Applied Mathematical Sciences, Vol. 7, 2013, no. 109, 5397 - 5408

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2013.37409

Solution of Non Linear Singular Perturbation Equation

Using Hermite Collocation Method

Happy Kumar1, Shelly Arora

2 and R. K. Nagaich

3

1Department of Mathematics, Guru Nanak College, Budhlada (Mansa), Punjab, India

2&3Department of Mathematics, Punjabi University, Patiala, Punjab, India

Copyright © 2013 Happy Kumar, Shelly Arora and R. K. Nagaich. 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

A numerical method for solving second order, transient, parabolic partial differential

equation is presented. The spatial discretization is based on Hermite collocation method

(HCM). It is a combination of orthogonal collocation method and piecewise cubic Hermite

interpolating polynomials. The solution is obtained in terms of cubic Hermite interpolating

basis. Numerical results have been plotted in terms of time space graphs to illustrate the

applicability and efficiency of the HCM in terms of convergence and stability analysis. The

present method has been compared with orthogonal collocation method for different value of

ε.

Introduction

The orthogonal collocation method is one of several weighted residual techniques.

Orthogonal collocation method for numerical solution of partial differential equations has

been followed various investigators (Ghanaei & Rahimpour (2010), Rohman et al. (2011),

Wu et al. (2011) & Zugasti (2012)) due to its compailibilty, simplicity and accuracy which

makes it different from other numerical methods such that Galerkin (Nadukandi et al.

(2010)), Least square method (Ren et al. (2012)) and finite difference (Jha (2013)). In this

method, an approximate solution is substituted into the differential equation to form the

5398 Happy Kumar, Shelly Arora and R. K. Nagaich

residual. Then, this residual is set to zero at collocation points. The choice of collocation

points plays an important role in collocation techniques for the convergence and efficiency.

Generally, zeros of Jacobi polynomials are taken as collocation points over the normalized

interval. Hermite collocation method (HCM) is combination of orthogonal collocation

method and piecewise cubic Hermite interpolating polynomials.

HCM can be expressed as a linear combination of Hermite basis functions. Hermite

functions have great advantages that functions and its slope are continuous at junction

points. Numerous investigators have used this technique in different manner to solve

different type of problem such as near-singular problems (Lang and Sloan (2002)),

convection-diffusion problems (Rocca et al. (2005)) and nonlinear Lane–Emden type

equations (Peirce (2010)).

Hermite collocation is considered for the discretization of second-order boundary value

problems, the usual choice of Hermite is either quadratic or cubic at one or two collocation

points. In the case of quadratic or cubic, Hermite collocation in second order problems, the

computed approximations exhibit up to fourth order convergence (Prenter (1975), Sun

(2000), Parand et al. (2010)).

Consider the second order non-linear parabolic boundary value problem:

x

y

x

y

t

z

t

y

∂

∂−

∂

∂=

∂

∂−+

∂

∂2

21ε

θ

θ (x,t)∈(0,1)×(0,T) (1)

)(yfz = (2)

021 =∂

∂+

x

yqyq at x = 0 (3)

043 =∂

∂+

x

yqyq at x = 1 (4)

1=y at t= 0 (5)

where ε,ϴ, q1, q2, q3, q4 are constants

Cubic Hermite polynomials

Hermite interpolating polynomials was first introduced by Charles Hermite (1822-1905). It

is an extension of Lagrange interpolating polynomials as in Hermite interpolating

polynomials, both the function and its derivative are to be assigned values at interpolating

point. An nth-order Hermite polynomial in x is a polynomial of order 2n+1 and therefore,

cubic Hermite interpolating polynomials particular case of general Hermite interpolating

polynomials for n=1. It consists of two node points and two tangents in cubic polynomial

and defined as

132)( 23

1 +−= xxxH xxxxH +−= 23

2 2)(

Solution of non linear singular perturbation equation 5399

23

3 32)( xxxH +−= 23

4 )( xxxH −=

Fig.1. Cubic Hermite interpolating polynomials

Figure.1. shows the behavior of the Cubic Hermite interpolating polynomials. The values of

H1,H2, H3 and H4 lies within [0, 1] as x goes from 0 to 1 and their derivatives are unity or

zero at the end points.

The Cubic Hermite approximation is defined as:

)(uy l =∑=

4

1

)()(i

ii uHtal Where l = 1, 2,…….,k (6)

Where, k is the number of elements and )(tai

l ’s are the continuous functions of ‘t’ in lth

element.{Hi(u)} are piecewise cubic Hermite polynomials as defined above.

The first and second order discretized derivatives of the trial function ly taken at jth

collocation point are defined by Aji and Bji respectively, where Aji=H’i(uj) and Bji=H”i(uj).

After applying Hermite collocation method, the following set of collocation equations is

obtained:

∑∑==

−=∂

∂−+

∂

∂ 4

1

4

12

)(1

)(1

i

jii

i

jii

jjAta

hBta

ht

z

t

yl

l

l

l

ll

ε

θ

θ; ℓ=1,2,…,k; j=2,3 (7)

)( l

j yfz =l (8)

0)(4

1

1

121

11 =+ ∑=i

ii Atah

qyq

l

at x = 0 (9)

5400 Happy Kumar, Shelly Arora and R. K. Nagaich

0)(4

1

44

43 =+ ∑=i

i

k

i

kAta

h

qyq

l

at x = 1 (10)

The equations from (7) to (10) can be put into the matrix form as:

Saa =' (11)

Where ( )ABHS −= − ε1 and H be the coefficient matrix of Cubic Hermite polynomials at jth

collocation points.

=−

kk QP

QP

QP

QP

AB

........

.......

33

22

11

ε

Figure 2: The resulting matrix AB −ε arises from HCM.

The system AB −ε has the block-tridiagonal structure. Where P1,Q1 and Pk, Qk are

collocation matrices of block 32 × in first element and last element, respectively.

Remaining collocation matrix Pi, Qi (i=2,3,…..k-1) with each block 42 × . The resulting stiff

system of 2k differential algebraic equations (DAE’s) is solved using MATLAB with the

ode15s system solver. This system solver uses backward differentiation formula to integrate

the set of DAE’s.

Hermite collocation method

The Hermite collocation method is a numerical technique for solution of partial differential

equations defined over the interval [0, 1]. It is a combination of orthogonal collocation

method and cubic Hermite interpolating polynomials that have been used as trial function

for the spatial domain and the coefficients are determined so that the differential equation is

satisfied by the interpolation function at collocation points. The roots of shifted Legendre

polynomials have been taken as collocation points. In this method, the domain is divided

into small elements and then orthogonal collocation is applied within each element. Two

interior collocation points have been used within each element. Therefore, Convergence and

stability of numerical solutions does not depend upon the number of collocation points,

rather it depends upon the number of elements to be taken in the domain of interest.

Solution of non linear singular perturbation equation 5401

Results and Discussion

To check the effect of different parameters on solution profiles via for different range of

parameters and number of elements.

Problem:

Consider the singularly perturbed nonlinear parabolic boundary value problem with

Richardson’s boundary conditions,

( ) x

y

x

y

t

y

y ∂

∂−

∂

∂=

∂

∂

+

−+

2

2

21

111 ε

ηθ

θ (x,t)∈(0,1)×(0,T)

(12)

0=∂

∂−

x

yy ε at x = 0, for all 0≥t (13)

0=∂

∂

x

y at x = 1, for all 0≥t (14)

1=y at t = 0, for all x (15)

The set of equations (12) to (15) has been solved using HCM. In figure 3, numerical results

are presented in the form of breakthrough curves for different values of ε ranging from 0.01

to 0.0025 at x=1. The values of η and ϴ are taken to be 0.263 and 0.931, respectively. It is

evident from this figure that the numerical values are converging to zero as increase t even

for ε >0.01, which show that the method is stable and convergent for any value of ε.

Fig. 3. Behavior of solution profiles at x = 1 for different values ofε .

5402 Happy Kumar, Shelly Arora and R. K. Nagaich

In figure 4 the behavior of solution profiles for different values of ϴ. The values of ε and η

are taken to be 0.025 and 0.263, respectively. It is evident from this figure that with the

increase the value of ϴ= 0.9, the breakthrough curves converges to zero more rapidly as

compared to small values of ϴ= 0.7 or 0.8 . Thus, desired numerical results can be achieved

for large value of ϴ.

Fig. 4. Behavior of solution profiles at x = 1 for different values of ϴ.

In figure 5 the solution profiles are presented graphically for ε=0.025, ϴ=0.942 at different

values of η. It is evident from this figure that the deflation of solution profiles for small to

large range of η is not great deal but even than higher values of η=80 slightly better than η=4

or 40. Hence From this analysis is observed that the numerical results are stable and accurate

for a wide range of η.

Solution of non linear singular perturbation equation 5403

Fig. 5. Behavior of solution profiles for different values of η.

In figure 6 the solution profile for ε = 0.0167 is presented. The fluctuations are observed and

the results are oscillating at the initial stage and also for large values of t. In this case the

values of y at x = 1 are increasing from1, causing instability to the numerical results. The

two interior collocation points have been taken in each element. The fluctuations decrease

by increase number of elements that convert instable results into stable results.

Fig. 6. Behavior of solution profiles for ε== 0.0167.

5404 Happy Kumar, Shelly Arora and R. K. Nagaich

In figure 7, behavior solution profiles for various numbers of elements for ε=0.05 η =0.263

and ϴ=0.931. However, for number of elements 7 or 17 the solution profiles converge to 0

for t ≥ 4, whereas for number of elements 35 the solution profile converges to 0 for t ≥ 2. It

shows that as number of elements increases, the time taken to converge to steady state

condition is decreases. Hence stable and accurate numerical results have been achieved as

increase number of elements and the graph becomes smoother.

Fig. 7. Behavior of solution profiles for different number of elements.

Comparison of OCM and HCM

In figure 8 HCM has been compared with OCM for ϵ=0.01667, ϴ=0.942 and η=0.263. The numerical results were obtained using ode15s solver in MATLAB. The similar

problem is solved by OCM for 6 interior collocation points. In this figure it is clear that

initial values oscillate in OCM that are increases from 1 not in case of HCM and

converging rate of OCM slightly slow than HCM. Hence, HCM has been better that OCM

in case of accuracy, efficiency and computationally and solution profiles have been

converging steady state condition smoothly for any value of ϵ.

Solution of non linear singular perturbation equation 5405

Fig. 8. Behavior of solution profiles for OCM and HCM

In figure 9 shows the solution profiles follow Gaussian shape for various values of t for

ε=0.0625, ϴ=0.942 and η=0.263. The peak position of solution profiles is 3.77, 2.77 and

1.93 at t=0.1, 0.2 and 0.4. The solutions become more accurate with high values of t. Hence,

HCM gives stable and convergent results for large time period.

Fig. 9. Behavior of solution profiles for different values of t.

5406 Happy Kumar, Shelly Arora and R. K. Nagaich

In Figure 10 displays numerical solution of the problem at t=0.1 for different values of ε. It is

observed that numerical results demonstrate the development of a sharp initial stage and becomes

smoother as x-axis progress. Therfore, the accuracy of the numerical solutions improves as the

value of ε is reduced. Hence, the results obtained from HCM are fairly accurate and stable.

Fig. 10. Behavior of solution profiles for different values of ε.

In figure 11, numerical results are presented in the form of breakthrough curves for ε ranging from

0.25 to 1 at t=0.1. It is obvious from this figure that the solution profiles are converging to zero very

smoothly for small values of ε as x increases, which show that the method is convergent and stable

for any value of ε.

Fig. 11. Behavior of solution profiles for different values of ε.

Solution of non linear singular perturbation equation 5407

Conclusion

The approximate solutions of non linear parabolic boundary-value problems using HCM

show that numerical results are better in the sense of accuracy and applicability. The

higher accuracy has been achieved by small value of parameters ϵ and large value of η, ϴ

and k. Hence it is evident from these figures that the method is stable and convergent.

Acknowledgement

Dr. Shelly is thankful to University Grant Commission for providing financial assistance

(vide No. F.41-786/2012 (SR)).

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Received: July 11, 2013