Solution of Non Linear Singular Perturbation Equation. Using Hermite Collocation Method

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1 Applied Mathematical Sciences, Vol. 7, 03, no. 09, HIKARI Ltd, Solution of Non Linear Singular Perturbation Equation Using Hermite Collocation Method Happy Kumar, Shelly Arora and R. K. Nagaich 3 Department of Mathematics, Guru Nanak College, Budhlada (Mansa), Punjab, India &3 Department of Mathematics, Punjabi University, Patiala, Punjab, India Copyright 03 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 (00), Rohman et al. (0), Wu et al. (0) & Zugasti (0)) due to its compailibilty, simplicity and accuracy which makes it different from other numerical methods such that Galerkin (Nadukandi et al. (00)), Least square method (Ren et al. (0)) and finite difference (Jha (03)). In this method, an approximate solution is substituted into the differential equation to form the

2 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 (00)), convection-diffusion problems (Rocca et al. (005)) and nonlinear Lane Emden type equations (Peirce (00)). 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 (975), Sun (000), Parand et al. (00)). Consider the second order non-linear parabolic boundary value problem: y θ z y y + = ε t θ t (x,t) (0,) (0,T) () z = f (y) () y q y + q = 0 at x = 0 (3) y q 3 y + q4 = 0 at x = (4) y = at t= 0 (5) where ε,θ, q, q, q 3, q 4 are constants Cubic Hermite polynomials Hermite interpolating polynomials was first introduced by Charles Hermite (8-905). 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 n+ and therefore, cubic Hermite interpolating polynomials particular case of general Hermite interpolating polynomials for n=. It consists of two node points and two tangents in cubic polynomial and defined as 3 3 H ( x) = x 3x H ( x) = x x + x +

3 Solution of non linear singular perturbation equation 5399 H ( + x 3 3 x) = x 3 H 3 4 ( x) = x x Fig.. Cubic Hermite interpolating polynomials Figure.. shows the behavior of the Cubic Hermite interpolating polynomials. The values of H,H, H 3 and H 4 lies within [0, ] as x goes from 0 to and their derivatives are unity or zero at the end points. The Cubic Hermite approximation is defined as: 4 y l (u) = a l i ( t) H i ( u) Where l =,,.,k (6) i= l Where, k is the number of elements and a i (t) s are the continuous functions of t in l th element.{h i (u)} are piecewise cubic Hermite polynomials as defined above. l The first and second order discretized derivatives of the trial function y taken at j th collocation point are defined by A ji and B ji respectively, where A ji =H i (u j ) and B ji =H i (u j ). After applying Hermite collocation method, the following set of collocation equations is obtained: l l y 4 4 j θ z j ε l l + = a ( ) i t B ji ai ( t) Aji ; l=,,,k; j=,3 (7) t θ t h h z = l j f ( y l ) l i= 4 q q y + ai ( t) A i = 0 h l i= l i= (8) at x = 0 (9)

4 5400 Happy Kumar, Shelly Arora and R. K. Nagaich 4 k q4 k q3 y4 + ai ( t) A4i = 0 at x = (0) hl i= The equations from (7) to (0) can be put into the matrix form as: a ' = Sa () Where S = H ( εb A) and H be the coefficient matrix of Cubic Hermite polynomials at j th collocation points. P B ε A= Q P Q P 3 Q P k Q k Figure : The resulting matrix ε B A arises from HCM. The system ε B A has the block-tridiagonal structure. Where P,Q and P k, Q k are collocation matrices of block 3 in first element and last element, respectively. Remaining collocation matrix P i, Q i (i=,3,..k-) with each block 4. The resulting stiff system of k differential algebraic equations (DAE s) is solved using MATLAB with the ode5s 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, ]. 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.

5 Solution of non linear singular perturbation equation 540 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, θ y y y + = ( y ) ε θ + η t (x,t) (0,) (0,T) () y y ε = 0 at x = 0, for all t 0 (3) y = 0 at x =, for all t 0 (4) y = at t = 0, for all x (5) The set of equations () to (5) 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 0.0 to at x=. The values of η and ϴ are taken to be 0.63 and 0.93, respectively. It is evident from this figure that the numerical values are converging to zero as increase t even for ε >0.0, which show that the method is stable and convergent for any value of ε. Fig. 3. Behavior of solution profiles at x = for different values ofε.

6 540 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 0.05 and 0.63, 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 = for different values of ϴ. In figure 5 the solution profiles are presented graphically for ε=0.05, ϴ=0.94 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.

7 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 = 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 = are increasing from, 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==

8 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 =0.63 and ϴ=0.93. However, for number of elements 7 or 7 the solution profiles converge to 0 for t 4, whereas for number of elements 35 the solution profile converges to 0 for t. 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=0.0667, ϴ=0.94 and η=0.63. The numerical results were obtained using ode5s 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 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.

9 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=0.065, ϴ=0.94 and η=0.63. The peak position of solution profiles is 3.77,.77 and.93 at t=0., 0. 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.

10 5406 Happy Kumar, Shelly Arora and R. K. Nagaich In Figure 0 displays numerical solution of the problem at t=0. 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. 0. Behavior of solution profiles for different values of ε. In figure, numerical results are presented in the form of breakthrough curves for ε ranging from 0.5 to at t=0.. 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.. Behavior of solution profiles for different values of ε.

11 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.4-786/0 (SR)). References. A. Peirce (00). A Hermite cubic collocation scheme for plane strain hydraulic fractures. Computer methods in applied mechanics and engineering; 99, A.L. Rocca, A.H. Rosales, & H. Power (005). Radial basis function Hermite collocation approach for the solution of time dependent convection- diffusion problems. Engineering analysis with boundary elements; 9(4), A.W. Lang & D.M. Sloan (00). Hermite collocation solution of near-singular problems using numerical coordinate transformations based on adaptivity. Journal of computational and applied mathematics; 40( ), B. Jumarhon, S. Amini & K. Chen (000). The Hermite collocation method using radial basis functions. Engineering analysis with boundary elements; 4(7 8), E. Ghanaei & M.R. Rahimpour (00). Evaluation of orthogonal collocation and orthogonal collocation on finite element method using genetic algorithm in the pressure profile prediction in petroleum reservoirs. Journal of petroleum science and engineering; 74( ), F.S. Rohman & N. Aziz (0). Optimization of batch electrodialysis for hydrochloric acid recovery using orthogonal collocation method. Desalination; 75 (-3),

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