The key objective of this paper is to construct exact traveling wave solutions of the conformable time second integro-differential Kadomtsev–Petviashvili (KP) hierarchy equation using the Exp-function method and the (2 + 1)-dimensional conformable time partial integro-differential Jaulent–Miodek (JM) evolution equation utilizing the generalized Kudryashov method. These two problems involve the conformable partial derivative with respect to time. Initially, the conformable time partial integro-differential equations can be converted into nonlinear ordinary differential equations via a fractional complex transformation. The resulting equations are then analytically solved via the corresponding methods. As a result, the explicit exact solutions for these two equations can be expressed in terms of exponential functions. Setting some specific parameter values and varying values of the fractional order in the equations, their 3D, 2D, and contour solutions are graphically shown and physically characterized as, for instance, a bell-shaped solitary wave solution, a kink-type solution, and a singular multiple-soliton solution. To the best of the authors’ knowledge, the results of the equations obtained using the proposed methods are novel and reported here for the first time. The methods are simple, very powerful, and reliable for solving other nonlinear conformable time partial integro-differential equations arising in many applications.

The study of solutions of nonlinear partial differential equations (NPDEs) attracts the attention of scientists because their solutions can be used to lucidly explain many physical phenomena in various scientific fields, such as fluid mechanics, quantum mechanics, plasma physics, biology, chemistry, fiber optics, and many other branches of engineering. Obtaining solutions for NPDEs is of great significance for analyzing and better understanding the behaviors of the considered problems. There are many robust, stable, and effective methods that have been developed for constructing exact, approximate analytical and numerical solutions for NPDEs. Particularly, the methods have been extensively used to find exact solutions for NPDEs, such as the

A partial integro-differential equation (PIDE) is a mathematical equation involving partial derivatives and integrals of an unknown function of two or more independent variables. In the recent times, partial integro-differential equations (PIDEs) have been of considerable importance because they are widely used to model real-world problems and describe several physical phenomena in engineering, finance, and other fields of science. The applications of PIDEs have been studied in many papers. Sachs and Strauss [

In 2014, Khalil et al. [

In this article, we are interested in using the Exp-function method [

1. The conformable time second integro-differential Kadomtsev–Petviashvili (KP) hierarchy equation of order

2. The (2 + 1)-dimensional conformable time partial integro-differential Jaulent–Miodek (JM) evolution equation of order

The remaining parts of this article are organized as follows: In

In this section, we provide a definition of the conformable derivative and its important properties as established by Khalil et al. [

Let

If, in addition,

([

The Exp-function method and the generalized Kudryashov method described in this section are applied to the proposed PIDEs in the next section. Now, we consider the following general nonlinear conformable partial differential equation:

The symbols

The common step of the two methods is to transform the conformable PDE in (

It is not difficult to ascertain that

In this section, we use the Exp-function method and the generalized Kudryashov method to obtain explicit exact solutions for Equations (

Equation (

Applying the following transformation:

Applying the Exp-function method to Equation (

Proceeding in a similar manner as illustrated above, we can determine the values of

For simplicity, we choose

Substituting Equation (

With the help of Maple, we can simultaneously solve system (

Since

Inserting

Using the transformation

Applying the traveling wave transform (

On the basis of Equation (

Choosing

Solving the above system with the help of Maple, we obtain two different results for the exact solutions for Equation (

where

As a result of

By replacing

where

Since

Inserting

In this section, we provide interesting graphical representations of the exact solutions of the conformable time second integro-differential KP hierarchy Equation (

In

Furthermore,

We can observe some effects of the variation of the time-fractional order

In summary, the Exp-function method, the generalized Kudryashov method, the use of transformation (

Applying the generalized Kudryashov method to obtain exact solutions for Equation (

All of the obtained exact solutions discussed in this paper were verified by substituting them back into their corresponding equations with the help of the Maple package program. The Exp-function method and the generalized Kudryashov method are straightforward, reliable, efficient, and pragmatic mathematical tools for solving the proposed equations because they produce uncomplicated exact solution forms. These two methods could be effectively applied to solve a wide range of nonlinear partial integro-differential equations arising in natural phenomena, giving their analytically extracted exact solutions.

Conceptualization, S.S. (Sekson Sirisubtawee) and S.S. (Surattana Sungnul); methodology, S.K. and S.S. (Sekson Sirisubtawee); software, S.K. and S.S. (Sekson Sirisubtawee); validation, S.K., S.S. (Sekson Sirisubtawee) and S.S. (Surattana Sungnul); formal analysis, S.K. and S.S. (Sekson Sirisubtawee); investigation, S.K. and S.S. (Sekson Sirisubtawee); resources, S.S. (Sekson Sirisubtawee) and S.S. (Surattana Sungnul); data curation, S.K.; writing—original draft preparation, S.K. and S.S. (Sekson Sirisubtawee); writing—review and editing, S.S. (Sekson Sirisubtawee) and S.S. (Surattana Sungnul); visualization, S.K., S.S. (Sekson Sirisubtawee) and S.S. (Surattana Sungnul); supervision, S.S. (Sekson Sirisubtawee) and S.S. (Surattana Sungnul); project administration, S.S. (Sekson Sirisubtawee) and S.S. (Surattana Sungnul); funding acquisition, S.K. and S.S. (Sekson Sirisubtawee). All authors have read and agreed to the published version of the manuscript.

The first author was funded by King Mongkut’s University of Technology North Bangkok under contract no. KMUTNB-61-PHD-014. The second and third authors were financially supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

Not applicable.

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Not applicable.

The authors are grateful to anonymous referees for the valuable comments which have significantly improved this article. In addition, the first author would like to acknowledge the partial support from the Graduate College, King Mongkut’s University of Technology North Bangkok.

The authors declare no conflict of interest.

This part demonstrates the expression of each coefficient

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^{2})-expansion method in mathematical physics

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Associated plots of

Associated plots of

Associated plots of