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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 [1], 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 [2]; Khalid used the Elzaki transform method to solve the equations [3]; in [4], Ziane used the fractional Elzaki variational iteration method to solve the equations; Jafari introduced the Iterative Laplace transform method in [57]; Tarig used the Sumudutrans form of the variational iteration method to solve linear homogeneous partial differential equations; Thabet [8] introduced a new analytic method to solve partial differential equations with fractional order, and El-Rashidy [9] used the method to obtain new traveling-wave solutions of the equations. Hosseini [10] used the modified Kudryashov method to obtain exact solutions of the coupled sine-Gordon equations. Mohammad Tamsir [11] 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 [1417], including the homotopy analysis approach [18]. 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 [21] 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 [24]

$$ E\bigl[D_{x}^{\alpha}w(x)\bigr]=v^{-\alpha}E \bigl[w(x)\bigr]-\sum_{k=0}^{m-1}v^{2- \alpha +k}w^{(k)}(0),\quad \text{where } m-1< \alpha < m. $$
(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 [8].

Test example

Example 1

Consider the linear fractional Klein–Gordon equation [25]

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

subject to the initial condition:

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

The Elzaki transform of the linear fractional Klein–Gordon equation [26] 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.

Figure 1
figure 1

Graph of \(u(x,t)\) at \(\alpha =0.01\), 0.02, and 0.1 of Example 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.

Figure 2
figure 2

The graph of \(u(x,t)\), \(v(x,t)\), \(and w(x,t)\) when \(\alpha =c_{1}=c_{0}=1\), \(k=0.01\) and \(\beta =1\)

Table 1 Approximate solutions, exact solutions, and error estimates for the Elzaki iterative transformation method when α is 1 for Example 2

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.

Availability of data and materials

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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Google Scholar 

  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

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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

    Article  MATH  Google Scholar 

  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

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Google Scholar 

  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

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

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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.

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

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He, Y., Zhang, W. Application of the Elzaki iterative method to fractional partial differential equations. Bound Value Probl 2023, 6 (2023). https://doi.org/10.1186/s13661-022-01689-9

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Keywords

  • Elzaki transform
  • Iterative method
  • Fractional differential equation