Solving Nonlinear Partial Differential Equations of Special Kinds of 3rd Order Using Balance Method and Its Models

. Most nonlinear partial diferential equations have many applications in the physical world. Finding solutions to nonlinear partial diferential equations is not easily solvable and hence diferent modifed techniques are applied to get solutions to such nonlinear partial diferential equations. Among them, we considered the modifed Korteweg–de Vries third order using the balance method and constructing its models using certain parameters. Te method is successfully implemented in solving the stated equations. We obtained kind one and two soliton solutions and their graphical models are shown using mathematical software-12. Te obtained results lead to shallow wave models. A few illustrative examples were presented to demonstrate the applicability of the models. Furthermore, physical and geometrical interpretations are considered for diferent parameters to investigate the nature of soliton solutions to their models. Finally, the proposed method is a standard, efective, and easily computable method for solving the modifed Korteweg–de Vries equations and determining its perspective models.


Introduction
It is signifcantly important in nonlinear phenomena to search for exact solutions to nonlinear partial diferential equations (NLPDEs). Exact solutions play a vital role in understanding various qualitative and quantitative features of nonlinear phenomena. Tere are diverse classes of interesting exact solutions, traveling wave solutions, and soliton solutions, but it often needs specifc mathematical techniques to construct exact solutions due to the nonlinearity present in their dynamical nature [1][2][3]. Te NLPDEs are widely used as models to depict many important complex physical phenomena in a variety of felds of science and engineering. Some nonlinear partial diferential equations can be written as follows: Burger's equation is as follows: Te modifed Korteweg-de Vries (KdV) equation is as follows: Te Kadomtsev-Petriashvili (KP) equation is as follows: Te NLPDEs are fundamentally important because lots of mathematical physics models are often described by such wave phenomena and the investigation of traveling wave solutions is becoming more and more attractive in nonlinear sciences nowadays. However, NLPDEs are very difcult to solve explicitly, specifcally, or in detail. As a result, many powerful methods have been proposed and developed for fnding analytical solutions to nonlinear problems.
Some of the researchers used to solve NLPDEs are the simple equation method [4], the tanh-function and balance methods [5], the inverse scattering transform method [6], Backland and transformation [7], and other mathematical procedures such as combinations of the equation, transformation procedure, bilinear method, integration, and so on. Of these methods, the balance method was based on the higher order of partial derivative terms, the highest nonlinear terms, and parameters of the intended technique to solve the proposed problem of the modifed KdV equation.
Te Kadomtsev-Petviashvili (KP) equation is a partial diferential equation that describes nonlinear wave motion [8], which is usually written as δ � ± 1 that can be applied to mathematical physics as a way to model water waves of long wavelengths. It is a twodimensional generalization of the one-dimensional KdV (2). Like the KdV equation, the KP equation is completely integrable.
KdV were among the scholars who derived the KdV equation and one of the most famous nonlinear PDEs that arise in a great number of physical situations. It was derived from fuid mechanics to describe shallow water waves in a rectangular channel.
Te positive parameter β refers to a dispersive efect. Te generalized modifed KdV equation is given by the following equation: x ∈ R, where p is a positive parameter. Formulated in the moving frame x � ξ − ct, the generalized modifed KdV equation reads as follows: where c denotes the wave speed. Te inverse scattering transform method is a method introduced that yields a solution to the initial value problem for an NLPDE with the help of the solutions to the direct and inverse scattering problems for an associated linear ordinary diferential equation. Te balance method (BM) is a powerful method for fnding exact or approximate solutions to given NLPDEs which was presented by Wang and other scholars in recent years and was improved by Fan with others to make it more straight forward and simple [9]. In addition to this, as El-Wakil and others [10], used the balance method and auto-Backlund transformation. We used balance method for solving nonlinear partial diferential equations, the modifed Korteweg-de Vries equation. It is well known the KdV equations describe the unidirectional propagation of shallow water waves and a number of generalizations.
Let us consider a general nonlinear PDE, say, in two variables, where P is a polynomial function of its arguments and the subscripts denote the partial derivatives. Te balance method consists of the following steps: Step 1. Suppose that the solution of (8) is in the form of the following equation: where u � u(x, t), ω � ω(x, t), ((ln ω)) k,j � z k+j (ln ω (x, t))/zx k zt j , and a k,j (k � 0, 1, 2, . . . .p, j � 0, 1, 2, . . . .., d) balance coefcients are constants to be determined. By balancing the highest nonlinear terms and the higher order partial derivative terms in this expression; p, d, and q can be determined.
Step 3. Solving the set of DAEs, ω and a kj (k � 0, 1, 2, . . . .p, j � 0, 1, 2, . . . .., d) can be determined. By substituting ω, p, d, q, and a kj into (9), the exact solutions of (8) can be obtained. Nonlinear type partial diferential equations can be solved by many diferent methods such as the Hirota method [11,12]. Inverse scattering transform method and similarity reductions method [13]. Among those methods, the modifed Korteweg-de Vries equation was solved by a few of them with their limitations as it is. Te limitation of complexities to fnd the solutions are one core of the problem. Tis study aimed to fnd an alternative solution to the modifed KdV equation which simplifes the complexity of solving NLPDEs showing the 1 and 2− soliton solutions of the modifed KdV equation, by applying the balance method.
A long wave characterizes geophysical fuid dynamics in shallow waters and deep oceans [14,15]. From another perspective, the KdV equation appeared for the frst time in 1895 as a one-dimensional evolution equation describing the waves of an along surface gravity propagation in a water shallow canal [16]. It also appeared in a numeral of diverse physical phenomena such as hydromagnetic collision-free waves, ion acoustic waves, stratifed waves interior, lattice dynamics, and physics of plasma [17]. By applying the diferential transform technique, the approximate result of coupled KdV has been studied in [5]. One of the most attractive and surprising wave phenomena is the creation of solitary waves or solitons. An adequate theory for solitary waves was developed, in the form of a modifed wave equation known as KdV [18][19][20][21]. Te modifed KdV equation has defned a wide variety of physical phenomena used to model the interaction and evolution of nonlinear waves [22][23][24]. Korteweg and de Vries pursued the work done by Rayleigh and including the efect of surface tension leading to the now famous KdV equation given as follows: zη/zτ + 3/2 ��� g/hz/zζ(1/2η 2 + 2/3αη + 1/3δz 2 η/zζ 2 ) � 0, where η is the surface elevation, of the wave, above the equilibrium level h, α is a constant related to the uniform motion of the liquid, with the unit of length, g is the gravitational acceleration and δ � 1/3h 3 − Th/ρg, with T is surface tension.
Te permanent profle of a soliton solution of the KdV equation results from the equilibrium between two efects; the nonlinearity which is proportional to uu x or ηη ζ and Dispersion which is proportional to u xxx or η ζζζ . Te nonlinearity tends to comprise, to constitute the wave and dispersions spread it out. Te dispersive term is proportional to h 3 , it decreases whereas, the nonlinearity term, is pro- , increases leading to a large wave. Te balance method has been developed for the analytical solutions of nonlinear partial diferential equations. Compared to other methods, the balance method gives high accuracy and has a wide range of applications in many mathematical problems and many physical phenomena. In this paper, we used the balance method in combination with the Hirota bilinear equation method to solve the modifed form of the KdV equation considered using the balance method and its respective models.

Preliminaries
2.1.1. Bilinear Operators. Bilinear diferential operator D mapping a pair of functions D(f.g). Unlike usual linear diferential operators like (z/zx) n , which maps a single function f into a single function z n f/zx n . Tat is D: where f is a function of x and t, whereas g is a function of x ′ and t ′ ; for which f ≠ g.

Fractional Derivative.
A fractional derivative is a derivative of any order, real or complex in applied mathematics or mathematical analysis.

Te ELzaki Transform.
Te Elzaki transform is a function of the form h(x, t) with respect to t where s is the complex number, t > 0.
Putting (22) into (5), we have the following results, where, a 20 � 12β and a 10 � 0. 8 12 Simplifying (25) and integrating once with respect to x, we get z zx Equations (20) and (26) are identical with where C(t) is an arbitrary function of t, and a 00 is an arbitrary constant. Especially, taking C(t) as zero in equation (27), we get the bilinear equation of (5) as follows: Equation (28) can be written concisely in terms of D -operator as where D m Applying Hirota's method, the bilinear equation of (5) can be written as Equation (30) is obtained by setting a 00 � 0 in equation (29). Obviously, equation (30) is a special case of equation (29). Terefore, a more general bilinear equation of the modifed KdV equation is obtained by using Hirota's method.
Let us consider the modifed KdV equation u t + 6uu x + u xxx � 0 by using Hirota's method.
which gives the following new equation: w xt + 6w x w xx + w xxxx � 0. Now integrating both sides with respect to x, we have; where C � g(t) is constant of integration. Likely we can get, without loss of generality, the integration "constant" with International Journal of Diferential Equations respect to x, g(t) can be observed by a redefnition of w that does not change the KdV feld, , dt ′ is the time derivative. Using the new w, we have Now considering the 1-soliton solution of the modifed KdV equation, let where v is the velocity of the wave propagating. It can be written as u � w x , with In fact, we can integrate the right-hand side of equation (36) once again, using tanh y � d/d y log cosh y. Terefore, equation (34) can be written as follows: (39) which is the 1-soliton solution of the modifed KdV equation, that we use in the following: Considering equations (31) and (33), the KdV equation in a bilinear form, and rewriting it in a quadratic form, we have the following equation. Inspired by equation (34), let us substitute where f � 1 + exp (2μ)(x− x 0 − 4μ 2 t) . Substituting in equation (33), we get Equation (43) is quadratic in f. Tus, equation (33) for w becomes Multiplying by f 2 throughout, we have Tis is the bilinear form of the modifed KdV equation. Despite the fact that some nontrivial cancellations took place it looks more complicated than the original problem, however, its special form makes it possible to solve using bilinear operators.
To express equation (45), we use the following expressions: Again this looks a bit like the diferentiation of a product. But, we have D 2 In fact, the right-hand side is meaningless, because D x (f.f) is a single function, but the outer D x needs to act on a pair of functions.
Again it looks just like a diferentiated product of f and g but with signifcant changes on the odd terms.
Here, we have to notice that (49) is like z 4 x (fg), but we have an alternative signs; So, the KdV equation is in a quadratic form in equation (45) and can be recast as (D t D x + D 4 x )(f.f) � 0 is a bilinear form of the KdV equation. (

1) Solutions of the Bilinear KdV Equation.
For example (1) and considering u � 2z 2 x log f, we have two cases.

Proof. By induction.
Let if θ i � a i x + b i t + c i , (i � 1, 2). Ten, Ten, and similarly for D t operator. Finally, the equation is obviously true for n � 1, m � 0 and n � 0, m � 1 so by induction must be true for all n and m. In particular, we fnd Unless m � n � 0 D m t D n x e θ .1 � b m a n e θ D m t D n x 1.e θ � (− 1) m+n b m a n 1, e θ . (56) Ten, the bilinear form of the KdV equation Since e θ ≠ 0, we have two opportunities. Tese two cases or opportunities are as follows: (1) a � 0. If a � 0, then f is independent of x. So u � 0, (which is a boundary condition).
(2) b � − a 3 . If b � − a 3 , then f � 1 + e ax− a 3 t+c and hence u � 2z 2 x log f, f � 1 + e ax− a 3 t+c , which is u � 2z 2 Tis is the one-soliton solution with v � a 2 . Figure 1 shows the disturbance of the water surface having kink-shaped traveling waves with amplitude and antibell soliton. Te follow of this wave is to the front. Figure 2 shows the propagation of waves having kink shaped and like some periodic soliton. And follow the wave from back to front. Figure 3 shows surface waves of long wave length having kink-shaped and bright periodic soliton. And the wave is from left to right. Figure 4 surface waves of long wave length having kink shaped and bright periodic soliton. And the wave is from left to right. Tis is the other form of Figure 3, but it is diferent by the value of x, that is why it is broken from the above.

14
International Journal of Diferential Equations With f 0 � 1 in which the series will terminate at some power of ε to give exact results rather than an infnite series in ε. So, we can take ε fnite ε � 1.
To begin with, we put f(x, t) � 1 + ∞ n�0 ε n f n (x, t) and collect powers of ε. We start with 1 as the ε 0 term as it appeared in the one-soliton solution. Defning We want to fnd solutions such that and gathering terms of the same degree n � n 1 + n 2 inε and also expanding as for the one soliton case we have for all n � 1, 2, . . .. Equation (63) can be written as (Expression involving in f 1 , f 2 , . . . ., f n− 1 ). Now we can solve (63) which is equivalent to (64) recursively to determine the coefcients f n . Tat is we begin with f 1 and solve iteratively to get f 2,3,.. hoping that it terminates at some point. To do this we need the following:

Theorem .
For any given f, we have; Proof. By considering to interchange primed and unprimed values as a simple label; i.e.,

International Journal of Diferential Equations
where f is any function of x and t Tus, (70) became For n � 1, it reduced to Beginning with f 1 we may iterate(repeat) to fnd all the f n and note that at any order we are now solving linear partial diferential equations. A simple particular solution to the last equation would be Using (63) for f 2 , we have z/zx(z/zt+ z 3 /zx 3 )f 2 � − 1/2B(f 1 .f 1 ) � 0, where this result follows (55). Here is the fact, with f 1 , the expansion of (59), terminated at order N or (the f 3...∞ all vanish trivially). Te series has terminated with f � 1 + εf 1 .
(74) Figure 5 shows follow of the presence of a solitary wave to the front, having two kinks shaped with long amplitude, dark of high speed. Figure 6 shows the solitary wave to the front, having two kinks shaped, a valley of the wave, and dark high speed. Figure 7 shows the solitary wave from left to right, having two kink-shaped solitons, a valley-shaped wave, long amplitudes, and dark high speed. Figure 8 shows follow of the solitary wave to the front, having two kink--shaped soliton profles, periodic, and dark and bell slow speed.