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The present research work is devoted to investigate fractional order Benjamin-Bona-Mahony (FBBM) as well as modified fractional order FBBM (FMBBM) equations under nonlocal and nonsingular derivative of Caputo-Fabrizio (CF). In this regards, some qualitative results including the existence of at least one solution are established via using some fixed point results of Krasnoselskii and Banach. Further on using an iterative method, some semianalytical results are also studied. The concerned tool is formed when the Adomian decomposition method is coupled with some integral transform like Laplace. Graphical presentations are given for various fractional orders. Also, the concerned method is also compared with some variational-type perturbation method to demonstrate the efficiency of the proposed method.

Fractional calculus is the generalized form of classical calculus. With the rapid change in science and technology, the aforesaid area has attracted the attention of many researchers. The mentioned branch has many applications in different areas of science like modeling, control theory, physics, signal processing, economics, and chemistry [

Let

The CF integral with

For the CF derivative of order

The considered method is used to compute the solution in an infinite series form. We consider the solution as

If

In the ongoing section, we discuss the existence of the considered problem.

Under Definitions (

The assumptions needed for our work are

(_{1})

(_{2}) For all

Furthermore,

In light of hypothesis (_{1}) and (_{2}), if

Using (2.5), and a bounded set defined as _{1})

This show that

Therefore,

This implies that

In view of assumption (_{2}) if

By using (

Suppose

Therefore,

To present the iterative solution of our considered problem, we first give a general procedure for the given problem as

Taking Laplace transform of (

Let us consider the solution in terms of a series as

Using (

After evaluation, the required solution is

Let

Then, the unique fixed point

(_{1}) By using mathematical induction for

Considering that the result for

Now consider

With the help of (_{1}), we have

Consider the following FBBM equation under the given condition as

Taking Laplace transform of (

Let us consider the solution in terms of a series as

After calculation, the solution of the considered problem (

Consider the following FBBM equation under the given condition as

Taking Laplace of (

Here, we consider the unknown solution as

In this way, the series solution of the proposed problem (

Consider the following FMBBM equation under the given condition

Taking Laplace of (

Here, we consider the unknown

Hence, in this case, the solution in same way may be computed.

Here, in the ongoing section, we find series solutions for (

Consider the following FBBM equation [

With the exact solution given below,

With the help of the procedure discussed in Case

And hence, the solution of (

The approximate solution graphs for various fractional orders are given in Figure

Consider the FBBM equation using CFFOD as

With the help of procedure discussed for Case

Here, we plot the approximate solution of the FBBM equation up to four terms in Figure

Consider the FBBM equation using CFFOD as

Here, we plot the approximate solution of FBBM equation up to four terms in Figure

Consider the modified FBBM equation using CFFOD as

With the help of the procedure mentioned in Case

Here, we plot the approximate solution of the FBBM equation up to four terms in Figure

Consider the modified FBBM equation using CFFOD as

With the help of the procedure discussed for Case

Here, we plot the approximate solution of FBBM equation up to four terms in Figure

Surface plots of the required solution up to four terms at different values of

Comparison between the absolute error at VHPM [

0.03 | 0.04 | 0.05 | ||||
---|---|---|---|---|---|---|

VHPM | LADM | VHPM | LADM | VHPM | LADM | |

0.01 | ||||||

0.02 | ||||||

0.03 | ||||||

0.04 | ||||||

0.05 |

Surface plots of the resultant solution up to four terms at different values of

Surface plots of the resultant solution up to four terms at different values of

Surface plots of the resultant solution up to four terms at different values of

Surface plots of the resultant solution up to four terms at different values of

In our work, some existence results about the solution to the nonlinear problem of BBM equations under nonsingular kernel-type derivative have been developed successfully. We have discussed different cases of the concerned equations for semianalytical results. For approximate analytical results, a novel iterative method of Laplace transform coupled with Adomian polynomials has been used. Further, by providing an example, we have computed the absolute errors in comparison with VHPM for first four-term solutions at different values of variables

Data availability is not applicable in this manuscript.

There is no competing interest regarding this work.

An equal contribution has been done by all the authors.

Prince Sultan University provided support through the research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM), group number RG-DES-2017-01-17.