# Solving Nonlinear Boundary Value Problems Using He's Polynomials and Padé Approximants

Authors: Syed Tauseef Mohyud-Din; Ahmet Yildirim
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

We apply He's polynomials coupled with the diagonal Padé approximants for solving various singular and nonsingular boundary value problems which arise in engineering and applied sciences. The diagonal Padé approximants prove to be very useful for the understanding of physical behavior of the solution. Numerical results reveal the complete reliability of the proposed combination.


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## Body

## 1. Introduction

With the rapid development of nonlinear sciences, many analytical and numerical techniques have been developed by various scientists for solving singular and nonsingular initial and boundary value problems which arise in the mathematical modeling of diversified physical problems related to engineering and applied sciences. The application of these problems involves physics, astrophysics, experimental and mathematical physics, nuclear charge in heavy atoms, thermal behavior of a spherical cloud of gas, thermodynamics, population models, chemical kinetics, and fluid mechanics see [1–68] and the references therein. Several techniques [1–68] including decomposition, variational iteration, finite difference, polynomial spline, differential transform, exp-function and homotopy perturbation have been developed for solving such problems. Most of these methods have their inbuilt deficiencies coupled with the major drawback of huge computational work. He [19–24] developed the homotopy perturbation method (HPM) for solving linear, nonlinear, initial and boundary value problems. The homotopy perturbation method was formulated by merging the standard homotopy with perturbation. Recently, Ghorbani and Saberi-Nadjafi [15, 16] introduced He’s polynomials by splitting the nonlinear term and also proved that He’s polynomials are fully compatible with Adomian’s polynomials but are easier to calculate and are more user friendly. The basic motivation of this paper is to apply He’s polynomials coupled with the diagonal Padé approximants for solving singular and nonsingular boundary value problems. The Padé approximants are applied in order to make the work more concise and for the better understanding of the solution behavior. The use of Padé approximants shows real promise in solving boundary value problems in an infinite domain; see [42, 50, 56–59]. It is well known in the literature that polynomials are used to approximate the truncated power series. It was observed [42, 50, 56–59] that polynomials tend to exhibit oscillations that may give an approximation error bounds. Moreover, polynomials can never blow up in a finite plane and this makes the singularities not apparent. To overcome these difficulties, the obtained series is best manipulated by Padé approximants for numerical approximations. Using the power series, isolated from other concepts, is not always useful because the radius of convergence of the series may not contain the two boundaries. It is now well known that Padé approximants [42, 50, 56–59] have the advantage of manipulating the polynomial approximation into rational functions of polynomials. By this manipulation, we gain more information about the mathematical behavior of the solution. In addition, the power series are not useful for large values ofx. It is an established fact that power series in isolation are not useful to handle boundary value problems. This can be attributed to the possibility that the radius of convergence may not be sufficiently large to contain the boundaries of the domain. It is therefore essential to combine the series solution with the Padé approximants to provide an effective tool to handle boundary value problems on an infinite or semi-infinite domain. We apply this powerful combination of series solution and Padé approximants for solving a variety of boundary value problems. Precisely the proposed combination is applied on boundary layer problem, unsteady flow of gas through a porous medium, Thomas-Fermi equation, Flierl-Petviashivili (FP) equation, and Blasius problem. It is worth mentioning that Flierl-Petviashivili equation has singularity behavior at x=0 which is a difficult element in this type of equations. We transform the FP equation to a first-order initial value problem and He’s polynomials are applied to the reformulated first-order initial value problem which leads the solution in terms of transformed variable. The desired series solution is obtained by implementing the inverse transformation. The fact that the proposed algorithm solves nonlinear problems without using Adomian’s polynomials is a clear advantage of this technique over the decomposition method.


## 2. Homotopy Perturbation Method and He’s Polynomials

To explain the He’s homotopy perturbation method, we consider a general equation of the type(2.1)L(u)=0,
where L is any integral or differential operator. We define a convex homotopy H(u,p) by
(2.2)H(u,p)=(1-p)F(u)+pL(u),
where F(u) is a functional operator with known solutions v0, which can be obtained easily. It is clear that, for
(2.3)H(u,p)=0,
we have
(2.4)H(u,0)=F(u),H(u,1)=L(u).
This shows that H(u,p)continuously traces an implicitly defined curve from a starting point H(v0,0) to a solution functionH(f,1). The embedding parameter monotonically increases from zero to unit as the trivial problem F(u)=0, continuously deforms the original problem L(u)=0. The embedding parameter p∈(0,1] can be considered as an expanding parameter [15, 16, 19–24, 41–50, 60, 63–68]. The homotopy perturbation method uses the homotopy parameter p as an expanding parameter [19–24] to obtain
(2.5)u=∑i=0∞piui=u0+pu1+p2u2+p3u3+⋯,
if p→ 1, then (2.5) corresponds to (2.2) and becomes the approximate solution of the form
(2.6)f=limp→1u=∑i=0∞ui.
It is well known that series (2.6) is convergent for most of the cases and also the rate of convergence is dependent on L(u); see [19–24]. We assume that (3.2) has a unique solution. The comparisons of like powers of p give solutions of various orders. In sum, according to [15, 16], He’s HPM considers the nonlinear term N(u)as(2.7)N(u)=∑i=0∞piHi=H0+pH1+p2H2+⋯,
where Hn’s are the so-called He’s polynomials [15, 16], which can be calculated by using the formula(2.8)Hn(u0,…,un)=1n!∂n∂pn(N(∑i=0npiui))p=0,n=0,1,2,…
of various orders.


## 3. Padé Approximants

A Padé approximant is the ratio of two polynomials constructed from the coefficients of the Taylor series expansion of a functionu(x). The [L/M] Padé approximants to a function y(x) are given by [42, 50, 56–59](3.1)[LM]=PL(x)QM(x),
where PL(x) is polynomial of degree at most L and QM(x) is a polynomial of degree at most M. The formal power series
(3.2)y(x)=∑i=1∞aixi,(3.3)y(x)-PL(x)QM(x)=O(xL+M+1)
determine the coefficients of PL(x)andQM(x) by the equation. Since we can clearly multiply the numerator and denominator by a constant and leave [L/M] unchanged, we imposed the normalization condition(3.4)QM(0)=1.0.
Finally, we require that PL(x)andQM(x) have noncommon factors. If we write the coefficient of PL(x)andQM(x) as(3.5)PL(x)=p0+p1x+p2x2+⋯+pLxL,QM(x)=q0+q1x+q2x2+⋯+qMxM,
then by (3.6) and (3.7), we may multiply (3.3) byQM(x), which linearizes the coefficient equations. We can write out (3.5) in more details as(3.6)aL+1+aLq1+⋯+aL-MqM=0,qL+2+qL+1q1+⋯+aL-M+2qM=0,⋮aL+M+aL+M-1q1+⋯+aLqM=0,(3.7)a0=p0,a0+a0q1+⋯+=p1,⋮aL+aL-1q1+⋯+a0qL=pL.
To solve these equations, we start with (3.6), which is a set of linear equations for all the unknown q’s. Once the q’s are known, then (3.7) gives and explicit formula for the unknown p’s, which complete the solution. If (3.6) and (3.7) are nonsingular, then we can solve them directly and obtain (3.8) [42, 50, 56–59], where (3.8) holds, and if the lower index on a sum exceeds the upper, the sum is replaced by zero:(3.8)[LM]=det[aL-M+1aL-M+2⋯aL+1⋮⋱⋮aLaL+1⋯aL+M∑j=MLaj-Mxj∑j=M-1Laj-M+1xj⋯∑j=0Lajxj]det[aL-M+1aL-M+2⋯aL+1⋮⋮⋱aLaL+1⋯aL+MxMxM-1⋯1].
To obtain diagonal Padé approximants of different order such as [2/2], [4/4], or [6/6], we can use the symbolic calculus software Maple.


## 4. Numerical Applications

In this section, we apply He’s polynomials for solving boundary layer problem, unsteady flow of gas through a porous medium, Thomas-Fermi equation, Flierl-Petviashivili equation, and Blasius problem. The powerful Padé approximants are applied for making the work more concise and to get the better understanding of solution behavior.Example 4.1 (see [51, 59]).
Consider the following nonlinear third-order boundary layer problem which appears mostly in the mathematical modeling of physical phenomena in fluid mechanics [51, 59]
(4.1)f′′′(x)+(n-1)f(x)f′′(x)-2n(f′(x))2=0,n>0,
with boundary conditions
(4.2)f(0)=0,f′(0)=1,f′(∞)=0,n>0
By applying the convex homotopy, we have
(4.3)f0+pf1+⋯=f0(x)-p∬0x((n-1)(f0+pf1+⋯)(f0′′pf1′′+⋯)-2n(f0′+pf1′+⋯)2)dxdx,n>0,
comparing the co-efficient of like powers of p, following approximants are made
(4.4)p(0):f0(x)=x,p(1):f1(x)=12αx2+13x3,p(2):f2(x)=124α(3n+1)x4+130n(n+1)x5,p(3):f3(x)=1120α2(3n+1)x5+1720α(19n2+18n+3)x6+1315n(2n2+2n+1)x7,p(4):f4(x)=15040α2(27n2+42n+11)x7+140320α(167n3+297n2+161n+15)x8+122680n(13n3+38n2+23n+6)x9,⋮
where f′′(0)=α<0andpis are He’s polynomials. The series solution is given as
(4.5)f(x)=x+αx22+nx33+(18nα+124α)x4+(130n2+140nα2+1120α2+130n)x5+(19720n2α+1240α+140nα)x6+(1120nα2+1315n+2315n3+115040α2+3560n2α2+2315n2)x7+(1140320α3+334480n2α+34480α3n2+235760nα+12688α+16740320n3α+1960α3n)x8+(13780n+527362880n3α2+1911340n3+709362880nα2+238064n2α2+2322680n2+1322680n4+43120960α2)x9+⋯.Example 4.2 (see [51, 57]).
Consider the following nonlinear differential equation which governsthe unsteady flow of gas through a porous medium(4.6)y′′(x)+2x1-αyy′(x)=0,0<α<1
with the following boundary conditions:
(4.7)y(0)=1,limx→∞y(x)=0.
By applying the convex homotopy method we have
(4.8)y0+py1+⋯=y0(x)-p∬0x(2x(1-α)(y0+py1+p2y2+⋯)-1/2)dxdx.
By comparing the coefficient of like powers of p, the following approximants are obtained:
(4.9)p(0):y0(x)=1,p(1):y1(x)=Ax,p(2):y2(x)=A31-αx3,p(3):y3(x)=-αA212(1-α)3/2x4+A10(1-α)x5,p(4):y4(x)=-3α2A380(1-α)5/2x5+αA215(1-α)2x6+⋯,⋮
where A=y′(0)andpis are He’s polynomials. The series solution is given as
(4.10)y(x)=1+Ax-A31-αx3-αA212(1-α)3/2x4+(A10(1-α)-3α2A380(1-α)5/2)x5+(αA215(1-α)2-α3A448(1-α)7/2)x6+⋯.
The diagonal Padé approximants [51, 57] can be applied to analyze the physical behavior. Based on this, the [2/2] Padé approximants produced the slope A to be
(4.11)A=-2(1-α)1/43α,
and by using [3/3] Padé approximants we find
(4.12)A=-(-4674α+8664)1-α-144γ57α,
where
(4.13)γ=5(1-α)(1309α2-2280α+1216).
Using (4.11)–(4.13) gives the values of the initial slope A=y′(0) listed in Table 3. The formulas (4.11) and (4.12) suggest that the initial slope A=y′(0) depends mainly on the parameterα, where0<α<1. Table 3 exhibits the initial slopes A=y′(0) for various values ofα. Table 4 exhibits the values of y(x) for α=0.5 forx=0.1to1.0.Table 1
Numerical values forα=f″(0)  for 0<n<1  by using diagonal Padé approximants [51, 59].
n[2/2][3/3][4/4][5/5][6/6]0.2−0.3872983347−0.3821533832−0.3819153845−0.3819148088−0.38191218541/3−0.5773502692−0.5615999244−0.5614066588−0.5614481405−0.5614419340.4−0.6451506398−0.6397000575−0.6389732578−0.6389892681−0.63897347940.6−0.8407967591−0.8393603021−0.8396060478−0.8395875381−0.83960567690.8−1.007983207−1.007796981−1.007646828−1.007646828−1.007792100Table 2
Numerical values forα=f″(0)  for n<1  by using diagonal Padé approximants [51, 59].
nα4−2.48395403210−4.026385103100−12.843343151000−40.655382185000−104.8420672Table 3
[51, 57].
αB[2/2]=y′(0)B[3/3]=y′(0)0.1−3.556558821−1.9572089530.2−2.441894334−1.7864755160.3−1.928338405−1.4782708430.4−1.606856838−1.2318018090.5−1.373178096−1.0255297040.6−1.185519607−0.84003460850.7−1.021411309−0.66120478930.8−0.8633400217−0.47766972860.9−0.6844600642−0.2772628386Table 4
[51, 57].
xykiddery[2/2]y[3/3]0.10.88165882830.86330606410.89791670280.20.76630767810.73012622610.79852281990.30.65653799950.60330541400.70411297030.40.55440240320.48488987170.61650379010.50.46136502950.37616038690.53705337960.60.37831093150.27773116280.46656256690.70.30559765460.18968433710.40624260330.80.24313254730.11171051650.35608016990.90.19046236810.043236732360.31799666141.00.15876898260.016467508470.2900255005Table 5
Padé approximants and initial slopesy′(0)  [56]
Padé approximantsInitial slopey′(0)Error (%)[2/2]−1.21141372923.71[4/4]−1.5505259192.36[7/7]−1.58602103712.9×10-2[8/8]−1.5880768203.66×10-4[10/10]−1.5880767793.64×10-4Table 6
Roots of the Padé approximants monopole [42] α,n=1.
DegreeRoots[2/2]−1.5[4/4]−2.50746[6/6]−2.390278[8/8]−2.392214Table 7
Roots of the Padé approximants monopole [42] α,n=3.
DegreeRoots[2/2]−2.0[4/4]−2.0[6/6]−2.0[8/8]−2.0Table 8
Roots of the Padé approximants monopole [42] α.
DegreeRoots[2/2]0.0[4/4]−.2197575908[6/6]−1.1918424398[8/8]−1.848997181Example 4.3 (see [56]).
Consider the following Thomas-Fermi (T-F) equation [6–13, 17, 31, 33, 34, 54] which arises in the mathematical modeling of various models in physics, astrophysics, solid state physics, nuclear charge in heavy atoms, and applied sciences:
(4.14)y′′(x)=y3/2x1/2,
with boundary conditions
(4.15)y(0)=1,limx→∞y(x)=0.
By applying the convex homotopy,
(4.16)y0+py1+p2y2+⋯=y0(x)+p∬0x(x-1/2(y0+py1+p2y2+⋯)3/2)dxdx.
Now, we apply a slight modification in the conventional initial value and take y0(x)=1,instead of y0(x)=1+Bx, where B=y′(0). By comparing the coefficient of like powers of p, the following approximants are obtained
(4.17)p(0):y0(x)=1,p(1):y1(x)=Bx+43x3/2,p(2):y2(x)=25Bx5/2+13x3,p(3):y3(x)=370B2x7/2+215Bx4+227x9/2+13x3,p(4):y4(x)=370B2x7/2+215Bx4+227x9/2-1252B3x9/2+1175B2x5+227x9/2+370B2x7/2.
The series solution is given as
(4.18)y(x)=1+Bx+Bx+43x3/2+25Bx5/2+13x3+370B2x7/2+215Bx4+227x9/2+370B2x7/2-1252B3x9/2+1175B2x5+227x9/2+370B2x7/2+11056B4x11/2+41575B3x6+557100100B2x13/2+4693Bx7+10152650x15/2-39152B5x13/2-2924255B4x7-512351000B3x15/2-4645045B2x8-1131178100Bx17/2+23473850x9⋯,⋮
Setting x1/2=t, the series solution is obtained as
(4.19)y(t)=1+Bt2+43t3+25Bt5+13t6+370B2t7+215Bt8+(-1252B3+227)t9+1175B2t10+(11056B4+311485B)t11+(41575B3+4405)t12+(-39152B5+557100100B2)t13+(-2924255B4+4693B)t14+(7499B6-623351000B3+10152650)t15+(68105105B4-4645045B2)t16+(-343520B7+153173116424000B4-1131178100B)t17+⋯.
The diagonal Padé approximants can be applied [56] in order to study the mathematical behavior of the potential y(x) and to determine the initial slope of the potentialy′(0).Example 4.4 (see [42]).
Consider the generalized variant of the Flierl-Petviashivili equation [37]
(4.20)y′′+1xy′-yn-yn+1=0,
with boundary conditions
(4.21)y(0)=α,y′(0)=0,y(∞)=0.
Using the transformation u(x)=xy′(x), the generalized FP equation can be converted to the following first-order initial value problem:
(4.22)u′(x)=x(∫0x(u(x)x)n+(u(x)x)n+1dx),
with initial conditions
(4.23)u(0)=0,u(0)=0.
By applying the convex homotopy, we have
(4.24)u0+pu1+p2u2+⋯=p∫0s(x(∫0s(1x(u0+pu1+p2u2+⋯))n+(1x(u0+pu1+p2u2+⋯))n+1dx))dx.
The series solution after four iterations is given by
(4.25)u(x)=(αn+αn+1)2x2+(αn+αn+1)(nαn+(n+1)αn+1)16αx4+(αn+αn+1)(2n(3n-1)α2n+2n(3n+1)α2n+1+(3n+1)(n+1)α2n+2)384α2x6+(αn+αn+1)(n(18n2-29n+12)α3n+n(54n2-33n+7)α3n+1+𝔄)18432α3x8+⋯,
where 𝔄 denote (18n2+7n+1)(3nα3n+2+(n+1)α3n+3), and the inverse transformation will yield
(4.26)y(x)=α+(αn+αn+1)4x2+(αn+αn+1)(nαn+(n+1)αn+1)64αx4+(αn+αn+1)(2n(3n-1)α2n+2n(3n+1)α2n+1+(3n+1)(n+1)α2n+2)2304α2x6+(αn+αn+1)(n(18n2-29n+12)α3n+n(54n2-33n+7)α3n+1+𝔄)147456α3x8+⋯,
where 𝔄 denote (18n2+7n+1)(3nα3n+2+(n+1)α3n+3).Diagonal Padé approximants can be applied [42] to find the roots of the FP monopole α forn≥1.
Table 9 shows that the roots of the monopole α converge to −1 as n increases.Table 9
Roots [42] of the Padé approximants [8/8] monopole  for several values of n.
n[8/8]  rootsn[8/8]  roots1−2.3922138667−1.0007082852−2.08−1.0006016153−1.8489971819−1.0005230054−1.28602589210−1.0004626365−1.00110114111−1.0002621376−1.000861533n→∞−1.0Example 4.5 (see [58, 59]).
Consider the two-dimensional nonlinear inhomogeneous initial boundary value problem for the integro-differential equation related to the Blasius problem(4.27)y′′(x)=α-12∫0xy(t)y′′(t)dt,-∞<x<0,
with boundary conditions
(4.28)y(0)=0,y′(0)=1,limx→∞y′(x)=0,
where the constant α is positive and defined by
(4.29)y′′(0)=α,α>0.
By applying the convex homotopy, we have
(4.30)y0+py1+⋯=y0(x)-p∬0x(∫0x(y0+py1+⋯)(d2y0dx2+pd2y1dx2+⋯))dxdx.
Proceeding as before, the series solution is given as
(4.31)y(x)=x+12αx2-148αx4-1240α2x5+1960αx6+1120160α2x7+(11161280α3+1960α)x8-43967680α2x9+(152960α-5387072α3)x10+(587212889600α2-54257792α4)x11+(-116220160α+17257792α3)x12+⋯,
and consequently
(4.32)y′(x)=1+αx-112αx3-148α2x4+1160αx5+112880α2x6(1120160α3-12688α)x7-43107520α2x8+10(1552960α-5387072α3)x9+11(587212889600α2-54257792α4)x10+12(-116220160α+1725760α3)x11+⋯.
Now, we apply the diagonal Padé approximants to determine a numerical value for the constant α by using the given condition. Padé approximant of y′(x)usually converges on the entire real axis [58, 59]. Moreover, y′(x) is free of singularities on the real axis. Substituting the boundary conditions y′(-∞)=0 in each Padé approximant which vanishes if the coefficient of x with the highest power in the numerator vanishes. By solving the resulting polynomials of these coefficients, we obtain the values of α listed in Table 10 [58, 59].Table 10
Padé approximants and numerical value ofα [53].
Padé approximantα[2/2]0.5778502691[3/3]0.5163977793[4/4]0.5227030798


## 5. Conclusion

In this paper, we applied a reliable combination of He’s polynomials and the diagonal Padé approximants for obtaining approximate solutions of various singular and nonsingular boundary value problems of diversified physical nature. The proposed algorithm is employed without using linearization, discretization, transformation, or restrictive assumptions. The fact that the suggested technique solves nonlinear problems without using Adomian’s polynomials is a clear advantage of this technique over the decomposition method.



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