Open access peer-reviewed chapter

Invariants of Generalized Fifth Order Non-Linear Partial Differential Equation

By Sachin Kumar

Submitted: November 30th 2017Reviewed: May 7th 2018Published: July 18th 2018

DOI: 10.5772/intechopen.78362

Abstract

The fifth order non-linear partial differential equation in generalized form is analyzed for Lie symmetries. The classical Lie group method is performed to derive similarity variables of this equation and the ordinary differential equations (ODEs) are deduced. These ordinary differential equations are further studied and some exact solutions are obtained.

Keywords

• generalized fifth order non-linear partial differential equation
• lie symmetries
• exact solutions

1. Introduction

The theories of modern physics mainly include a mathematical structure, defined by a certain set of differential equations and extended by a set of rules for translating the mathematical results into meaningful statements about the physical work. Theories of non-linear science have been widely developed over the past century. In particular, non-linear systems have fascinated much interest among mathematicians and physicists. A lot of study has been conducted in the area of non-linear partial differential equations (NLPDEs) that arise in various areas of applied mathematics, mathematical physics, and many other areas. Apart from their theoretical importance, they have sensational applications to various physical systems such as hydrodynamics, non-linear optics, continuum mechanics, plasma physics and so on. A large variety of physical, chemical, and biological phenomena is governed by nonlinear partial differential equations (NLPDEs). A number of methods has been introduced for finding solutions of these equations such as Homotopy method [1], G/Gexpansion method [2, 3], variational iteration method [4], sub-equation method [5], exp. function method [6], and Lie symmetry method [7, 8, 9, 10]. Although solutions of such equations can be obtained easily by numerical computation. However, in order to obtain good understanding of the physical phenomena described by NLPDEs it is important to study the exact solutions of the NLPDEs. Exact solutions of mathematical equations play an major role in the proper understanding of qualitative features of many phenomena and processes in different areas of natural and applied sciences. Exact solutions of non-linear differential equations graphically demonstrate and allow unraveling the mechanisms of many complex non-linear phenomena. However, finding exact solutions of NLPDEs representing some physical phenomena is a tough task. However, because of importance of exact solutions for describing physical phenomena, many powerful methods have been introduced for finding solitons and other type of exact solutions of NLPDEs [2, 11, 12, 13]. Comparing to other approximate and numerical methods, which provides approximate solutions [14, 15, 16], the Lie group method provides the exact and analytic solutions of the differential structure (Figures 13).

Lie group method is one of the most effective methods for finding exact solutions of NLPDEs [17, 18]. This method was basically initiated by Norwegian mathematician Sophus Lie [19]. He developed the theory of “Continuous Groups” known as Lie groups. This method is orderly used in various fields of non-linear science. Shopus Lie was the first who arranged differential equations in terms of their symmetry groups, thereby analyzing the set of equations, which could be integrated or reduced to lower order equations by group theoretic algorithms. The Lie group analysis is a mathematical theory that synthesizes symmetry of differential equations. In this method, the differential structure is studied for their invariance by acting one or several parameter continuous group of transformations on the space of dependent and independent variables. We observe a plenty books and research article about Lie group method [17, 18, 20, 21, 22].

Wazwaz [23] introduced a fifth order non-linear evolution equation as follows:

utttutxxxx4uxutxx4uxuxtx=0.E1

In this chapter, he obtained multiple soliton solutions of this equation.

We will consider the generalized fifth order non-linear evolution equation of the form:

utttutxxxxαuxutxxβuxuxtx=0,E2

where α,βare parameters.

In this chapter, we will study the Eq. (2) by the Lie classical method. Firstly, Lie classical method will be used to obtain symmetries of generalized fifth order non-linear evolution Eq. (2). Symmetries will be used to reduce the Eq. (2) to ordinary differential equations (ODEs) and corresponding exact solutions of the generalized fifth order non-linear evolution Eq. (2) will be obtained.

2. Symmetry analysis

Lie classical method of infinitesimal transformation groups reduces the number of independent variables in partial differential equations (PDEs) and reduces the order of ODEs. Lie’s method has been widely used in equations of mathematical physics and many other fields [11, 24]. In this chapter, we will perform Lie symmetry analysis [17, 18, 19, 24] for the generalized fifth order non-linear evolution Eq. (2).

Let the group of infinitesimal transformations be defined as:

t=t+ετxtu+Oϵ2x=x+εξxtu+Oϵ2u=u+εηxtu+Oϵ2,E3

which leaves the Eq. (2) invariant. The infinitesimal transformations (3) are such that if uis solution of Eq. (2), then uis also a solution.

Herein, on invoking the invariance criterion as mentioned in [18], the following relation is deduced:

ηtttηtxxxxαηxxxut+ηtuxxx2α+βηxxuxt+ηxtuxxα+βηxuxxt+ηxxtux=0,E4

where ηx,ηt,ηxt,ηxx,ηxxx,ηttt,ηtxxxxand ηxxtare extended (prolonged) infinitesimals acting on an enlarged space corresponding to ux,ut,uxt,uxx,uxxx,uttt,utxxxxand uxxt, respectively, given by:

ηx=DxηuxDxξutDxτ,ηt=DtηuxDtξutDtτ,ηxx=DxηxuxxDxξuxtDxτ,ηxt=DtηxuxxDtξuxtDtτ,ηxxx=DxηxxuxxxDxξuxxtDxτ,ηttt=DtηttuxttDtξutttDtτ,ηxxxxt=DtηxxxxuxxxxxDtξuxxxxtDtτ,E5

where Dxand Dtare total derivative operators with respect to xand trespectively given as:

Dx=x+uxu+uxxux+,Dt=t+utu+uttut+.

Now, after computing (5) we get:

ηx= ηx+ηuξxuxτxutξuux2τuuxut,ηt= ηt+ηuτtutξtuxτuut2ξuuxut,ηxx= ηxx+ux2ηxuξxxutτxx+x2ηuu2ξxu+uxxηu2ξx2uxtτx2utuxτxuux3ξuuux2utτuu2uxuxtτuuxxutτu3uxuxxξu,ηxt= ηxt+uxηtuξxt+ηtηxuτxtux2ξtuut2τxuuxxξtuxtτt+ξxηuuttτx+uxutηuuξxuuxuxtξuuxtuxξuuxtutτuuttuxτuux2utξuuut2uxτuuutuxτtuutuxtτuuxxutξu,ηxxx= ηxxx+x3ηxxuξxxxutτxxx+uxx3ηxu3ξxx3uxtτxx3uxutτxxu3uxxtτx+ux23ηxuu3ξxxu+uxuxx3ηuu9ξxu+ux3ηuuu3ξxuu+uxxxηu3ξx2uxuxtτxuux4ξuuu6ux2uxxξuu3uxx2ξu4uxuxxxξu3utux2τxuu3utuxxτxu4uxutxτxuux3utτuuu3uxutuxxτuu3ux2uxtτuu3utxuxxτu3uxuxxtτuutuxxxτu,ηttt= ηtttuxξttt+ut3ηttuτttt+ut23ηtuu3τttu+ut3ηuuu3τtuuut4τuuuutt23τu3uxutξttu3ux2utξtuu3uxuttξut6uxtutξtu3uxuttξtu+uxt4ηxxxu3ξttξxxxx+3uttηtuτttutttτt2uxttξ+2τxxxt+uxxxηu2τtuxttξtuxttutξuut3uxξuuu3ut2uxtξuu3uxtuttξu2uxttutξuututttτu+ξuututt9τtu3ηuu6ut2uttτuu3ututttξu,ηxxxxt= ηxxxxt+ux4ηxxxtuξxxxxt+utτxxxxtux2τxxxxu+ux36ηxxtuu4ξxxxtu+ux34ηxtuuu6ξxxtuu+ux4ηtuuuu4ξxtuuuuux5ξtuuuu4uxtτxxxt+uxx6ηxxtu4ξxxxt+2uxxxt2ηxu3ξxx2τxt+uxxx4ηxtu6ξxxt+uxxxxηtu4ξxt+uxxt6ηxxu6τxxt4ξxxx+uxxxuxt4ηuu16ξxu+6uxxuxxtηuu4ξxu+uxxxxutηuu4ξxu+4uxuxxxtηuu4ξxu6ux2uttτxxuu24uxutuxtτxxuu6uxxut2τxxuu4uxuttτxxxu8uxtutτxxxu10ux2uxxxtξuu5uxutuxxxxξuu30uxxuxxtuxξuu20uxxxuxtuxξuu15uxx2uxtξuu5uxuxxxxtξu10uxxuxxxtξu5uxtuxxxxξu10uxxtuxxxξu5uxuxxxxξtu10uxxuxxxξtu6ux2uxxtτtuu12uxxuxtuxτtuu4uxutuxxxτtuu3uxx2utτtuu12uxuxxtτxtu4utuxxxτxtuuxxuxt12τxtu+18ξxxu18uxuxxtξxxu6utuxxxξxxu4uxxxuxttτu6uxxt2τuuttuxxxxτu8uxtuxxxtτu6uxxuxxttτuutuxxxxtτu4uxuxxxttτu12ux2uxttτxuu24uxutuxxtτxuu12uxuxxuttτxuu24uxt2uxτxuu4ut2uxxxτxuu24utuxxuxtτxuu4ux3uxttτuuu12utux2uxxtτuuu

6uxxuttux2τuuu12uxt2ux2τuuu4ut2uxxxτuuu24uxutuxxuxtτuuu3ut2uxx2τuuu12uxuxxttτxu8utuxxxtτxu12uxxuxttτxu24uxxtuxtτxu4uxxxuttτxuuttτxxxx+utηxxxxu10ux3uxxξtuuu24ux2uxxξxtuu+uxuxx12ηxtuu18ξxxtu6ux2ut2τxxuuuux2ut4ξxxxuu6τxxtuu+ux3ut4ηxuuuu6ξxxuuu4τxtuuu+ux4utηuuuuu4ξxuuuuτtuuuu+6ux2utηxxuuu4ux3ut2τxuuuu10ux3uxxutξuuuu5ux4uxtξuuuu+ux2uxxut6ηuuuu24ξxuuu6τtuuu+ux3uxt4ηuuuu16ξxuuu4τtuuu4ux3uttτxuuu24ux2utuxtτxuuu12uxxut2τxuuu10ux2uxxxξxuu15uxuxx2ξxuu+6ux2uxt2ηxuuu2τxtuu3ξxxuu+6uxutuxx2ηxuuu2τxtuu3ξxxuu10ux3uxxtξuuu10utux2uxxxξuuu30uxxuxtuux2ξuuu15uxx2uxutξuuuux4ut2τuuuuu8ux3utuxtτuuuu+uxut4ηxxxuuξxxxxu4τxxxtuux5utξuuuuuux4uxtτuuuu6ux2ut2uxxτuuuu4uxut2τxxxuu+6ux2uttηtuuu24ux2uxxtξxuu+12uxxuxtuxηuuu4ξxuu16uxutuxxxξxuu12uxx2utξxuu+uxuxxt12ηxuu+uxxxut4ηxuu+uxxuxtηxuu6ux2uxxxttτuu8uxutuxxxtτuu12uxuxxuxttτuu24uxuxtuxxtτuu4uxuttuxxxτuuut2uxxxxτuu12uxxuxxtutτuu8uxxxuxtutτuu3uxx2uttτuu12uxxuxt2τuu+6ux2uxxtηuuu+4uxutuxxxηuuu+3uxx2utηuuu12uxuxtτxxtu6uxxutτxxtu+uxuxxxηtuu16ξxtu+uxx2ηtuu12ξxtu8uxuxtξxxxu4uxxutξxxxu+12uxuxtηxxuu+6uxxutηxxuu12uxxtutτxxu12uxuxttτxxu6uxxuttτxxu12uxt2τxxu4uxxxuxtτxu6uxxuxxtτtuutuxxxxτtu4uxuxxxtτtu6uxxttτxx4uxxxttuxxxxxξt+uxxxxtηuτt4ξxuxxxxtutτuuxxxxxutξu.τxE6

The Lie classical method for determining the similarity variables of (2) is mainly consists of finding the infinitesimals τ,ξ,and η,which are functions of x,t,u.After substituting the values of ηx,ηt,ηxt,ηxx,ηxxx,ηttt,ηtxxxxand ηxxtfrom (5) to (4) and equating the coefficients of different differentials of uto zero, we get a number of PDEs in τ,ξ, and η, that need to be satisfied. Solving these system of PDEs, we obtain the infinitesimals τ,ξ, and ηas follows:

τ=C1+tC4ξ=C2+x2C4η=C3u2C4,E7

where C1,C2,C3, and C4are arbitrary constants.

Corresponding vector fields can be written as:

V1=t,V2=x,V3=u,V4=x2x+ttu2u.E8

3. Symmetry reductions and invariant solutions

To obtain the symmetry reductions of Eq. (2), we have to solve the characteristic equation:

dxξ=dtτ=duη,E9

where ξ,τand ηare given by Eq. (7).

To solve Eq. (9), following cases will be considered: (i) V1+μV2+λV3and (ii) V4,where μ,λare arbitrary constants.

Case (i) V1+μV2+λV3

On solving Eqs. (9) we have,

ρ=xμtu=λt+Fρ,E10

where ρis new independent variables and Fρis new dependent variable. Substituting (10) into Eq. (2), we obtain the reduced ODE which reads:

μ2α+βFμ3αλF′″+μ2α+βF2+F′″″=0,E11

where primes denotes derivative with respect to ρ.

Let assume the solution of ODE (11) in following form:

F=a0+a1ρ+a2ρ,E12

where a0,a1, and a2needs to be determined. Substituting (12) into ODE (11) and equating coefficients of the different powers of ρequal to zero, we obtain:

a0=arbitrarya1=μ3+αλμ2α+βa2=122α+β.E13

Corresponding solution of ODE (11) can be written as:

F=a0+μ3+αλμ2α+βρ+122α+βρ,E14

where β2α.

Corresponding solution of main Eq. (2) is given by:

uxt=λt+a0+μ3+αλμ2α+βxμt+122α+βxμt,E15

with β2α.

Some more solutions of ODE (11) are given by:

iFρ=b3±6μαλ+μ3μ2α+βtanhb1μαλ+μ3ρ2μwithβ2α,iiFρ=b4+b5coshb2±μαλ+μ3ρμwithβ=2α,iiiFρ=b3+b4cothb1+12μαλ+μ3ρμwithβ=2b4b4α+3μαλ+μ3μ,E16

where b1,b2,b3,b4and b5are arbitrary constants.

Corresponding solutions of main Eq. (2) are given by:

iuxt=λt+b3±6μαλ+μ3μ2α+βtanhb1μαλ+μ3xμt2μwithβ2α,E17
iiuxt=λt+b4+b5coshb2±μαλ+μ3xμtμwithβ=2α,E18
iiiuxt=λt+b3+b4cothb1+12μαλ+μ3xμtμwith β=2b4b4α+3μαλ+μ3μ,E19

where b1,b2,b3,b4, and b5are arbitrary constants.

Case (ii) V4

On solving Eq. (9) for vector field V4, we have:

ϕ=tx2u=Gϕx,E20

where ϕis new independent variables and Gϕis new dependent variable. Substituting (20) into Eq. (2), we obtain the reduced ODE which reads

138α+54βϕG2+8beta+2αϕ3G′′′+30α+18βG+148α+68βϕ2G′′360G+8β+2αϕ3G′′2+4α+βϕ2GG′′′+26α+22βϕGG′′16ϕ4G′′′′′+11020ϕ2G′′′1320ϕG′′240ϕ3G′′′′=0,E21

where primes denotes derivatives with respect to ϕ.

Let assume the solution of ODE (21) in following form:

G=b2ϕ2+b1ϕ+a0+a1ϕ+a2ϕ2,E22

where b1,b2,a0,a1and a2needs to be determined.

Substituting (22) into ODE (21) and equating coefficients of the different powers of ϕequal to zero, we obtain:

ia0=arbitrary,a1=a2=b1=0,b2=15α+3βiia0=a1=a2=0,b1=arbitrary,b2=15α+3βE23

Corresponding solution of ODE (21) can be written as:

iG=15α+3βϕ2+a0,iiG=15α+3βϕ2+b1ϕ,E24

where b1is arbitrary constant.

Corresponding solution of main Eq. (2) can be written as:

iuxt=1xx45α+3βt2+a0,iiuxt=x35α+3βt2+b1xt,E25

where b1is arbitrary constant.

4. Conclusion

In this chapter, we derived the symmetry variables and symmetry transformations of the generalized fifth order non-linear partial differential equation. We applied Lie symmetry analysis for investigating considered nonlinear partial differential equation and using similarity variables, given equation is reduced into ordinary differential equations. We derived explicit exact solutions of considered partial differential equation corresponding to each ordinary differential equation obtained by reduction.

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Sachin Kumar (July 18th 2018). Invariants of Generalized Fifth Order Non-Linear Partial Differential Equation, Nonlinear Systems - Modeling, Estimation, and Stability, Mahmut Reyhanoglu, IntechOpen, DOI: 10.5772/intechopen.78362. Available from:

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