Solitary Wave Solutions of Nonlinear Integro-Partial Differential Equations of <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.9939pt" id="M1" height="15.2545pt" version="1.1" viewBox="-0.0657574 -12.2606 46.3673 15.2545" width="46.3673pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,5.937,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"/></g><g transform="matrix(.017,0,0,-0.017,18.01,0)"><path id="g117-36" d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"/></g><g transform="matrix(.017,0,0,-0.017,31.918,0)"><path id="g113-50" d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"/></g><g transform="matrix(.017,0,0,-0.017,40.154,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>-</span>Dimensional and Its Models

Gusu, Daba Meshesha; Diro, Shelama

doi:https://doi.org/10.1155/2022/9954649

International Journal of Differential Equations

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Abstract Introduction Results and Discussion Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 9954649 | https://doi.org/10.1155/2022/9954649

Solitary Wave Solutions of Nonlinear Integro-Partial Differential Equations of -Dimensional and Its Models

Daba Meshesha Gusu¹and Shelama Diro¹

Academic Editor: Wen-Xiu Ma

Received18 Nov 2021

Revised20 Jan 2022

Accepted18 Mar 2022

Published29 May 2022

Abstract

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 [1–5].

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 [14–16], methods of expansions [17, 18], methods of expansion [19–21], 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 [30–32]. 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, 35–37].

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)

By using (3), equation of becomes

Finally, by using equations (3)

By using (3) and (35), is equivalent to

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

By using (3) and (43), is equal to the following form:

By using equations (3) and (43), becomes

By using equations (47), (49), (51), and (53), becomes the following:

Inserting (31) in (29) and applying (4) as well as (3), we have polynomials in the variable form of and . Then, we obtain equations of algebraic sets by applying the coefficients of the polynomials set to zero. Hence, the following algebraic expressions are obtained:

By solving the previous algebraic equations using the Mathematica software package 9.1, we get the following results.

Result 1. By substituting (58) into (31) with (13), (16), and (4), we get the exact solutions of (18) as follows:By substituting the value of and from (3) we get = .
When we put the value of from equation (3), we getThen, the exact solution of -dimensional Sawada–Kotera equation of (18) is as follows:By letting , , and in (61), the particular solution of (18) isThen, by letting , , , , and , the graph of (63) has the following forms.
Figure 1 indicates the soliton of (63) in three and two dimensions. If we put and in (61), we produce a solitary solution:The particular solution of -dimensional equation for hyperbolic function, if we put result on (61) when and , has the following form.
The graph of the particular solution where , , , is as follows.
Figure 2 shows the soliton of (65) in three and 2 dimensions by considering certain parameters.

Result 2. By substituting (66) into (31) with (13) by using (4) and (16), we get the exact solutions of equation (18) as follows:By substituting the value in place of and from (3), we get = ; when we put the value of from (3), we getThen, the exact solution of -dimensional equation from (18) is as follows:By letting , , and in (69), the particular solution of (18) isThen, by letting , , , , and , the graph of (71) has the following forms.
Figure 3 represents the soliton of (71) in two and three dimensions using certain parameters.
If we put and in (61), we produce a solitary solution:The particular solution of (2 + 1)-dimensional equation for hyperbolic function, if we put result on equation (51) when and , has the following form.
The graph of the particular solution where , , , is as follows.
Figure 4 illustrates the soliton of (73) in three-dimensional space and two-dimensional plane.

3.2. Case B: —Nature of Trigonometric Solutions

From (7), (29) and (39) become

Finally, by using (7), (29) and (35) become

From (7), (29) and (41) become

Finally, by using (7), (29) and (41) become

From (7), (29) and (45) become

By using (7), (29) and (45) become

From (7), (29) and (35) become

Again considering (7), (29) and (35) become

From (7), (29) and (43) become

Using (7), (29) and (43) become

By using (75), (77), (79), (81) and (83), becomes the following forms.

Substituting (31) into (29) and using (4) with (3), we get the polynomials in and , setting the coefficients of this polynomial to zero to obtain the following set of algebraic equations:

Solving the above equations of algebraic type by Mathematica software, we have.

Result 3. By substituting (86) into (31) with (13) by using (6) and (16), we get the exact solutions of (18) as follows:By substituting the value of and from (3), we getWhen we put the value of from (3), we getThen, the exact solution of -dimensional equation from (18) is as follows:By letting , = and in (90), the particular solution of (18) isThen, by letting , , , , and , the graph of (92) has the following forms.
Figure 5 shows periodic soliton in two and three dimensions for (92). When we put and in (89), we produce a solitary solution:The particular solution of -dimensional equation for trigonometric function solutions, if we put result on (90) when and , has the following form.
The graph of the particular solution where , , , , is as follows.
Figure 6 represents a periodic soliton nature of (94) in three and two dimensions.

Result 4. By substituting (95) into (31) with (6), (13), and (16), we get the exact solutions of (18) as follows:By substituting the value of and from (3), we get
When we put the value of from (3), we getThen, the exact solution of -dimensional equation from (18) is as follows:By letting , = 0, and in (99), the particular solution of (18) isThen, by letting , , , , and , the graph of (101) has the following forms.
Figure 7 shows the periodic nature for (101) in three and two dimensions. But, if we put and in (98), we produce a solitary solution:The particular solution of (2 + 1)-dimensional equation for trigonometric function solutions, if we put result on (103) when and , has the following form.
The graph of the particular solution of (103) where , , , is as follows.
Figure 8 shows periodic nature for (101) in three-dimensional space and two-dimensional plane.

3.3. Case C: —Nature of Rational Solutions

From (9), (29) and (39) become

Finally by using (9), (29 and (39) become

From (9) and (29), (41) becomes

Finally, by using (9) and (29), (41) becomes

From (9), (45) becomes

Finally by using (9), (45) becomes

From (9) and (29), (35) becomes

By using (9), (29), and (35), we get

From (9), (29), and (43),

At the end, by using (9) and (29) in , we get

From (105), (107), (109), (111), and (113), equation is equivalent to

Solving the above equation of algebraic type by Mathematica software, we have

By substituting (119) into (31) with (8), (13), and (16), we get the exact solutions of (18) as follows:

By substituting the value of and from (3), we get ; when we put the value of from (3), we get

When we put the value of from (8) in (120),

Hence, we get the exact solution of (2 + 1)-dimensional integro-partial equation with respect to result of (120) as follows: by letting , , and in (120), the particular solution of (1) is

The particular solution of (2 + 1)-dimensional for rational function solutions and putting the result in equation (119) when M_2 > 0 and M_1 >0 have the form of graphical representation for the particular solution of Equation (123):

Figure 9 shows soliton behaviour for (120) in three-dimensional space and in two-dimensional plane.

4. Conclusion

In this paper, the new special expansion of the and method of expansion has been applied to discover solutions of solitary waves of NLIPDEs by the help of Mathematica software 9.1. We applied successfully the obtained soliton wave solutions (2+1) -dimensional space time using certain parameters. These parameters were taken as special values in which the solutions of wave of the solitary and periodic are developed from the exact solutions. The solution of NLIPDEs of (2+1) dimensional equation is analytically solved and it is an easy and direct way to solve nonlinear partial differential equations. To use the method effectively, we applied the combined expansion methods in the form of and . The combined methods to solve the NLIPDEs for integro-differential equations led us to get exact solutions for three different cases of which has hyperbolic, trigonometric, and rational nature. We get different results for each case. The obtained results were illustrated by graphical plots. It can be concluded that the obtained solutions will serve as a very important milestone in the study of nonlinear phenomena arising in mathematical physics and engineering fields. Finally, the method is a powerful and effective method to solve soliton wave solutions for NLIPDEs.

Data Availability

The data used to support the findings of the study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2022 Daba Meshesha Gusu and Shelama Diro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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