Available online at www.ispacs.com/cna
Volume 2013, Year 2013 Article ID cna-00127, 6 Pages
doi:10.5899/2013/cna-00127
Research Article
Approximations of the nonlinear Painlev
e transcendents
´
Najeeb Alam Khan1∗, Muhammad Jamil2, Nadeem Alam Khan1
(1)Department of Mathematical Sciences, University of Karachi, Karachi 75270, Pakistan (2)Department of Mathematics, NED University of Engineering & Technology, Karachi-75270, Pakistan
Copyright 2013 c⃝Najeeb Alam Khan, Muhammad Jamil and Nadeem Alam Khan. 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.
Abstract
In this communication, we apply the new homotopy perturbation method (NHPM) to solve the first and second Painlev´e transcendent. The NHPM for solving differential equations based on two component procedure and poly-nomial initial condition. The numerical results are compared with existing analytic results obtained by Adomian decomposition method (ADM), homotopy perturbation method (HPM), analytic continuation method and Legendre Tau method. The outcomes confirm that the scheme yields accurate and excellent results even when two components are used.
Keywords:Painlev´e transcendent; new homotopy perturbation method (NHPM); differential equation; nonlinear equations.
1 Introduction
The Painlev´e equations [1, 2, 3] were discovered by P. Painlev´e, B. Gambier and their colleagues during studying a nonlinear second order differential equation
d2u
dx2 =R(x,u,u ′),
whereR(x,u,u′) is a function rational inuandu′and analytic inx. It was shown by P. Painlev´e (1900) and B. Gambier (1910) that all equations of this type whose solutions do not have movable critical points (but are allowed to have fixed singular points and movable pole) can be reduced to 50 classes of equations. Moreover, 44 classes out of them are in-tegrable by quadrature or admit reduction of order. The remaining 6 equations are irreducible; these are known as the Painlev´e equations or Painlev´e transcendent, and their solutions are known as the Painlev´e transcendental functions. It is significant that the Painlev´e equations often arise in mathematical physics. Some connections are given as follows:
The Boussineseq equation in canonical form
∂2w ∂t2 +
∂ ∂x
(
w∂w
∂x
)
+∂ 4w
∂x4 =0. (1.1)
This equation arises in several physical applications: propagation of long waves in shallow water, one dimensional nonlinear lattice-waves, vibrations in a nonlinear string, and ion sound waves in plasma. The connection of Boussi-neseq equation with first Painlev´e equation is due to Clarkson and Kruskal [4]. Assuming that a solution of Eq. (1.1) may be written in the form
w(x,t) = (a0+a1t)2U(z)−
(
a1x+b1
a1t+b0
)2 ,
wherez=x(a1t+a0) +b1t+b0, anda0,a1,b0 and b1are arbitrary constants. The functionU(z)is determined by second order differential equation
U′′(z) +1 2U
2=c 1z+c2.
Ifc1̸=0 the equation is reduced to the first Painlev´e equation.
The Korteweg-de Vries equation in canonical form
∂w
∂t +
∂3w ∂t3 −6w
∂w
∂x =0. (1.2)
It is used in many sections of nonlinear mechanics and theoretical physics for describing one-dimensional nonlinear dispersive non-dissipative waves. Also, the mathematical modeling of moderate-amplitude shallow water surface waves is based on this equation. Assuming that a solution of Eq. (1.2) may be written in the form
w(x,t) = [3(t−t0)]−2/3U(z),
wherez= [3(t−t0)]−1/3(x−x0), the functionU(z)is determined by third order ordinary differential equation
U′′′(z)−zU′−2U−6U′U=0. (1.3)
A solution of Eq. (1.3) can be represented as
U(z) =g′(z) +g2(z),
where the functiong(z)is any solution of the second Painlev´e equation
g′′(z)−2g3−zg=A,
whereAis any arbitrary constant. In this paper, NHPM is applied to solve the nonlinear first and second Painlev´e equations with Cauchy-Dirchelet conditiony|x=0 =A, y′|x=0 =B. Unlike the ADM [5, 6], the NHPM is free from the need to use Adomian polynomials. In this method we do not need the Lagrange multiplier, correction functional, stationary conditions, and calculating integrals, which eliminate the complications that exist in the VIM [7]. In contrast to the HPM [8, 9, 10, 11, 12], in this method, it is not required to solve the functional equations in each iteration. HPM is an especial case of HAM (Homotopy Analysis Method) and the comparison of HAM and HPM can be find in [13, 14, 15].
2 Analysis of new homotopy perturbation method
Let us consider the nonlinear differential equation
A(u) =f(z), z∈Ω, (2.4)
whereA is operator, f is a known function anduis a sought function. Assume that operatorA can be written as:
A(u) =L(u) +N (u),
whereL(u)is the linear operator andN(u)is the nonlinear operator. Hence, Eq. (2.4) can be rewritten as follows:
We define an operatorH(u)as:
H(v;p) = (1−p)(L(v)−L(u0)) +p(A(v)−f), (2.5)
wherep∈[0,1]is an embedding or homotopy parameter, andv(z;p):Ω×[0,1]→Ris an initial approximation of solution of the problem in Eq. (2.4). Eq. (2.5) can be written as:
H(v;p) =L(v)−L(u0) +pL(u0) +p(N (v)−f(z)) =0. (2.6)
Clearly, the operator equationsH(v,0) =0 and H(v,1) =0 are equivalent to the equations L(v)−L(u0) =0 andA(v)−f(z) =0, respectively. Thus, a monotonous change of parameterpfrom zero to one corresponds to a continuous change of the trivial problemL(v)−L(u0) =0 to the original problem. OperatorH(v;p)is called a homotopy map. Next, we assume that the solutionH(v;p)can be written as a power series in embedding parameter
p, as follows:
v=v0+pv1. (2.7)
Now let us write the Eq. (2.6) in the following form
L(v) =u0(z) +p(f−N (v)−u0(z)).
By applying the inverse operatorL−1to both sides of the Eq. (2.4), we have
v=L−1u0(z) +p(L−1f−L−1N (v)−L−1u0(z)). (2.8)
Suppose that the initial approximation of Eq. (2.4) has the form
u0(z) =
∞
∑
n=0
anPn(z), (2.9)
wherean, n=0,1,2,· · ·are unknown coefficients andPn(z), n=0,1,2,· · ·are specific functions on the problem. By substituting Eqs. (2.7) and (2.9) into the Eq. (2.8), we get
v0+pv1=L−1
(∞
∑
n=0
anPn(z) )
+p
(
L−1f−L−1N (v)−L−1
( ∞
∑
n=0
anPn(z) ))
.
Equating the coefficients of like powers ofp, we get following set of equations:
Coefficient ofp0:v0=L−1(∑∞n=0anPn(z)).
Coefficient ofp1:v1=L−1f−L−1(∑n∞=0vnpn)−L−1N (v0).
Now, we solve these equations in such a way thatv1(z) =0. Therefore, the approximate solution may be obtained as
yi(z) =Yi,0(z) =L−1 ( ∞
∑
n=0
ai,nPn(z) )
.
3 Approximations of the nonlinear Painleve transcendent´
3.1 Solution of the first Painleve transcendent´
The first Painlev´e equation has the form
d2u
dx2 =6u
2+x,
with Cauchy data
y|x=0 =A,y′|x=0 =B.
To obtain the solution of Eq. (3.10) by NHPM, we construct the following homotopy:
(1−p)
(
d2U(x)
dx2 −u0(x)
)
+p
(
d2U(x)
dx2 −u0(x)−
(
6U2(x) +x)
)
=0. (3.11)
Applying the inverse operatorL−1(•) =∫x
0
∫τ
0(•)dξdτto the both sides of the above Eq. (3.11), we obtain
U(x) =U(0) +xU′(0) +
∫ x
0
∫ τ
0 (
U0(ξ))dξdτ−
∫ x
0
∫ τ
0 (
U0(ξ)(
6U2(ξ) +ξ)
)dξdτ. (3.12)
Suppose the solution of Eq. (3.12) to have the following form
U(x) =U0(x) +pU1(x). (3.13)
Substituting Eq. (3.13) in Eq. (3.12) and equating the coefficients of like powers ofp, we get following set of equations
U0(x) =U(0) +xU′(0) +
∫ x
0
∫ τ
0
U0(ξ)dξdτ,
U1(x) =
∫ x
0
∫ τ
0 ((−
U0(ξ) +6U02(ξ) +ξ))dξdτ.
Assumingu0(x) =Σ30
n=0anPn, Pk=xk, , U0=u0and solving the above equation forU1(t)leads to the result
U1(x) =(3A2−a0 2
)
x2+
(
1−a0 6 +2AB
)
x3+
( Aa0 2 − a2 12+ B2 2 )
x4+· · ·.
VanishingU1(t)lets the coefficientsai, i=1,2,3,· · ·, the following values
a0=6A2,a1=1+12AB,a2=6(6A3+B3),a+3=2(A+30A2B),· · ·
Therefore, we obtain the solution of Eq. (3.10) as
u(x) =A+Bx+3A2x2+1
6(1+12AB)x 3+1
2(6A
3+B2)x4+ 1
10(A+30A
2B)x5+· · ·.
3.2 Solution of the second Painleve transcendent´
The second Painlev´e equation [16] has the form
d2u
dx2=2u
3(x) +xu(x) +λ, (3.14)
with Cauchy data
y|x=0=A, y′|x=0=B.
To obtain the solution of Eq. (3.14) by NHPM, we construct the following homotopy:
(1−p)
(d2u
dx2−u0(x)
)
+p
(d2u
dx2−(2U
3(x) +xU(x) +λ))=
0. (3.15)
Applying the inverse operator,L−1=∫x
0
∫τ
0(•)dξdτto the both sides of the above Eq. (3.15), we obtain
U(x) =U(0) +xU′(0) +
∫ x
0
∫ τ
0
(u0(ξ))dξdτ−p ∫ x
0
∫ τ
0
Suppose the solution of Eq. (3.16) to have the following form
U(x) =U(0) +pU1(x). (3.17)
Substituting Eq. (3.17) in the Eq. (3.16) and equating the coefficients of like powers of p, we get following set of equations
U0(x) =U(0) +xU′(0) +
∫ x
0
∫ τ
0
u0(ξ)dξdτ,
U1(x) =
∫ x
0
∫ τ
0
(−u0(ξ) + (2U03(ξ) +ξU0(ξ) +λ))dξdτ.
Assumingu0(t) =∑30n=0anPn,Pk=tk, U0=u0and solving the above equation forU1(t)leads to the result
U1(x) =(3A2−a0 2
)
x2+
(
1−a0 6 +2AB
)
x3+
(Aa
0 2 −
a2 12+
B2
2
)
x4+· · ·.
VanishingU1(t)lets the coefficientsai, i=1,2,3,· · ·, the following values
a0=6A2,a1=1+12AB,a2=6(6A3+B3),a+3=2(A+30A2B),· · ·
Therefore, we obtain the solution of the Eq. (3.10) as
u(x) =A+Bx+3A2x2+1
6(1+12AB)x 3+1
2(6A
3+B3)x4+ 1
10(A+30A
2B)x5+· · ·.
4 Concluding remarks
New homotopy perturbation solutions [17] have been presented for the first and second Painlev´e transcendents. The results show a very good agreement with the other methods. The major advantage of the NHPM is that it is simple, straightforward and systematic when compared with other techniques. To avoid the cumbersome of the computational work that usually arises from the traditional strategies; we present alternative numerical strategies that proved to be effective, reliable and easy to use. In HPM and ADM we reach to a set of recurrent differential equations, which must be solved consecutively to give the approximate solution of the problem. The computations associated with the solutions of first and second Painlev´e transcendent in this work were performed using MATHEMATICA 7.
Acknowledgements
The author Najeeb Alam Khan is highly thankful and grateful to the Dean of Faculty of Sciences, University of Karachi, Karachi-75270, Pakistan for facilitating this research work, and grateful to the anonymous referees for their comments which substantially improved the quality of this paper.
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