Spline Computation for Solving Magnetohidrodynamics Free Convection Flow

158 | P a g e M a y J u n e , 2 0 1 3 Spline Computation for Solving Magnetohidrodynamics Free Convection Flow Faraidun K. Hamasalh, Joseph G. andNajmadeen G. 1 Department of Mathematics,Collegeof Science Education, University of Sulaimani  faraidunsalh@gmail.com 2 Faculty of Science, University of Zakho, Kurdistan -Iraq.  Joseph1955@yahoo.com 3 College of Science Education, University of Sulaimani, Kurdistan-Iraq. Najm1961@gmail.com


INTRODUCTION
Many problems in applied sciences and engineering are modeled as system of differential equations such as spring-mass systems, bending of beams, Magnetohidrodynamics free convection flow (MHD), chemical reactions and so forth can be formulated in terms of differential equations. Since the system of differential equations has wide applications in scientific research, so we consider the convective flow in fluid saturated porous medium which has been the subject of several recent papers. Therefore faster and accurate numerical solutions to this problem is very important, see( [1][2][3], [5], and [12]).
There are several methods that can be used to solve the nonlinear problems numerically. A broad class of analytical solutions methods, such as Runge-Kutta of order six, Taylors series, Hirota's bilinear scheme and Hereman's method as [6][7][8], [10] and [12], were used to handle these problems. However, some of spline approximation had been proposed by ( [2], [4] and [11]]) solved the system of differential equations and some initial value problems.
In the present paper, we discussed the convergence analysis of the ninth spline method for system of differential equations with new constraint and boundary conditions, also we apply this new model for solving magnetohidrodynamics free convection flow and mass transfer over a stretching sheet which has been analyzed numerically including the dufour and sort effects. We will use the function, two, fourth and seventh boundary conditions, to constructed the ninth spline with two initial conditions.

DESCRIPTION OF THE METHOD
We present a ninth spline interpolation for one dimensional and for a given sufficiently smooth function where are unknown values to be determine.

CONVERGENCE ANALYSIS
In this section, we investigate the convergence analysis of the method for ninth degree spline function which are developed by [2], [4], [10] and [11]. The equation (2) yields the two initial conditions M a y -J u n e , 2 0 1 3 Now the coefficient matrixof the above system of equations can be found, in the unknowns , i=1, 2,…,n-1which is a non-singular matrix and hence all the coefficients are determined uniquely.
By taking the eight derivatives and subtracting the function,using Taylor's series expansion on ) ( Also, from (2) and bytaking the seventh derivatives, we get (7) (7) and i c depend on the numbers of intervals.
Also, from (3) and taking the seventh derivatives, we get .
And by taking the sixth derivatives with using Taylor's series expansion on ) ( by the same process way we can find that: (4) and (5)

NUMERICAL ILLUSTRATION
In this section, the nonlinear differential system of Magnetohidrodynamics free convection flow is presented and this problem is referred to [8] and [9]. The problems are tested to the efficiency of the development solutions, and to demonstrate its convergence computationally. The problems have been solved by using our method with different values of step size h; it's tabulated in Tables 1. These show that our results are more accurate.  The absolute of maximum error with respect to derivatives defined as [4]:

CONCLUSION
The approximate solutions of the nonlinear problems as differential system of Magnetohidrodynamics free convection flow by using ninth spline interpolation show that our method is better in the sense of accuracy and applicability. These have been verified by maximum absolute errors given in the tables it changes with respect to the step size of the tolerance and various problems. Some properties of spline are obtained which are required in proving the uniqueness, existences and convergence analysis of the present method.