The findings indicate an application of a new method of expansion of the forms and to determine the solutions for wave of the solitary nature in the -dimensional modified form for nonlinear integro-partial differential equations. By using this strategy, we acquired solutions of wave which has a solitary nature that have been solved for three different kinds: hyperbolic, trigonometric, and rational functions. As a result, we obtained different forms of solutions which are new, effective, and powerful to illustrate the solitary nature of waves. The physical and geometrical interpretations have been shown using software in 2 and 3-dimensional surfaces. The obtained results have applications in mathematical and applied sciences. It can also solve different nonlinear integro-partial differential equations which have different applications in physical phenomena using this new method. It has many applications to solve the nonlinear nature of the physical world.

1. Introduction

The real-world models can be illustrated by nonlinear partial differential equations (NLPDEs). They are decisive in phenomena of the world which is full of complexity in areas of applied mathematics, engineering, and physics. Solutions of nonlinear evolution equations (NLEEs) were acclimated to obtain phenomena of nonlinear wave nature. This event is an important and critical issue in occurrence of scientific nature, and hence critical analysis of solutions of NLEEs has a great advantage in understanding nonlinear physical phenomena. In nonlinear phenomena of physical systems, the solutions of NLIPDEs have important applications to describe the natural problems. These include traveling wave of solitary nature and propagation and vibrations in certain restricted space and time. Some scholars have studied the solitary wave solutions and their applications in nonlinear physical phenomena [15].

Finding the exact solutions of NLIPDEs is not an easy way to determine, and hence certain scholars have constructed different methods to reduce the existing complexity: balancing homogeneous equation [6, 7], transform Darboux method [8, 9], first kind integral [10, 11], and function of tanh [12]. The methods like elliptc function of Jacobi and Kudryshov have been used for the presentation of truncated expansion to analysis nonlinear equations having analytical type solutions. One of the main advantages of determining exact solutions is that it allows scholars to create concrete physical world and innovate more experiments by creating suitable conditions with certain parameters. Hence, analyzing wave solutions of solitary type is a current issue in physical nonlinear systems. But, most of the equations of nonlinear physical phenomena of NLIPDES are easily solvable to illustrate physical models. As a result, many mathematicians have developed different methods. The solutions of NLIPDEs have a great applications in nonlinear sciences in wave dispresion, diffusion, reaction, dissipation, and convection. They provide us more information about the physical nature which can be advanced for better implementaions. These nonlinear integro-differential equations are solved by many different methods such as modification of simple equation [13], equations of auxiliary [1416], methods of expansions [17, 18], methods of expansion [1921], and -dimensional space time of Kudryashov method of generalization. The method of expansion for hierarchy equations of an integro differential determine solutions can be transformed to reduce differential nature Different scholars have investigated NLIPDEs to get solutions of such equations in an exact form [3032]. Solitary wave and compaction type in a generalized form were investigated. Their exact solutions have different in nature: kink, complex, soliton, and solitary. The obtained solutions can also be incorporated in the form of periodic, solitary bell and kink [32, 3537].

As a result, some new exact solutions were obtained including triangular periodic wave solutions, exponential solutions, and complex traveling solutions. The -expansion method was used to construct some new traveling wave solutions including hyperbolic function, trigonometric function, and rational function solutions of the -dimensional Jaulent–Miodek equation formulated in [38, 39]. The technique factor of integrating to solve -dimensional space time was applied in [35, 40, 41]. Based on the previous results, the authors believe that these studies will be useful for understanding physical phenomena in many areas of applied mathematics and physics, especially nonlinear systems. Even if, those equations solved by many different methods as far as the researchers go through the existing literature, among those methods, we are concentrated in this article determining the wave solutions of solitary type in -dimensional space time of NLIPDEs.

Generally, in this article, we are going to find the solitary wave solutions of NLIPDEs of -dimensional which is given bywhere .

2. Mathematical Formulation and Method

2.1. Description of Special Expansion Method

In order to carry out this study successfully, we used two variables . Here, the following notes will be considered to proceed to the next steps.

Remark 1. If we see the second-order linear ordinary differential equation

Remark 2. If , then the general solution of equation (2) has the following form:where and are arbitrary constants.where .

Remark 3. If , then the general solution of equation (2) has the following form:where  = 

Remark 4. If , then the general solution of equation (2) has the following form: , which becomes .
Then, integrating both sides with respect to , it becomes the following.
, where is constant of integration.
Again integrating with respect to , we getwhere and are constants. and subsequently, the relation between and isApplying the transformation on equation (1),and hence (1) is transformed to

3. Results and Discussion

The following are the main steps of the special expansion methods by using two variables .

Step 1. First we change the given equation nonlinear partial integro-differential equation to nonlinear partial differential equation.
Suppose we have the following NLPDEs:where is a polynomial of unknown function and its total derivatives with respect to . And its various partial derivatives in which the highest order derivatives and nonlinear terms are involved.

Step 2. We use wave transformation to convert NLPDES to ODEs:where is arbitrary constant. The traveling wave transformation (13) and considering the resulting equation with respect to , (12) transformed to ordinary differential equations.

Step 3. By assuming solutions of (14) transformed into variables of two polynomials of the form and ,where (i = 0, 1, 2, 3, …, N) and (i = 1, 2, 3, …, N) are arbitrary constants.
Also, and satisfy the following.

Step 4. Determine the positive integer in equation (15) by using the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in equation (14). More precisely, if the degree of is deg  = , then the degree of other term will be expressed as follows:where degree highest order of the function, the highest order, is degree of coefficient of nonlinear term, is degree of nonlinear term, and is order of nonlinear term.

Step 5. From the above steps, we get the value of and we write the form of solution of equation (14) by equation (15).
Then, substitute (15) into (14) along with (3), and the left-hand side of (14) can be converted into the polynomial in and in which the degree of is no longer than 1.
Equating each coefficient of this polynomial to zero yields a system of algebraic equation which can be solved by using Maple or Mathematica to get the values of each coefficient ( ) with leading coefficient variable and the value of constant number .
We use  =  and substitute it into (1) to reduce the NLIPDE (1) to the following NLPDEs of the equivalent to form of (18) as follows:From the traveling wave transformation of (13), we have the following.
, , where represent the space variables, ttime variable, and  −  is the speed of traveling wave, which permits us to transform (12) into ODE:Depending on (19)–(27), (18) becomes the following ODE forms:where  =  . By integrating (28) with respect to and taking the transformation  =  of equation (28), we obtainwhere is the integration constant. Considering the homogeneous balance between the highest order derivative and nonlinear term in (29) by (17), we get the following.
+  =  .
Consequently, the solution of (29) has the following form:where are constants.
To solve for (29), first we have to find , and by using (31).
From (31), , and by using (3),Finally, substituting (3) in place of and in equation (31), we get the following equation.From (33), we can get equal to the following results:From (33) and second partial derivatives of (31), by substituting into second partial derivative of equations in place of w’ as follows and finally adding according to the same power coefficient:From (35), is equivalent toFrom (35) and third partial derivatives of (31) with substitute (3) into third partial derivatives of equation in place of and equation (31) equivalent form of (34) as follows and finally adding according to the same power coefficient:From (37), is equivalent toFrom (37) and fourth partial derivatives of (31) with substitute (3) into fourth partial derivatives of equation in place of and equation (31) equivalent to form of (35) as follows and finally adding according to the same power coefficient:When (31) and (33) are multiplied with each other, is equivalent toFrom (31) and (35), when they multiply each other, the result of them has equivalent form of (41) as follows according to the same power coefficient:From (31), when multiplying two times itself by the form of  =  , thenFinally, are as follows:From (31) and (43),The product of (31) and (43) is the following:Also, we have .

3.1. Case A: —Nature of Hyperbolic Solutions

By using (3), equation is equivalent to

Finally, by using equation (3), equation (35) becomes

By using (3), equation is equivalent to

Finally, by using (3)