A Quintic Spline method for fourth order singularly perturbed Boundary Value Problem of Reaction-Diffusion type

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1 A Quintic Spline method for fourth order singularly perturbed Boundary Value Problem of Reaction-Diffusion type Jigisha U. Pandya and Harish D. Doctor Asst. Prof., Department of Mathematics, Sarvajanik College of Engineering & Technology, Surat Professor, Department of Mathematics, Veer Narmad South Gujarat Unersity, Surat ABSTRACT We consider quintic spline collocation for fourth order singularly perturbed boundary value problem in which the highest order derate is multiplied by a small parameter. The quintic spline collocation method, finite difference method and Spline method due to Blue are tested on an example. The results obtained using quintic spline collocation method are found to be in agreement with the exact solution gen in the literature. KEY WORDS: Quintic spline collocation, Singular perturbation, Fourth order ordinary differential equation, Upper triangular matrix.. INTRODUCTION The approximate solution of boundary value problems with a small parameter affecting highest derate of the differential equation is described. This class of problems has recently gained importance in the literature for two main reasons. Firstly, they occur frequently in many areas of science and engineering, for example, combustion, chemical reactor theory, nuclear engineering, control theorylasticity, fluid mechanicstc. Secondly, the occurrence of sharp boundary-layers as ε, the coefficient of highest derate, approaches zero creates difficulty for most standard numerical schemes. There exist a variety of techniques for solving singularly perturbed boundary value problems [-]. The numerical solution of two point boundary value problems using splines has been considered by many authors [,6-9] and references therein. Most of the papers concerning computational aspects are confined to second order equations. Only a few results are available for higher order equations. Singularly perturbed higher-order problems are classified on the basis of how the order of the original differential equation is affected if one sets by Roos et al [] where, is a small posite parameter assigned with order of the differential equation. We say that if the order of the differential equation is reduced by one, then the Singular Perturbation Problem (SPP) is said to be of convection diffusion type and if the order is reduced by two, then it is called reaction diffusion type. There are a variety of techniques for solving singularly perturbed boundary value problems. O Malley [], Howes [], and Zhao [3] dered analytical results of third order nonlinear SPPs. Feckan [] considered higher order problems and his approach is based on the nonlinear analysis involving fixed-point theory, Leray-Schauder theorytc. A FEM for convectionreaction type problems is described by G. Sun et.al. [,6] and Semper [7]. Roberts [8] suggested a numerical method of finding an approximate solution for third order ODEs. He introduced a technique called Boundary Value Technique (BVT). In section, we suggest spline collocation method to obtain solutions of the Boundary Value Problems (BVPs) of higher order SPODEs. The boundary value technique to find numerical solution for third order SPODEs subject to certain types of boundary conditions is suggested by Valarmathi et al [9]. This work is an extension from third order to fourth order. In the present paper, we solve the fourth order singularly perturbed boundary value problem using quintic spline collocation method.. SINGULARLY PERTURBED FOURTH ORDER ORDINARY DIFFERENTIAL EQUATION OF REACTION DIFFUSION TYPE Consider the boundary value Problem y ( x) b( x) y"( x) c( x) y( x) f ( x), x D, () y() p, y() q, y"() r, y"() s () where is a small posite parameter, and b(x), c(x) and f(x) are sufficiently smooth functions satisfying the following conditions: 3- May B.V.M. Engineering College, V.V.Nagar,Gujarat,India

2 bx ( ), cx ( ),,, is arbitrary close to, for some D (, ), D [, ], and y c ( D) c D The above problem comes from the reference [], which states that the deflection of an elastic beam with small flexural rigidity under tension subject to a specified load and stability problems in fluid dynamics leads to the Orr-Sommerfield equation presented by O'Malley[] and B.Semper [7]. This yield fourth-order singularly perturbed differential equations, whose order decreases by two when one sets. Section 3 describes the quintic spline collocation method to solve such type of singularly perturbed fourth-order ODE of reaction-diffusion type. 3 Quintic Spline Collocation : The fifth degree spline is used to find numerical solutions to the boundary value problems discussed in equation () together with equation (). A detailed description of spline functions generated by subdision is gen by De Boor []. Consider equally spaced knots of a partition π: a xo x x... xn b on [a,b]. Let S [π] be the space of continuously differentiable, piecewise, Quartic polynomials on π. That is, S [π] is the space of Quartic polynomials on π. The Quintic spline is gen by Bickley [8] and by Micula et al []. 3 s( x) a b ( x x ) c ( x x ) d ( x x ) (3) n e( x x) Fk( x xk) k where the power function (x x k ) + defined as (x x k ) + = x - x k, if x > x k =, if x x k Consider a third order linear boundary value problem of the form ''' '' ' y ( x) p( x) y ( x) q( x) y ( x) r( x) y ( x) t( x) y( x) m( x); a x b subject to the boundary conditions y yn ' yn " yn "' y ' yn yn " yn "' y " yn yn " yn "' 3 y "' 3 yn 3 yn " 3yn "' 3 where y( x), p( x), q( x), r( x), t( x), m( x ) are continuous functions defined in the interval x [ a, b] ;,,, 3 are finite real constants. Let equation (3) be an approximate solution of equation (), where a, b, c, d, F,F,, F n- are real coefficients to be determined. Let x,..., x x n be n+ grid points in the interval [a,b], so that xi a ih, i=,, n; x a, xn b, h=(b-a)/n. () It is required that the approximate solution (3) satisfies the differential equation at the point x=x i. Putting (3) with its successe derates in (), we obtain the collocation equations as follows: n xi xk p ( xi ) xi xk ek k 3 q( xi ) xi xk r( xi ) xi x k 6 p( xi )( xi x) q( xi ) xi x d 3 r ( xi ) xi x 6 c p( xi ) q( xi ) xi x r( xi ) xi x ( i ) ( i ) i ( i ) b p x r x x x a r x From boundary conditions, n k mx ( i ), i,,,... n (6) 3 Fk ( b xk ) ( b xk ) e ( b a) ( b a) d b a c b ( ) a ( ) 7 3- May B.V.M. Engineering College, V.V.Nagar,Gujarat,India

3 n k 3 Fk ( b xk ) ( b xk ) ( b xk ) e( b a) ( b a) ( b a) 3 d ( b a) ( b a) 6 c ( b a) b ( b a) a ( ) 8 3 n ( b xk) ( b xk) Fk k ( b x ) k e ( b a) ( b a) ( b a) 3 d ( b a) ( b a) 6 c ( b a) ( b a) b b a a ( ) ( ) n ( b xk) ( b xk) Fk k 3 ( b x ) k 3 3 e3( b a) ( b a) ( b a) 3 3 d 3 3( b a) ( b a) c 3 ( b a) b 3( b a) a ( ) 3 3 Using the power function (x x k ) + in the above equations a system of n+ linear equations in n+ unknowns a, b, c, d, n is thus obtained. This system can be written in matrixvector form as follows Where AX=B () X= [e n- n-,, d, c, b, a ] T B,,, m( x ), ( ),..., ( ), ( ) T n m xn m x m x The coefficient matrix A is an upper triangular Hessenberg matrix with a single lower sub diagonal, principal and upper diagonal having non-zero elements. Because of this nature of matrix A, the determination of the required quantities becomes simple and consumes less time. The values of these constants ultimately yield the Quintic spline s(x) in equation (3). In case of nonlinear boundary value problem, the equations can be converted into linear form by quasilinearization method [3] and hence this method can be used as iterate method. The procedure to obtain a spline approximation of y i (i=,, j; where j denotes the number of iteration) by an iterate method starts with fitting a curve satisfying the end conditions and this curve is designated as y i. We obtain the successe iterations y i s with the help of an algorithm described as above till desired accuracy. NUMERICAL ILLUSTRATION AND DISCUSSION Consider the boundary value problem y ( x) y"( x) y( x) f ( x), y(), y(), y"(), y"(), x( x) where f( x) 8 6 x / ( x) / ( x) / ( x) / e e e e 6 / e The solutions of above problem are obtained through the Quintic spline collocation method, Blue's method and Finite difference method which are exhibited in table-.blue's method and Finite difference method ge good results, are reflected in table-. Exact solutions, which are presented by S May B.V.M. Engineering College, V.V.Nagar,Gujarat,India

4 Shanthi et.al.[] are also shown in table- and also in figure- so that one can easily find the closed form solution of the above problems. Table Numerical Solutions of Fourth Order SPODE (h =.,.) x y(x) Quintic Spline Blue's Finite Difference Exact Solution Figure-:Numerical Solutions of Fourth Order SPODE As mentioned in section, the second order singularly perturbed differential equations have been studied extensely, but only few results on the higher-order problems are available in the literature. We presented numerical methods, like spline collocation method, finite difference method, Blue s method to solve boundary value problems for fourth-order SPODE. In section, the singularly perturbed fourth order ordinary differential equation of reaction diffusion problem is solved using spline collocation method. The comparison table- shows that Quintic spline ges better approximation than Blue's method & Finite difference method and the convergence is also faster. In addition, spline method is easy to implement for boundary value problems for higher order singularly perturbed ordinary differential equations. Figure- provide a good agreement of results presenting actual as well as approximate solutions, which proves the reliability of the spline collocation methods. REFERENCES [] Kadalbajoo M.K., Bawa R. K., 996, Variable mesh difference scheme for singularly perturbed boundary valye problems using splines, J. OpTIM. Theory Appl. 9 (), pp. -6. []Kadalbajoo M.K., Reddy Y.N., 989, Asymptotic and numerical analysis of singular perturbation problems: A survey, Appl. Math. Comput. 3, pp [3] Natesan S., Ramanujam N., 999, A Booster method for singular perturbation problems arising in chemical reactor theory, Appl. Math. Comput.,pp []Stojanovic M., 996, Spline difference methods for a singularly perturbation problem, Appl. Numer. Math. (3) pp [] Taiwo O.A., Evans D.J., 999, On the solution of singularly-perturbed boundary value problem by a Chebyshev method, Int. J. Comput. Math. 7, pp. -9. [6] Al-Said E. A., 998, Cubic spline method for solving two point boundary value problem, Korean J. Comput. Appl. Math. (3) pp [7] Aziz A.K., 97, Numerical Solution of Two Point Boundary-Value Problem, Blaisdal, New York. [8]Bickley W.G., 968, Piecewise cubic interpolation and two point boundary value problems, Comput. J., pp [9] Fyfe D.J., 969, The use of cubic splines in the solution of two point boundary-value problems, Comput. J., pp []Roos H.G., Stynes M.and Tobiska L., 996, Numerical methods for Singularly Perturbed Differential Equations- Convection-Diffusion and Flow Problems, Springer, Berlin. [] O Malley R.E., 99, Singular Perturbation s for Ordinary Differential Equations, Springer-Verlag, New York,. [] Howes F.A., 983, The asymptotic solution of a class of third-order boundary value problem arising in the theory of thin film flow, SIAM J. Appl. Math 3(), pp [3] Zhao W., 99, Singular Perturbations of boundary value problems for a class of third order nonlinear ordinary differential equations, Journal of Differential Equations 88(), pp [] Feckan M., 99, Singularly perturbed higher order boundary value problems, J. Differential Equations III () pp May B.V.M. Engineering College, V.V.Nagar,Gujarat,India

5 [] Sun G., Stynes M., 99, Finite element methods for singularly perturbed higher order elliptic two point boundary value problems I: reaction-diffusion type, IMA J. Numer. Anal. pp [6] Sun G., Stynes M., 99, Finite element methods for singularly perturbed higher order elliptic two point boundary value problems II: convection-diffusion type, IMA J. Numer. Anal. pp [7] Semper B.,99, Locking in finite element approximation of long thin extensible beams, IMA J. Numer. Anal., pp [8] Roberts S.M., 98, A boundary value technic for singular perturbation problems, J. Math. Ana. Appl. 87 pp [9] Valarmathi S., Ramanujam N.,, A computational method for solving boundary value problems for third-order singularly perturbed ordinary differential equations, Appl. Math. and Comput. 9 pp [] Shanthi V. and Ramanujam N., 3, Asymptotic numerical methods for singularly perturbed fourth order ordinary differential equations of reaction-diffusion type, Comput. And Math. With Appl., 6, pp [] De Boor. C; 978, A practical Guide to Splines, Springer-Verlag, New York. [] Micula G., Micula S., 998, Handbook of splines, Kluwar academic press, Netherlands. [3]Bellman R.E., Kalaba, 96, Quasilinearization and nonlinear boundary value problems, American Elsevier Publishing Company, INC, New York. 3- May B.V.M. Engineering College, V.V.Nagar,Gujarat,India