Solution of non linear singular perturbation equation using Hermite collocation method

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 e. 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 2 1 e θ θ (x,t)∈(0,1)×(0,T) (1) ) (y f z = (2) 0 2 1 = ∂ ∂ + x y q y q at x = 0 (3) 0 4 3 = ∂ ∂ + x y q y q at x = 1 (4) 1 = y at t= 0 (5) where e,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 1 3 2 ) ( 2 3 1 + − = x x x H x x x x H + − = 2 3 2 2 ) ( Solution of non linear singular perturbation equation 5399 2 3 3 3 2 ) ( x x x H + − = 2 3 4 ) ( x x x H − = 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: ) (u y l =∑ = 4 1 ) ( ) ( i i i u H t a l Where l = 1, 2,.......,k (6) Where, k is the number of elements and ) (t ai l ’s are the continuous functions of ‘t’ in l th element.{Hi(u)} are piecewise cubic Hermite polynomials as defined above. The first and second order discretized derivatives of the trial function l y taken at j th 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