Advances in Engineering Software 40 (2009) 1104–1111

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Advances in Engineering Software

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Computational method for finding various solutions for a Quasilinear EllipticEquation of Kirchhoff type

Manoj Kumar, Prashant Kumar *

Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad 211 004, UP, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 26 February 2009Received in revised form 21 April 2009Accepted 16 June 2009

Keywords:Kirchhoff equationForward differenceFinite element methodScaling Iterative AlgorithmNewton method

0965-9978/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.advengsoft.2009.06.003

* Corresponding author.E-mail addresses: manoj@mnnit.ac.in (M. Ku

(P. Kumar).

This paper is concerned with the various solutions to the class of nonlocal boundary value problem withthe help of finite element method for the class of Kirchhoff-type problem

�MZ

Xjruj2dx

� �Du ¼ f ðx;uÞ in X; u ¼ 0 on @X

where X � RN is a bounded smooth domain, M : Rþ ! R is continuous and f : X� R! R hassubcritical growth. Kirchhoff equation does not appear to have been previously studied in detailcomputationally and it is hope that this paper will help to provide a new idea in this direction.Here we solved the nonlinear function f ðx; uÞwith the help of Newton Raphson method and qua-si-linear term by using Scaling Iterative Algorithm, forward difference and with the help of theseschemes; we used finite element method to solve the Quasilinear Elliptic Equation of KirchhoffType and discussed the behaviour. In order to numerically confirm our theoretical results and todemonstrate the performance of the algorithm, we develop a MATLAB program for solving Kir-chhoff-type problem.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper, we considered the quasilinear elliptic boundaryvalue problem of the type

� aþ bZ

Xjruj2dx

� �Du ¼ f ðx; uÞ in X

u ¼ 0 on @Xð1Þ

where X is a smooth bounded domain of RN , a and b are positivefunctions, and f : X� R! R is a carathéodory function. Since theequation contains an integral over X, it is no longer a pointwiseidentity, and therefore is often called Nonlocal Problem.

Problem (1) is related to the stationary version of the Kirchhoff[17] equation

q@2u@t2 �

P0

hþ E

2L

Z L

0

@u@x

��������

2

dx

!@2u@x2 ¼ 0 ð2Þ

This equation is an extension of the classical D’Alembert’s waveequation by considering the effects of the changes in the length ofthe string during the vibrations. It is worth mentioning that Eq. (2)

ll rights reserved.

mar), kam3545@gmail.com

received much attention after the work of Lions [18], where a func-tional analysis framework was proposed to the problem. Someinteresting results and further references are given in [4,14,23].

A key work on nonlocal elliptic problems is the paper by Chipotand Rodrigues [10], cited in [1,2,8,13,20]. They studied nonlocalboundary value problems and unilateral problems with severalapplications. However, Vasconcellos [25] gave one of the firstworks on the Kirchhoff equation (1) in a functional analysis setting.He considered the equation Nðx; aðx;uÞÞLu ¼ f in an unbounded do-main of RN , by using Galerkin approximations and weighted Sobo-lev Spaces. The existence of solutions of problem (1), with f ¼ f ðxÞ,with the solutions of a suitable algebraic equation, based on thesolutions of the linear problem �Dw ¼ f in H1ðXÞ (see Theorem 3in [21]). We mention that existence of positive solutions was con-sidered in [2], with arguments motivated by [10,13], and argu-ments from critical point theory (see Theorems 6 and 9 in [21]).

The variational approach to the problem (1) was given in Maet al. [20], where minimization arguments were used. The Kirchhoff’stime-retarded potential boundary integral equation used in thenumerical calculation of the transient response in an enclosurewith rigid or absorbent boundary. The sound pulse reflected fromthe absorbent boundary, consisted of a porous layer on a rigidbacking, is calculated by using the time-dependent ‘‘impulseadmittance”. Kirchhoff equation [3,5,19] can used as a string vibra-tion model that takes into account tension growth due to string

M. Kumar, P. Kumar / Advances in Engineering Software 40 (2009) 1104–1111 1105

extension as given in Eq. (1.1) in [24], yet no essential approximatealgorithms is exist for this equation and investigation of their accu-racy is remains in the shadow.

In Chipot [9] consider the following model problem given,where find a function uðx; tÞ such that

ut �MR

X uðx; tÞdx� �

Du ¼ f in X� ð0; TÞuðx; tÞ ¼ 0 on C� ð0;TÞuðx;0Þ ¼ u0ðxÞ on X

8><>:

Here X is a bounded open subset in Rn; n P 1 with smoothboundary C. T is some arbitrary time. M is some function from R

into ð0;þ1Þ. In [11], Kirchhoff equation for the string with movingends is solved by means of the explicit Lax–Friedrichs differencescheme and the stability of this scheme is studied assuming thatthe nonlinear term is neglected. Finally, we propose to solve thesystem obtained by discretization of the system of nonlinear equa-tions by means of the finite element scheme.

This work is organized as follows: In Section 2, we present theexistence of solution of Kirchhoff-type problem. In Section 3, wediscussed our numerical scheme for solving nonlinear problem(1) for finding various solutions (Positive, Negative and Sign-Changing solution) with the help of finite element method anddemonstrated computational result with the help of example. InSection 4, conclusion in the field of computational solution of non-linear problem are discussed. In last the Matlab codes are given inSection 5.

2. Existence of solution

In particular, let a ¼ 0; b ¼ M in given Elliptic Eq. (1) is to find u,can be represented as

�MZ

Xjruj2dx

� �Du ¼ f ðx;uÞ in X; u ¼ 0 on @X ð3Þ

Let us assume that f : X� R! R is a Carathéodory functionsatisfying

jf ðx; sÞj 6 Cð1þ jsjpÞ 8 x 2 X; 8 s 2 R ð4Þ

where C P 0; 1 < p < ðN þ 2Þ=ðN � 2Þ if N P 3 and 1 < p <1 ifN ¼ 1; 2. With this subcritical growth, u! f ð�; uÞ defines a contin-uous Nemytskii mapping Nf : LpðXÞ ! Lðpþ1Þ0 ðXÞ (see DeFigueiredo[15]). A typical assumption for M is the following

9 m0 > 0 such that MðsÞP m0 8 s P 0 ð5Þ

We emphasize that M represents the expression in parenthesesof Eq. (2), and therefore, in its original motivation, it is a line withpositive slope. Under these assumptions, a function 8 u 2 H1

0ðXÞ iscalled weak solution of (1) if

Mðkuk2ÞZ

Xrurudx�

ZX

f ðx;uÞudx ¼ 0 8 u 2 H10ðXÞ

where kuk2 ¼R

X jruj2dx. If f is Locally Lipschitz continuous in X*

�R,then weak solutions are classical solutions.

The arguments of Theorem 3 from [21] can be adapted to findpositive solutions of the problem

�Mðkuk2ÞDu ¼ up in X

u ¼ 0 on @Xð6Þ

where p – 1.

3. Numerical scheme

Here we present the finite element method that is one of thediscretization methods. The purpose of discretization method is

reducing of continuous system to a simple discrete system that isequivalent with it. To describe our solution we assume the follow-ing: (i) Let f be a continuous function prescribed on the boundaryof a finite region X. (ii) There exists a function u 2 C2 which takesthe prescribed value f on @X.

Consider a quasilinear elliptic problem with zero Dirichletboundary condition. i.e.

�MR

X jruj2dx� �

DuðxÞ ¼ f ðx;uÞ in X

u ¼ 0 on @X

(ð7Þ

with the weak form

qðUÞ ¼ �MZ

Xjruj2dx

� �Aðu;vÞ þ ðf ðuÞ;vÞ ¼ 0; 8 v 2 S0 ð8Þ

where in particular, let f ðuÞ ¼ u3 and X is a two-dimensionalbounded domain with Lipschitz boundary S0 ¼ fu 2 H1ðXÞ; u ¼0 on @Xg and the bilinear form Aðu;vÞ ¼ �

RXðriu � rjvÞdx is as-

sumed to be bounded and S0-coercive.Let the domain X subdivide into a finite number of elements s

with the subdivision Jh and zh ¼ fxjgm1 be the set of all interior

nodes. The n-degree finite element subspace denoted by Sh � S0

and fNjðxÞgm1 be the bases of Sh. It is well known that standard finite

element solution [6,12,16,20] uh 2 Sh of (4) can be expressed as

uhðxÞ ¼Xm

j¼1

UjNjðxÞ 2 Sh; Uj ¼ uhðxjÞ; and satisfies

Jðu;vÞ ¼ �MZ

Xjruj2dx

� �ZXrurvdxþ

ZX

u3vdx ¼ 0; 8 v 2 Sh

ð9Þ

By taking v ¼ Ni; i ¼ 1;2; . . . ;m, Eq. (6) leads to a nonlinear sys-tem of equations

�MZ

Xjruhj2dx

� �Xm

j¼1

AðNj;NiÞUj � fXm

j¼1

NjðxÞUj

!;Ni

!¼ 0;

i ¼ 1;2; . . . ;m ð10Þ

This nonlinear equation is solved by the Gauss–Newton itera-tive method. Assume that we have a initial guess UðnÞ of the solu-tion. If UðnÞ is close enough to the exact solution, an improvedapproximation Uðnþ1Þ is obtained by solving the linearized problem.In Eq. (10), we use the Gauss–Newton method to linearized thenonlinear problem [22] as follows:

@qðUðnÞÞ@U

!ðUðnþ1Þ � UðnÞÞ ¼ �aqðUðnÞÞ ð11Þ

where a is a positive number. It is well known that for sufficientlysmall a; kqðUðnþ1ÞÞk < kqðUðnÞÞk and pn ¼ @qðUðnÞ Þ

@U

� ��1qðUðnÞÞ is called

a descent direction for kqðUÞk, where k � k is the L2-norm. The iter-ation is

Uðnþ1Þ ¼ UðnÞ þ apn ð12Þ

where a 6 1 is chosen such that the step has a reasonable descent.We aim to solve (9) with Newton Raphson’s method, starting

with some Uð0Þ in each iteration step. We computeUn � Unþ1 2 H1

0ðXÞ satisfying

DJðUn;v; Un � Unþ1Þ ¼ JðUn;vÞ 8 v 2 H10ðXÞ ð13Þ

where

DJðu;v ; wÞ ¼ �MZ

Xjruj2dx

� �ZXrvrwdxþ

ZX

3u2vwdx ¼ 0

ð14Þ

1106 M. Kumar, P. Kumar / Advances in Engineering Software 40 (2009) 1104–1111

Using Eqs. (9), (14) in Eq. (13), we obtain

�MZ

XjrUnj2dx

� �ZXrvrðUn � Unþ1Þdx

þZ

X3ðUnÞ2vðUn � Unþ1Þdx ¼ �M

ZXjrUnj2dx

� �ZXrUnrvdx

þZ

XðUnÞ3vdx

On solving above equation, we get

�MZ

XjrUnj2dx

� �ZXrvrUnþ1dx� 3

ZXðUnÞ2vUnþ1dx

¼ �2Z

XðUnÞ3vdx ð15Þ

Fig. 1. Positive solution w

Fig. 2. Negative solution w

�MZ

X

@Un

@x1

� �2

þ @Un

@x2

� �2( )

dx1dx2

!ZXrvrUnþ1dx� 3

�Z

XðUnÞ2vUnþ1dx ¼ �2

ZXðUnÞ3vdx

multiplicative of first term is solved by forward

�MZ

X

Uniþ1;j � Un

i;j

Dx1

!2

þUn

i;jþ1 � Uni;j

Dx2

! !28<:

9=;dx1dx2

0@

1A

�Z

XrvrUnþ1dx� 3

ZXðUnÞ2vUnþ1dx ¼ �2

ZXðUnÞ3vdx

ith one and two peak.

ith one and two peak.

Fig. 3. Sign-Changing solution.

M. Kumar, P. Kumar / Advances in Engineering Software 40 (2009) 1104–1111 1107

In Eq. (15), multiplicative of first term is solved by forward-differ-ence formula and Scaling Iterative Algorithm [7] is used to solvethe quasi-linear term in Eq. (1).

The Gauss–Newton method is local, and convergence is assuredonly when Uð0Þ is close enough to the solution. In general, the firstguess may be outside the region of convergence. To improve ourconvergence from bad initial guesses, a damping strategy theArmijo–Goldstein line search is implemented here for choosing a.It chooses the largest damping coefficient a out of the sequence1, 1/2, 1/4, . . . Such that the following inequality holds:

kqðUnÞk � kqðUn þ apnÞkPa2kqðUnÞk ð16Þ

which guarantees a reduction of the residual norm by at least1� a=2. Here in this approach each step of the line-search algo-rithm requires an evaluation of the residual qðUðnÞ þ apnÞ.

An important point of this strategy is that when UðnÞ approachesthe solution, then a! 1 and thus convergence rate increases. Ifthere is a solution to qðUÞ ¼ 0, the scheme ultimately recoversthe quadratic convergence rate of the standard Newton iteration.

Another strategy is this, here we solve the nonlinear functionf ðx;uÞ with the help of Newton Raphson method and quasi-linearterm is solved by using Scaling Iterative Algorithm, forward differ-ence and with the help of these two schemes we used finite ele-ment method to solve the Quasilinear Elliptic Equation ofKirchhoff type. The various solution of (1) obtained by our presentscheme are given in Figs. 1–3.

4. Conclusions

To our best knowledge, Kirchhoff equation does not appear tohave been previously studied in detail computationally; variationalapproach to the problem was given in 2003 by Ma and Rivera,where minimization arguments were used. In the present paperwe develop an algorithm in which there is no need of finding Jaco-bian of the nonlinear terms, it reduces the cost of computation incomparison to standard finite element method. We also discussedthe different behaviours (Positive, Negative and Sign-Changingsolution) of the Kirchhoff-type problem with the help of Figs. 1–3.

1108 M. Kumar, P. Kumar / Advances in Engineering Software 40 (2009) 1104–1111

5. Matlab code for the Quasilinear Elliptic Kirchhoff-Type

problem

M. Kumar, P. Kumar / Advances in Engineering Software 40 (2009) 1104–1111 1109

1110 M. Kumar, P. Kumar / Advances in Engineering Software 40 (2009) 1104–1111

M. Kumar, P. Kumar / Advances in Engineering Software 40 (2009) 1104–1111 1111

References

[1] Alves CO, Corrêa FJSA. On existence of solutions for a class of probleminvolving a nonlinear operator. Commun Appl Nonlinear Anal 2001;8:43–56.

[2] Alves CO, Corrêa FJSA, Ma TF. Positive solutions for a quasilinear ellipticequation of Kirchhoff type. Comput Math Appl 2005;49:85–93.

[3] Andrade D, Ma TF. An operator equation suggested by a class of stationaryproblems. Commun Appl Nonlinear Anal 1997;4:65–71.

[4] Arosi A, Panizzi S. On the well-posedness of the Kirchhoff string. Trans AmMath Soc 1996;348:305–30.

[5] Attigui, F. Controle et stabilisation de cables non lineaires. Repport de stagescientifique, Laboratoire des Materiaux et des Structures du Genie Civil,Universite Paris XIII, France; 1999. p. 158.

[6] Brenner SC, Scott LR. The mathematical theory of finite elementmethod. Springer Verlag; 2002.

[7] Chen G, Zhou J. Algorithms and visualization for solutions of nonlinear ellipticequations. Int J Bifurcat Chaos 2000;10(7):1565–612.

[8] Chipo M, Valente V, Caffarelli G, Vergara. Remarks on a nonlocal probleminvolving the Dirichlet energy. Rend Sem Mat Padova 2003;110:199–220.

[9] Chipot M, Lovat B. Some remarks on nonlocal elliptic and parabolic problems.Nonlinear Anal Theory Methods Applicat 1997;30(7):4619–27.

[10] Chipot M, Rodrigues JF. On a class of nonlocal nonlinear elliptic problems.RAIRO Modélisation Math Anal Numér 1992;26:447–67.

[11] Christie I, Sanz-Serna JM. A Galerkin method for a nonlinear integro-differential wave system. Comput Method Appl Mech Eng 1984;44:229–37.

[12] Ciarlet PG. The finite element method for elliptic problems. North-HollandPublishing Company; 2002.

[13] Corrêa FJSA, Menezes SDB. Existence of solutions to nonlocal and singularelliptic problems via Galerkin method. Electron J Differ Equations 2004:1–10.

[14] Cousin AT, Frota CL, Lar’kin NA, Medeiros LA. On the abstract model of theKirchhoff–Carrier equation. Commun Appl Anal 1997;1:389–404.

[15] DeFigue iredo DG. Ekeland variational principle with applications anddetours. Heidelberg: Tata Institute; 1989.

[16] Feistauer Miloslav, Zenisek Alexander. Finite element solution of nonlinearelliptic problems. Numer Math 1987;50:451–75.

[17] Kirchhoff G. Mechanik. Leipzig: Teubner; 1883.[18] Lions JL. On some questions in boundary value problems of mathematical

physics. In: de la Penha G, Medeiros LA, editors. International symposium oncontinuum, mechanics and partial differential equations, Rio de Janeiro 1977,mathematics studies, vol. 30. Amsterdam: North-Holland; 1978. p. 284–346.

[19] Liu IS, Rincon MA. Effect of moving boundaries on the vibrating elastic string.ELSGMLTM (APNUM): m2, v 1, vol. 162; 2003. p. 1–14.

[20] Ma TF, Rivera JEMuñoz. Positive solutions for a nonlinear nonlocal elliptictransmission problem. Appl Math Lett 2003;16:243–8.

[21] Ma TF. Remarks on an elliptic equation of Kirchhoff type. Nonlinear Anal2005;63:e1967–77.

[22] Mittelmann Hans Detlef. On the efficient solution of nonlinear finite elementequations I. Numer Math 1980;35:227–91.

[23] Ono K. On global solutions and blow-up solutions of nonlinear Kirchhoffstrings with nonlinear dissipation. J Math Anal Appl 1997;216:321–42.

[24] Peradze Jemal. A numerical algorithm for the nonlinear Kirchhoff stringequation. Numer Math 2005;102:311–42.

[25] Vasconcellos CF. On a nonlinear stationary problem in unbounded domains.Rev Mat Univ Comput Madrid 1992;5:309–18.