=E^{-1}\bigl[v^{\alpha}E[-{v_{0}}_{xxx}+3{u_{0}} {v_{0}}_{x}]\bigr], \end{aligned}
(43)
\begin{aligned}& w_{1}(x,t)=E^{-1}\bigl[v^{\alpha}E[-{w_{0}}_{xxx}+3u_{0}{w_{0}}_{x}] \bigr], \\& u_{1}(x.t)=4k^{3}\beta \operatorname{sech}^{2}(kx)\tanh(kx) \frac{t^{t^{\alpha}}}{\Gamma (1+\alpha )}, \\& v_{1}(x,t)=4k^{3}\beta \bigl(\beta +k^{2}\bigr)\operatorname{sech}^{2}(kx) \frac{t^{\alpha}}{\Gamma (1+\alpha )}, \end{aligned}
(44)
\begin{aligned}& w_{1}(x,t)=c_{1}k\beta \operatorname{sech}^{2}(kx) \frac{t^{\alpha}}{\Gamma (1+\alpha )}, \\& u_{2}(x,t)=E^{-1} \biggl[v^{\alpha}E\biggl[\frac{1}{2}{(u_{0}+u_{1})}_{xxx}-3u_{0}{(u_{0}+u_{1})}_{x}+3 \bigl((v_{0}+v_{1}) (w_{0}+w_{1}) \bigr)_{x}\biggr]\biggr], \\& v_{2}(x,t)=E^{-1}\bigl[v^{\alpha}E \bigl[-{(v_{0}+v_{1})}_{xxx}+3{(u_{0}+u_{1})} {(v_{0}+v_{1})}_{x}\bigr]\bigr], \end{aligned}
(45)
\begin{aligned}& w_{2}(x,t)=E^{-1}\bigl[v^{\alpha}E \bigl[-{(w_{0}+w_{1})}_{xxx}+3(u_{0}+u_{1}){(w_{0}+w_{1})}_{x} \bigr]\bigr] , \\& \begin{aligned}[b]u_{2}(x,t)={}&\bigl[96k^{7} \beta \operatorname{sech}^{4}(kx)\tanh(kx)-144k^{7} \beta ^{2}\tanh(kx)\operatorname{sech}^{6}(kx) \\ & {}- 16c_{1}k^{7}\beta ^{2} \operatorname{sech}^{4}(kx)\tanh(kx)-16c_{1}k^{5} \beta ^{3}\operatorname{sech}^{4}(kx)\tanh(kx)\bigr] \\ & {}\times \frac{\Gamma (2\alpha +1)t^{3\alpha}}{\Gamma (3\alpha +1)\Gamma (1+\alpha )\Gamma (1+\alpha )}\\ & {}+\biggl[-48k^{6} \beta \operatorname{sech}^{4}(kx)+ \biggl(\frac{c_{0}}{3c_{1}}-8c_{0}\biggr)k^{6}\beta \tanh(kx)\operatorname{sech}^{2}(kx) \\ & {}+ (12c_{1}+72)k^{6}\operatorname{sech}^{4}(kx)+ \biggl(\frac{-116}{3}+8C_{1}\biggr)k^{6}\beta \operatorname{sech}^{6}(kx)\\ & {}+8 \beta ^{2}k^{4} \operatorname{sech}^{4}(kx) (c_{1}-1) \\ & {}+ 8\biggl(\frac{c_{0}k^{4}\beta ^{2}}{3c^{1}}+\frac{k^{2}\beta c_{1}}{3}-c_{0}k^{4} \beta ^{2}\biggr)\operatorname{sech}^{2}(kx)\tanh(kx)\\ & {}-8 \operatorname{sech}^{2}(kx) \biggl( \frac{4k^{4}\beta ^{2}+\beta ^{2}}{3}+\beta ^{2}k^{4}c_{1}\biggr)\biggr] \\ & {}\times \frac{t^{2\alpha}}{\Gamma (1+2\alpha )}+ \biggl[\frac{68}{3}k^{5}\tanh(kx) \operatorname{sech}^{2}(kx)-16k^{5}\tanh(kx) \\ & {}-4k^{3} \beta \tanh(kx)- 4k^{3}\beta \operatorname{sech}^{2}(kx)\tanh(kx)\biggr] \frac{t^{\alpha}}{\Gamma (1+\alpha )}, \end{aligned} \end{aligned}
(46)
\begin{aligned}& \begin{aligned}[b]v_{2}(x,t)={}&\bigl(-96k^{9} \beta ^{2}\operatorname{sech}^{2}(kx)\tanh^{2}(kx)-96k^{7} \beta ^{3}\operatorname{sech}^{4}(kx)\tanh^{2}(kx) \bigr) \\ & {}\times \frac{\Gamma (2\alpha +1)t^{3\alpha}}{\Gamma (3\alpha +1)\Gamma (1+\alpha )\Gamma (1+\alpha )} \\ & {}+\biggl[\frac{16k^{8}}{c_{1}}\beta \operatorname{sech}^{4}(kx) \tanh(kx)-144k^{8}\beta \operatorname{sech}^{4}(kx)\tanh(kx) \\ & {}- 152k^{6}\beta ^{2}\operatorname{sech}^{2}(kx) \tanh(kx)-96k^{6}\beta ^{2}\operatorname{sech}^{4}(kx) \tanh(kx) \\ & {}+ \frac{16k^{6}\beta ^{2}}{c_{1}}\operatorname{sech}^{4}(kx)\tanh(kx)-8k^{4} \beta ^{3}\operatorname{sech}^{2}(kx)\tanh(kx)\biggr] \frac{t^{2\alpha}}{\Gamma (2\alpha +1)} \\ & {}\times \biggl[\frac{24k^{7}}{c_{1}}-\frac{24k^{7}}{c_{1}}\operatorname{sech}^{4}(kx)+ \frac{8k^{5}\beta \operatorname{sech}^{4}(kx)}{3c_{1}}- \frac{8k^{5}\beta \operatorname{sech}^{2}(kx)}{3c_{1}} \\ & {}+ \frac{32k^{5}\beta}{c_{1}}\operatorname{sech}^{2}(kx)\tanh^{2}(kx)+ \frac{4k^{3}\beta ^{2}}{3c_{1}}\operatorname{sech}^{2}(kx)\\ & {}-4k^{3}\beta \bigl(\beta +k^{2}\bigr)\operatorname{sech}^{2}(kx)\biggr] \frac{t^{\alpha}}{\Gamma (1+\alpha )}, \end{aligned} \end{aligned}
(47)
\begin{aligned}& \begin{aligned}[b]w_{2}(x,t)={}&\bigl(-96k^{7} \beta ^{2}\operatorname{sech}^{4}(kx)\tanh(kx)+144k^{7} \beta ^{2}\operatorname{sech}^{6}(kx)\tanh(kx)\bigr) \\ & {}\times \frac{\Gamma (2\alpha +1)}{\Gamma (3\alpha +1)\Gamma (1+\alpha )\Gamma (1+\alpha )}t^{3 \alpha}+ \bigl(16k^{6}\beta \operatorname{sech}^{3}(kx)-24k^{6}\operatorname{sech}^{4}(kx) \\ &{}+72k^{6}\operatorname{sech}^{4}(kx)\tanh^{2}(kx)-8c_{1}k^{4} \beta \operatorname{sech}^{2}(kx)\tanh(kx)\\ & {}-24c_{1}k^{4} \beta \operatorname{sech}^{4}(kx)\tanh(kx) \\ & {}- 8\beta ^{2}k^{4}\operatorname{sech}^{2}(kx)+12 \beta ^{2}k^{4}\operatorname{sech}^{4}(kx)\bigr) \frac{t^{2\alpha}}{\Gamma (2\alpha +1)} \\ & {}\times \bigl[-8k^{5}\tanh(kx)+8c_{1}k^{3} \operatorname{sech}^{4}(kx)-4c_{1}k^{3} \operatorname{sech}^{2}(kx)+k^{3} \beta \tanh(kx) \\ & {}- 24k^{3}\tanh(kx)\operatorname{sech}^{2}(kx)-c_{1}k \beta \operatorname{sech}^{2}(kx)\bigr] \frac{t^{\alpha}}{\Gamma (1+\alpha )}, \end{aligned} \end{aligned}
(48)
\begin{aligned}& \dots \\ & \begin{aligned}[b]u_{n}(x,t)={}&E^{-1} \biggl[v^{\alpha}E\biggl[\frac{1}{2}{(u_{0}+u_{1}+ \cdots +u_{n})}_{xxx}-3u_{0}{(u_{0}+u_{1}+ \cdots +u_{n})}_{x} \\ & {}+ 3\bigl((v_{0}+v_{1}+\cdots +v_{n}) (w_{0}+w_{1}+\cdots +w_{n}) \bigr)_{x}\biggr]\biggr]\\ & {}-E^{-1}\biggl[v^{ \alpha}E \biggl[\frac{1}{2}{(u_{0}+u_{1}+\cdots +u_{n-1})}_{xxx} - 3u_{0}{(u_{0}+u_{1}+\cdots +u_{n-1})}_{x}\\ & {}+ 3\bigl((v_{0}+v_{1}+ \cdots +v_{n-1}) (w_{0}+w_{1}+ \cdots +w_{n-1})\bigr)_{x}\biggr]\biggr], \end{aligned} \end{aligned}
(49)
\begin{aligned}& \begin{aligned}[b]v_{n}(x,t)={}&E^{-1} \bigl[v^{\alpha}E\bigl[-{(v_{0}+v_{1}+\cdots +v_{n})}_{xxx}\\ &{}+3{(u_{0}+u_{1}+ \cdots +u_{n})} {(v_{0}+v_{1}+\cdots +v_{n})}_{x}\bigr]\bigr] \\ &{}-E^{-1}\bigl[v^{\alpha}E\bigl[-{(v_{0}+v_{1}+ \cdots +v_{n-1})}_{xxx}\\ &{}+3{(u_{0}+u_{1}+ \cdots +u_{n-1})} {(v_{0}+v_{1}+\cdots +v_{n-1})}_{x}\bigr]\bigr], \end{aligned} \end{aligned}
(50)
\begin{aligned}& \begin{aligned}[b]w_{n}(x,t)={}&E^{-1} \bigl[v^{\alpha}E\bigl[-{(w_{0}+w_{1}+\cdots +w_{n})}_{xxx}\\ &{}+3(u_{0}+u_{1}+ \cdots +u_{n}){(w_{0}+w_{1}+\cdots +w_{n})}_{x}\bigr]\bigr] \\ &{}-E^{-1}\bigl[v^{\alpha}E\bigl[-{(w_{0}+w_{1}+ \cdots +w_{n-1})}_{xxx}\\ &{}+3(u_{0}+u_{1}+ \cdots +u_{n-1}){(w_{0}+w_{1}+\cdots +w_{n-1})}_{x}\bigr]\bigr]. \end{aligned} \end{aligned}
(51)

The series-form solution is given as

\begin{aligned}& \begin{aligned} &u(x,t)=u_{1}(x,t)+u_{2}(x,t)+u_{3}(x,t)+ \cdots +u_{n}(x,t), \\ &v(x,t)=v_{1}(x,t)+v_{2}(x,t)+v_{3}(x,t)+ \cdots +v_{n}(x,t), \\ &w(x,t)=w_{1}(x,t)+w_{2}(x,t)+w_{3}(x,t)+ \cdots +w_{n}(x,t), \end{aligned} \end{aligned}
(52)
\begin{aligned}& \begin{aligned}[b]u(x,t)={}&\frac{1}{3}\bigl(\beta -2k^{2}\bigr)+2k^{2}\tanh^{2}(kx)+ \bigl[96k^{7} \beta \operatorname{sech}^{4}(kx)\tanh(kx) \\ &{}-144k^{7}\beta ^{2}\tanh(kx)\operatorname{sech}^{6}(kx) -16c_{1}k^{7}\beta ^{2} \operatorname{sech}^{4}(kx)\tanh(kx)\\ &{}-16c_{1}k^{5} \beta ^{3}\operatorname{sech}^{4}(kx)\tanh(kx)\bigr] \\ &{}\times \frac{\Gamma (2\alpha +1)t^{3\alpha}}{\Gamma (3\alpha +1)\Gamma (1+\alpha )\Gamma (1+\alpha )}+\biggl[-48k^{6} \beta \operatorname{sech}^{4}(kx)\\ &{}+ \biggl(\frac{c_{0}}{3c_{1}}-8c_{0}\biggr)k^{6}\beta \tanh(kx)\operatorname{sech}^{2}(kx) \\ & {}+ (12c_{1}+72)k^{6}\operatorname{sech}^{4}(kx)+ \biggl(\frac{-116}{3}+8C_{1}\biggr)k^{6}\beta \operatorname{sech}^{6}(kx)\\ &{}+8 \beta ^{2}k^{4} \operatorname{sech}^{4}(kx) (c_{1}-1) \\ & {}+ 8\biggl(\frac{c_{0}k^{4}\beta ^{2}}{3c^{1}}+\frac{k^{2}\beta c_{1}}{3}-c_{0}k^{4} \beta ^{2}\biggr)\operatorname{sech}^{2}(kx)\tanh(kx)\\ &{}-8 \operatorname{sech}^{2}(kx) \biggl( \frac{4k^{4}\beta ^{2}+\beta ^{2}}{3}+\beta ^{2}k^{4}c_{1}\biggr)\biggr]\frac{t^{2\alpha}}{\Gamma (1+2\alpha )} \\ &{}+ \biggl[\frac{68}{3}k^{5}\tanh(kx) \operatorname{sech}^{2}(kx)-16k^{5}\tanh(kx)\\ &{}-4k^{3} \beta \tanh(kx)\biggr] \frac{t^{\alpha}}{\Gamma (1+\alpha )}+\cdots, \end{aligned} \end{aligned}
(53)
\begin{aligned}& \begin{aligned}[b]v(x,t)={}&{-}\frac{4k^{2}c_{0}(\beta +k^{2})}{3c_{1}^{2}}+ \frac{4k^{2}(\beta +k^{2})}{3c_{1}} \tanh(kx) \\ &{}\times \bigl(-96k^{9}\beta ^{2}\operatorname{sech}^{2}(kx) \tanh^{2}(kx)-96k^{7}\beta ^{3} \operatorname{sech}^{4}(kx)\tanh^{2}(kx)\bigr) \\ & {}\times \frac{\Gamma (2\alpha +1)t^{3\alpha}}{\Gamma (3\alpha +1)\Gamma (1+\alpha )\Gamma (1+\alpha )}\\ & {} +\biggl[\frac{16k^{8}}{c_{1}}\beta \operatorname{sech}^{4}(kx) \tanh(kx)-144k^{8}\beta \operatorname{sech}^{4}(kx)\tanh(kx) \\ & {}- 152k^{6}\beta ^{2}\operatorname{sech}^{2}(kx) \tanh(kx)-96k^{6}\beta ^{2}\operatorname{sech}^{4}(kx) \tanh(kx) \\ & {}+ \frac{16k^{6}\beta ^{2}}{c_{1}}\operatorname{sech}^{4}(kx)\tanh(kx)-8k^{4} \beta ^{3}\operatorname{sech}^{2}(kx)\tanh(kx)\biggr] \frac{t^{2\alpha}}{\Gamma (2\alpha +1)} \\ &{}\times \biggl[\frac{24k^{7}}{c_{1}}-\frac{24k^{7}}{c_{1}}\operatorname{sech}^{4}(kx)+ \frac{8k^{5}\beta \operatorname{sech}^{4}(kx)}{3c_{1}}- \frac{8k^{5}\beta \operatorname{sech}^{2}(kx)}{3c_{1}} \\ & {}+ \frac{32k^{5}\beta}{c_{1}}\operatorname{sech}^{2}(kx)\tanh^{2}(kx)+ \frac{4k^{3}\beta ^{2}}{3c_{1}}\operatorname{sech}^{2}(kx)\biggr] \frac{t^{\alpha}}{\Gamma (1+\alpha )}+\cdots, \end{aligned} \end{aligned}
(54)
\begin{aligned}& w(x,t)=c_{0}+c_{1}\tanh(kx)+ \bigl(-96k^{7}\beta ^{2}\operatorname{sech}^{4}(kx) \tanh(kx) \\& \hphantom{w(x,t)=} {}+ 144k^{7}\beta ^{2}\operatorname{sech}^{6}(kx) \tanh(kx)\bigr) \frac{\Gamma (2\alpha +1)}{\Gamma (3\alpha +1)\Gamma (1+\alpha )\Gamma (1+\alpha )}t^{3 \alpha} \\& \hphantom{w(x,t)=} {}+ \bigl(16k^{6}\beta \operatorname{sech}^{3}(kx)-24k^{6} \operatorname{sech}^{4}(kx)+72k^{6}\operatorname{sech}^{4}(kx) \tanh^{2}(kx) \\& \hphantom{w(x,t)=}{}- 8c_{1}k^{4}\beta \operatorname{sech}^{2}(kx) \tanh(kx)-24c_{1}k^{4}\beta \operatorname{sech}^{4}(kx) \tanh(kx) \\& \hphantom{w(x,t)=}{}- 8\beta ^{2}k^{4}\operatorname{sech}^{2}(kx)+12 \beta ^{2}k^{4}\operatorname{sech}^{4}(kx)\bigr) \frac{t^{2\alpha}}{\Gamma (2\alpha +1)} \\& \hphantom{w(x,t)=}{}\times \bigl[-8k^{5}\tanh(kx)+8c_{1}k^{3} \operatorname{sech}^{4}(kx)-4c_{1}k^{3} \operatorname{sech}^{2}(kx)+k^{3} \beta \tanh(kx) \\& \hphantom{w(x,t)=} {}- 24k^{3}\tanh(kx)\operatorname{sech}^{2}(kx)\bigr] \frac{t^{\alpha}}{\Gamma (1+\alpha )}+ \cdots . \end{aligned}
(55)

Figure 2 shows the analytic solution of $$u(x,t)$$, $$v(x,t)$$, and $$w(x,t)$$ where $$\alpha =c_{1}=c_{0}=1$$, $$k=0.01$$, and $$\beta =1$$. We can see the exact solution and the approximate solution of the Elzaki iterative method for the case of $$\alpha =1$$ from Table 1.

### Remark

The reasons for the complexity of the solution are as follows:

1. Selection of initial values; 2. More parameters.

## Conclusion

In this article, we use the Elzaki transform with an iterative method to solve fractional partial differential equations. We find that the results using the homotopy perturbation method and the method in this article to the Klein–Gordon problem are the same. We see that the errors were not significant by picking specific values. Therefore, employing the Elzaki transform and the iterative method to solve fractional partial differential equations is effective.

Not applicable

## Abbreviations

ES:

denotes the exact solution

AS:

denotes the approximate solution

EE:

denotes the error estimate

## References

1. Li, C., Chen, A.: Numerical methods for fractional partial differential equations. Int. J. Comput. Math. 95(6–7), 1048–1099 (2018)

2. Gepreel, K.A.: The homotopy perturbation method applied to the nonlinear fractional Kolmogorov–Petrovskii–Piskunov equations. Appl. Math. Lett. 24(8), 1428–1434 (2011)

3. Khalid, M., Sultana, M., Zaidi, F., Arshad, U.: Application of Elzaki transform method on some fractional differential equations. Math. Theory Model. 5, 89–96 (2015)

4. Ziane, D., Elzaki, T.M., Hamdi Cherif, M.: Elzaki transform combined with variational iteration method for partial differential equations of fractional order. Fundam. J. Math. Appl. 1(1), 102–108 (2018). https://doi.org/10.33401/fujma.415892

5. Jafari, H., Nazari, M., Baleanu, D., Khalique, C.M.: A new approach for solving a system of fractional partial differential equations. Comput. Math. Appl. 66(5), 838–843 (2013)

6. Hilal, E., Elzaki, T.M.: Solution of nonlinear partial differential equations by new Laplace variational iteration method. J. Funct. Spaces 2014 (2014)

7. Mohamed, M.Z., Elzaki, T.M., Algolam, M.S., Elmohmoud, E.M.A., Hamza, A.E.: New modified variational iteration Laplace transform method compares Laplace Adomian decomposition method for solution time-partial fractional differential equations. J. Appl. Math. 2021, 1–10 (2021). https://doi.org/10.1155/2021/6662645

8. Thabet, H., Kendre, S., Chalishajar, D.: New analytical technique for solving a system of nonlinear fractional partial differential equations. Mathematics 5(4) (2017)

9. El-Rashidy, K.: New traveling wave solutions for the higher Sharma-Tasso-Olver equation by using extension exponential rational function method. Results Phys. 17, 103066 (2020). https://doi.org/10.1016/j.rinp.2020.103066

10. Hosseini, K., Mayeli, P., Kumar, D.: New exact solutions of the coupled Sine-Gordon equations in nonlinear optics using the modified Kudryashov method. J. Mod. Opt. 65(3), 361–364 (2018)

11. Tamsir, M., Srivastava, V.K.: Analytical study of time-fractional order Klein-Gordon equation. Alex. Eng. J. 55(1), 561–567 (2016). https://doi.org/10.1016/j.aej.2016.01.025

12. El-Sayed, S.M.: The decomposition method for studying the Klein–Gordon equation. Chaos Solitons Fractals 18(5), 1025–1030 (2003)

13. Kragh, H.: Equation with the many fathers. The Klein–Gordon equation in 1926. Am. J. Phys. 52(11), 1024–1033 (1984)

14. Tam, H.-W., Ma, W.-X., Hu, X.-B., Wang, D.-L.: The Hirota–Satsuma coupled kdv equation and a coupled Ito system revisited. J. Phys. Soc. Jpn. 69(1), 45–52 (2000)

15. Fan, E.: Soliton solutions for a generalized Hirota–Satsuma coupled kdv equation and a coupled mkdv equation. Phys. Lett. A 282(1–2), 18–22 (2001)

16. Wu, Y., Geng, X., Hu, X., Zhu, S.: A generalized Hirota–Satsuma coupled Korteweg–de Vries equation and Miura transformations. Phys. Lett. A 255(4–6), 259–264 (1999)

17. Abazari, R., Abazari, M.: Numerical simulation of generalized Hirota–Satsuma coupled kdv equation by rdtm and comparison with dtm. Commun. Nonlinear Sci. Numer. Simul. 17(2), 619–629 (2012)

18. Abbasbandy, S.: The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled kdv equation. Phys. Lett. A 361(6), 478–483 (2007)

19. Elzaki, T.M., Ezaki, S.M.: On the Elzaki transform and ordinary differential equation with variable coefficients. Adv. Theor. Appl. Math. 6(1), 41–46 (2011)

20. Mohamed, M.Z., Elzaki, T.M.: Applications of new integral transform for linear and nonlinear fractional partial differential equations. J. King Saud Univ., Sci. 32(1), 544–549 (2020). https://doi.org/10.1016/j.jksus.2018.08.003

21. Daftardar-Gejji, V., Jafari, H.: An iterative method for solving nonlinear functional equations. J. Math. Anal. Appl. 316(2), 753–763 (2006)

22. Elzaki, T.M.: The new integral transform Elzaki transform. Glob. J. Pure Appl. Math. 7(1), 57–64 (2011)

23. Elzaki, T.M.: On the connections between Laplace and Elzaki transforms. Adv. Theor. Appl. Math. 6(1), 1–11 (2011)

24. Hilal, E., Elzaki, T.M.: Solution of nonlinear partial differential equations by new Laplace variational iteration method. J. Funct. Spaces 2014 (2014)

25. Scott, A.C.: A nonlinear Klein-Gordon equation. Am. J. Phys. 37(1), 52–61 (1969)

26. Alderremy, A.A., Elzaki, T.M., Chamekh, M.: New transform iterative method for solving some Klein-Gordon equations. Results Phys. 10, 655–659 (2018)

## Acknowledgements

This work was supported by Hainan Provincial NSF of China with No. 120MS001, the authors would like to express sincere thanks to the referees for their valuable suggestions and comments.

## Funding

Supported by Hainan Provincial NSF of China with No. 120MS001.

## Author information

Authors

### Contributions

These authors contributed equally to this work.

### Corresponding author

Correspondence to Yong He.

## Ethics declarations

### Ethics approval and consent to participate

The manuscript is original, has not already been published, and is not currently under consideration by another journal.

Not applicable

### Competing interests

The authors declare no competing interests.

## Rights and permissions 