This paper applies a powerful scheme, namely Bernoulli sub-equation function method, to some partial differential equations with high non-linearity. Many new travelling wave solutions, such as mixed dark-bright soliton, exponential and complex domain, are reported. Under a suitable choice of the values of parameters, wave behaviours of the results obtained in the paper – in terms of 2D, 3D and contour surfaces – are observed.
Keywords
- Integro-partial differential equation
- Fifth-Order integrable model
- Analytical method
- Rational function solution
- Complex Solution
- Contour surface
- Travelling wave solutions
- Mixed dark-bright soliton
Mathematical models have been used to explain many real-world problems, in the past decade. In this sense, Qi et al. [1] have investigated some important models describing certain waves in physics. Colucci et al. [2] have introduced a new partial differential equation to define the ice crystal size delivery. Another novel model considered to explain the nucleation of spherical agglomerates using the immersion mechanism has been developed by Tash et al. [3]. Baleanu et al. [4] have presented a new study about people's liver using Caputo–Fabrizio fractional model. Pignotti et al. [5] have given another novel differential model related to the project of extraction in mines. Pompa et al. [6] proposed some important models about gastrointestinal absorption for availability of drugs to biological mechanisms. Compression of main electrocardiography signals using a new genetic programming-based mathematical modelling algorithm has been studied by Feli and Abdali-Mohammadi [7]. With the aim of assessing the Bang-Bang model related to hysteresis influences on heat and mass transmit in spongy building material, another important article has been proposed by Berger et al. [8]. Tsur et al. [9] studied the reaction of melanoma patients to the immune checkpoint surrounding (including understandings collected) in an assessment of a new mathematical mechanistic sample. Camaraza-Medina et al. [10] presented a new study on the mathematical inference of computation of heat transmission by thickenings inside tubes. Another powerful model involving chemical reaction systems has been proposed by Amin et al. [11]. Kortcheva et al. [12] explored new ways and differential equations related to peripheral risk administration in harbours. Aiming to get data using rubrics, Sahin and Baki have developed a new model for measuring mathematical success [13]. Meena et al. [14] composed a new mathematical model about the influencing agent in biofilms under toxic situations to discuss the values of parameters. Hamzehlou et al. have explored a unique way to model and predict the active progress of particle morphology mathematically [15]. There are many other such studies [16,17,18,19,20, 29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44].
The remainder of this current paper is constructed in the following parts. In Section 2, we introduce the Bernoulli sub-equation function method (BSEFM) in detail. In Section 3, as a first application, we apply BSEFM to the (1+1)-dimensional integro-differential Ito equation (ITOE) defined as follows [21]:
Gepreel et al. [21] have applied the modified simple equation method to Eq. (1) for getting some important properties. Wazwaz has investigated the physical meaning of Eq. (1) [22]. Further, Eq. (1) has been investigated by using meshless discrete collocation method, numerically in another paper [23].
As a second application, we consider the (2+1)-dimensional fifth-order integrable equation (FOIE) given as follows [24]:
In this sub-section of the paper, the scheme considered herein is introduced [25,26,27].
According to the balance principle, we can obtain a relationship between
Solving this system, the values of
When we use a complete discrimination system for polynomial, we get the solutions to Eq. (5) through computational programs and classify certain solutions to Eq. (5). For a better understanding of the results obtained in this manner, we can draw two- and three-dimensional surfaces of solutions by taking into consideration appropriate values of parameters.
This section applies BSEFM to the governing models, such as ITOE and FOIE models, to find new travelling wave solutions.
If we take
If we consider the travelling wave transformation as
When
Balancing,
From Eq. (15), we can get many entirely new travelling wave solutions to Eq. (1).
Putting these into Eq. (16) by considering Eq. (10), we get the following new exponential function solution for Eq. (1):
Here,
Fig. 1
The 3D and contour surfaces of Eq. (20) when the values are

Fig. 2
The 2D graph of Eq. (20) when the values are

Putting these variables into Eq. (16) by taking into account Eq. (10), we get the following new exponential function solution for Eq. (1):
Fig. 3
The 3D and contour surfaces of Eq. (22) when the values are

Fig. 4
The 2D graph of Eq. (22) when the values are

Fig. 5
The 3D and contour surfaces of Eq. (24) when the values are

Fig. 6
The 2D graph of Eq. (24) when the values are

Fig. 7
The 3D and contour surfaces of Eq. (26) when the values are

Fig. 8
The 2D graph of Eq. (26) when the values are

Fig. 9
The 3D and contour surfaces of Eq. (28) when the values are

Fig. 10
The 2D graph of Eq. (28) when the values are

Fig. 11
The 3D and contour surfaces of Eq. (30) when the values are

Fig. 12
The 2D graph of Eq. (30) when the values are

This sub-section applies BSEFM to the FOIE model for finding some new travelling wave solutions. First of all, considering the travelling wave transformation as
For simplicity, if we take
With the help of the balance principle, we obtain the following:
This gives many new travelling wave solutions to Eq. (2).
Fig. 13
The 3D and contour surfaces of Eq. (39) when the values are

Fig. 14
The 2D graph of Eq. (39) when the values are

Fig. 15
The 3D and contour surfaces of Eq. (41) when the values are

Fig. 16
The 2D graph of Eq. (41) for

Putting these variables into Eq. (35) by taking into account Eq. (10), we get the following exponential function solution for Eq. (2):
Fig. 17
The 3D and contour surfaces of Eq. (43) when the values are

Fig. 18
The 2D graph of Eq. (43) for

Fig. 19
The 3D and contour surfaces of Eq. (45) when the values are

Fig. 20
The 2D graph of Eq. (45) for

Here,
Fig. 21
The 3D and contour surfaces of Eq. (47) for

Fig. 22
The 2D graph of Eq. (47) for

Putting these variables into Eq. (35) by considering Eq. (10), we get the following exponential function solution for Eq. (2):
Here,
Fig. 23
The 3D and contour surfaces of Eq. (49) when the values are

Fig. 24
The 2D graph of Eq. (49) when the values are

Fig. 25
The 3D and contour surfaces of Eq. (51) when the values are

Fig. 26
The 2D graph of Eq. (51) when the values are

Fig. 27
The 3D and contour surfaces of Eq. (53) when the values are

Fig. 28
The 2D graph of Eq. (53) when the values are

In this paper, we have successfully applied BSEFM to some powerful non-linear models, such as the integro-partial differential equation and fifth-order integrable model. We have reported some strain conditions for the validity of the obtained results. Moreover, the travelling wave solutions, such as Eqs. (16, 18, 20, 29), obtained by using BSEFM are the new exponential function solutions for Eq. (1), compared with the paper previously found in literature [28]. Using powerful computational package programs, we observe that all solutions verify Eqs. (1, 2). Under specific values of parameters, we revise our results into existing solutions. Moreover, we have found many other entirely new analytical and complex travelling wave solutions for governing models. As far as we know, BSEFM has not been applied to Eq. (1) earlier. The projected method in this paper may be used to seek more travelling wave solutions of non-linear evolution equations for some applications, such as easy calculations, writing programs for obtaining variables, and so on.
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