Taylor Series Method with Numerical Derivatives …

1 Taylor Series Method with Numerical Derivatives forNumerical Solution of ODE Initial Value ProblemsE. MileticsG. Moln arkaDepartment of Mathematics, Sz echenyi Istv an University, Gy ISSN 1418-7108:HEJ Manuscript no.: ANM-030110-BAbstractThe Taylor Series Method is one of the earliest analytic-numeric algorithms forapproximate solution of initial value problems for ordinary differential this algorithm is not applied frequently. This is because when one solvessystems of ordinary differential equations, calculating the higher order derivativesformally is an over-elaborate task, this is true even if one uses the computer algebraicsystems such as MATHEMATICA or MAPLEV. The other reason is that only theexplicit versions of this algorithm are known. The main idea of the rehabilitation ofthese algorithms is based on the approximate calculation of higher Derivatives us-ing well-known technique for the partial differential equations.

2 In some cases suchalgorithms will be much more complicated than a R-K methods , because it will re-quire more function evaluation than well-known classical algorithms. However theseevaluations can be accomplished fully parallel and the coefficients of truncated Tay-lor Series can be calculated with matrix-vector operations. For large systems theseoperations suit for the parallel computers. These algorithms have several advanta-geous properties over the widely used classical methods . The approximate solutionis given as a piecewise polynomial function defined on the subintervals of the wholeinterval and the local error of this solution at the interior points of the subintervalis less than that one at the end point. This property offers different facility foradaptive error control. We remark that for the explicit Taylor Series methods ispossible to give its implicit extension.

3 Using the fact, that the approximate solutionis a continuous function, (in the case of implicit version it is a continuously differ-entiable function), adaptive examination and control of some qualitative propertiesof algorithms will be more simple than the case when the approximate solution aregiven only at discrete grid points. This paper describes several above-mentionedalgorithms and examines its consistency and stability properties. It demonstratessome Numerical test results for systems of equations herewith we attempt to provethe efficiency of these new-old : ANM-030110-B21 IntroductionThe Taylor Series algorithm is one of the earliest algorithms for the approximate solutionfor initial value problems for ordinary differential equations. Newton used it in his calcu-lation [1] and Euler describe it in his work [2].

4 Since then one can find many mentions ofit such as J. Liouville [3], G. Peano [4], E. Picard [5]. Many authors have furter developedthis algorithm, see for example A. Gibbons [12] and R. E. Moore [13]. The basic idea ofthese developments was the recursive calculation of the coefficients of the Taylor Numerical algorithms for the solution of ordinary differential equations arealso based on the Method of the Taylor Series . Each algorithm, such as the Runge-Kuttaor the multistep methods are constructed so that they give an expression depending ona parameter (h) called step size as an approximate solution and the first terms of theTaylor Series of this expression must be identical with the terms of the Taylor Series ofthe exact solution. These are the consistency and the order conditions for the expressions potentially can be evaluated at any value of the parameter (h), butpractically the evaluation is realized only at grid points.

5 Such algorithms give the value ofthe approximate solution at grid points. For such algorithm main cost is the number of thefunction evaluation. (Remark: some algorithm contains more than one parameter.) Thesealgorithms differ from others in its order, stability properties and its cost of overview of the modern algorithms on can find in the monograph of E. Harier, S. and G. Wanner [6] - [7]. A possible implicit extension of the Taylor Series algorithmis given in [9]. Actually, in the qualification of the algorithm become important theirquality properties such as conservativity, pozitivity preserving [8], monotonity preserving,detecting and following the bifurcation appearing the parallel computers as the complexity of algorithm new cost functionsmust be defined because in this case the main goal is to minimize of the execution timeand not the number of function evaluations.

6 From this point of view several variant ofthe Method of the Taylor Series could be an effective family of such algorithm can be derived from the classical Method of the Taylorseries by approximating the Derivatives in Taylor coefficients with Numerical Formulation of the problemThe problems to be solved are as follows:y (x) =f(x, y(x)), y(x0) =y0,(1)wherex [x0,x0+T], y(x) = [y1(x), y2(x), .., yn(x)] :R Rn,yi(x) Cp+1([x0, x0+T]), i= 1, .., nfor a givenpandf(x, y(x)) = [f1(x, y(x)), f2(x, y(x)), .., fn(x, y(x))] us introduce the following notations:Y(x) = [x, y1(x), .., yn(x)]T, F(Y(x)) = [1, f1(Y(x)), .., fn(Y(x))] this notation the equation (1) is as follows:Y(x) =F(Y(x)), Y(x0) = [x0, y1(x0), y2(x0), .., yn(x0)]T.(2)HEJ: ANM-030110-B3 Assume, that the functionsfi(x,(y(x)), i= 1, .., nare (p+1)-times continuously dif-ferentiable functions with respectx.)

7 Then the Method of the Taylor Series is based onthe following Taylor s Series expansion of the solution. Using our compact notation theTaylor Series of the solution of (2) is:Y(x0+s) =Y(x0) +Y (x0)s+12!Y (x0)s2++13!Y (x0)s3+..+1(p+ 1)!Y(p+1)(x0+ s),(3)where could be different for the components and the value ofpmight be chosen depend-ing on the smoothness of the right hand side in (1) and the desired order of the methodto be Numerical approximation of the derivativesThe main idea of the construction if the Method of Taylor algorithm with numericalderivatives is the Numerical approximation of the derivativesY(k)(x0),k= 2,3,4, .. The first Derivatives are given by the right hand side of (1), so its approxi-mation is only an Numerical approximation of the second derivativesThe second derivative of the right hand side of (2) called its Jacobian is:Y (x0) =F (x0) = [0, f 1(Y(x0)).]

8 , f n(Y(x0))]T== [0,( f1(x0) Y, F(x0)), ..,( fn(x0) Y, F(x0))]T= F YF,(4)where the Derivatives fk(x0) Y, k= 1,2, .., nare to be approximated by Numerical us introduce the following notations:Fk,i(Y(x0)) = [1, f1(.., yk(x0) +ih, ..), .., fn(.., yk(x0) +ih, ..)]T,k= 1, .., n, i= 0, 1, 2, ..(5)wherek= 0 means the first coordinate of the vectorYwhich is simplyxandh >0 is agiven small calculation of the valuesFk,i(Y(x0)) means the evaluation of the right hand sideof (2) at the given points. Using the finite difference formulae to approximate the firstderivatives we get that the followingA, B, C, andDmatrices approximate the generalizedJacobi matrix of the right hand side of (2) with different degree of us define theA, B, CandDmatrices by the following formulae:A=1h[(F0,1 F0,0),(F1,1 F1,0), ..,(Fn,1 Fn,0)],(6)B=12h[(F0,1 F0, 1),(F1,1 F1, 1).

9 ,(Fn,1 Fn, 1)],(7)HEJ: ANM-030110-B4C=16h[(F0, 2 6F0, 1+ 3F0,0+ 2F0,1), ..,(Fn, 2 6Fn, 1+ 3Fn,0+ 2Fn,1)],(8)D=112h[(F0, 2 8F0, 1+ 8F0,1 F0,2), ..,(Fn, 2 8Fn, 1+ 8Fn,1 Fn,2)].(9)Lemma 1(The order of accuracy of approximation.) Assume that every components ofthe functionFis five-times continuously differentiable with respect to all its variables insome neighbourhood ofxandyi(x), i= 1,2, .., n. Then the matrixAapproximatesthe generalized Jacobi matrix F Yby first order degree of accuracy while the matrixBbysecond order degree of accuracy, the matrixCby third order degree of accuracy and thematrixDby fourth order degree of accuracy at the pointxin the following sensemax0 i,j n(| Fi Yj Aij|) c1h,max0 i,j n(| Fi Yj Bij|) c2h2, ..,max0 i,j n(| Fi Yj Dij|) c4h4,wherec1, c2, .., c4are constants not depending on the only outline for the matrixA.

10 For the otherB,CandDmatrices theproof is similar. The expressions Fi Yjdenote the partial Derivatives fi(x, y1, y2, .., yn) yj, i, j= 1,2, .., n andAij=1h(fi(.., yj+h, ..) fi(.., yj, ..)).From this by the (6) we get thatAij= fi(x,y1,y2,..,yn) yj+cijh, i, j= 1,2, .., we choose max0 i,j n|cij|=c1, which gives the appropriate constant for the first 2 The different accuracy order approximations of the generalized Jacobi matrixwe need for construction of the different accuracy order Numerical algorithm but for higherorder methods we have to approximate the higher order Derivatives of the right hand sideof (2). Numerical approximation of the third derivativesThe third derivative of the right hand side of (2) is the coefficient of the third term in theTaylor expansion (3) is:Y =F = ( F YF) = ( 2F Y2F F) + F Y( F YF).(10)HEJ: ANM-030110-B5 The second term in (10) can be approximated by using the matricesA,B,CandD.