MULTIPOINT BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS. Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University

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1 MULTIPOINT BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University HU-1117 Budapest, Pázmány Péter sétány 1/c. Abstract Here we present a numerical method to solve MBVP s of ODE s and its computer adaptation developed as software packages written in Turbo Pascal and C languages. 1. Introduction Boundary Value Problems of Ordinary Differential Equations appear in a large domain of sciences and many practically important problems lead to Multipoint Boundary Value Problems (MBVP s). Such is the case, e.g., for modelling and analysing problems arising from electric power networks, electric railway systems, telecommunication lines and also in chemistry, analysing kinetical reaction problems. Applications for difference frequency domain problems, e.g. harmonic penetration in unbalanced power networks, railway circuits with auto- or booster transformers, telecommunication circuits have been studied at the Technical University in Budapest and at the Swedish Transmission Research Institute in Sweden and problems have been solved ([1],[2]) using the algorithm and computer programs presented in this article. 1

2 The class of problems studied here has the following form: (1) x (t) = f(t, x(t)), t [t 1, t m ] R, x : [t 1, t m ] R n, f : [t 1, t m ] R n R n, n, m N +, m > 1, subject to multipoint boundary conditions (2) r(x(t 1 ), x(t 2 +),..., x(t m 1 +), x(t m )) = 0, r : (R n ) m R n, t 1 < t 2 <... < t m, (3) g j (x(t j ), x(t j +)) = 0, g j : (R n ) 2 R n, (j =2, 3,..., m 1), where x(t j ) and x(t j +) mean the left and right limits of x at point t j, respectively (j =2, 3,..., m 1). We assume that f is continuous and twice continuously differentiable with respect to its second variable on the regions [t j, t j+1 ] R n (j = 1, 2,..., m 1), and the functions r and g j are continuously differentiable. These requierements can be weakened, but for simplicity and since the applications generally satisfy these assumptions, we suppose them to be fulfilled. 2. Numerical algorithm To solve the MBVP (1) (2) (3) we apply an iteration method, based on a modified shooting method ([3],[4]). In each iteration step, we solve the initial value problems (4) x (t) = f(t, x(t)), t [t j, t j+1 ], (5) x(t j ) = s j, (j =1, 2,..., m 1), where the initial values s j R n are given in the first iteration step using one of the strategies described below (at paragraph 3.), and determined in the following iteration steps by the formula (12). 2

3 Let us denote the solutions of (4) (5) by ξ j ( ; s j ), j =1, 2,..., m 1. We have to determine the elements s j in such a way that the function ξ : [t 1, t m ] R n, defined by (6) ξ(t) := ξ j ( t ; s j ), t [ t j, t j+1 ), ξ(t m ) := ξ m 1 ( t m ; s m 1 ), satisfies the boundary conditions (2) (3), that is (7) r( s 1, s 2,..., s m 1, ξ m 1 ( t m ; s m 1 ) ) = 0, (8) g j ( ξ j 1 ( t j ; s j 1 ), s j ) = 0 (j =2, 3,..., m 1), hence ξ is a solution of the problem (1) (2) (3). Thus we have to find a solution of the equation F (s 1, s 2,..., s m 1 ) = 0, where F is defined by F : (R n ) m 1 (R n ) m 1, (9) F (s 1, s 2,..., s m 1 ) := r( s 1, s 2,..., s m 1, ξ m 1 ( t m ; s m 1 ) ), g 2 ( ξ 1 ( t 2 ; s 1 ), s 2 ),.. g m 1 ( ξ m 2 ( t m 1 ; s m 2 ), s m 1 ) To solve this equation we use the general Newton s iteration (10) (s 1, s 2,..., s m 1 ) (i+1) := (s 1, s 2,..., s m 1 ) (i) (DF ((s 1, s 2,..., s m 1 ) (i) )) 1 F ((s 1, s 2,..., s m 1 ) (i) ). Therefore, we have to compute F (s 1, s 2,..., s m 1 ), DF (s 1, s 2,..., s m 1 ) in each iteration step, and to solve the system of linear equations (11) DF (s 1, s 2,..., s m 1 ) ( s 1, s 2,..., s m 1 ) = = F (s 1, s 2,..., s m 1 ) and finally, using the solution of the above system we get the initial values s j R n for the next iteration step by 3

4 (12) s (i+1) j := s (i) j + s j (j =1, 2,..., m 1). To compute DF (s 1, s 2,..., s m 1 ) we determine the partial derivatives of F with respect to the k th components of its j th variable s j, (j = 1, 2,..., m 1, k = 1, 2,..., n). Since the functions ξ j ( ; s j ) do not depend on s l if j l and 2k ξ ji ( t j ; s j ) = δ ki, the [(j 1) n + k] th column of the Jacobian matrix can be written (13) for j =1 as 1k F (s 1, s 2,..., s m 1 ) = 1k r( s 1, s 2,..., s m 1, ξ m 1 ( t m ; s m 1 ) ) n 1i g 2 ( ξ 1 ( t 2 ; s 1 ), s 2 ) 2k ξ 1i (t 2 ; s 1 ) i=1 0 n }.. (m 3) 0 n (14) for j =2,..., m 2 as jk F (s 1, s 2,..., s m 1 ) = jk r( s 1, s 2,..., s m 1, ξ m 1 ( t m ; s m 1 ) ) 0 n }.. (j 2) 0 n 2k g j ( ξ j 1 ( t j ; s j 1 ), s j ) n 1i g j+1 ( ξ j ( t j+1 ; s j ), s j+1 ) 2k ξ ji (t j+1 ; s j ) 0 n }.. (m 2 j) 0 i=1 4 n

5 (15) and for j =m 1 as (m 1)k F (s 1, s 2,..., s m 1 ) = (m 1)k r( s 1, s 2,..., s m 1, ξ m 1 ( t m ; s m 1 ) ) + + n mi r( s 1, s 2,..., s m 1, ξ m 1 ( t m ; s m 1 ) ) i=1 2k ξ (m 1)i (t m ; s m 1 ) 0 n }. (m 3) 0 n 2k g m 1 ( ξ m 2 ( t m 1 ; s m 2 ), s m 1 ) where [ 0 ] is the zero column of n Rn ; ji r, 1i g j and 2i g j are the partial derivatives of r and g j with respect to the i th component of their j th, first and second variables, respectively. To compute the partial derivatives 2k ξ ji (t j+1 ; s j ) for j =1, 2,..., m 1, k = 1, 2,..., n, i = 1, 2,..., n, we solve the following linear initial value problems (16) y (t) = 2 f( t, ξ j ( t ; s j ) ) y(t), t [ t j, t j+1 ], (17) y(t j ) = e k, (j =1, 2,..., m 1), (k =1, 2,..., n), where e k is the k th unit vector in R n. Denoting the solutions by η j,k we have 2k ξ ji ( ; s j ) = (η j,k ) i ( j = 1, 2,..., m 1, k = 1, 2,..., n, i=1, 2,..., n). Let us summarize the main steps of the algorithm: We have to prepare the first iteration by choosing the starting elements s j, (the initial values of the Cauchy problems (4) (5)), then 5

6 in each iteration step we have to (i) solve the Cauchy problems (4) (5) and (16) (17) simultaneously, (ii) compute F (s 1, s 2,..., s m 1 ) using (9) and the solutions of the Cauchy problems (4) (5), (iii) compute DF (s 1, s 2,..., s m 1 ) using (13), (14), (15) and the solutions of the Cauchy problems (4) (5) and (16) (17), (iv) determine the elements s 1, s 2,..., s m 1 by solving the system of linear equations (11), (v) compute the new elements s j by the formula (12), (vi) test if these new elements s j can be taken as good approximations for the solution of the problem (1) (2) (3) or initial values (5) for the next iteration step. Since the Newton method is only locally convergent, we replace (12) by the correspondent formula of the modified Newton method ([4]), that is, in the step (v) we compute the new elements s j by (18) s (i+1) j := s (i) j + λ s j (j =1, 2,..., m 1), where λ is determined so that the method converge to the solution of the problem (1) (2) (3), provided it exists ([4],[5]). 3. Computer adaptation The main steps of crucial importance in the algorithm which have to be worked out very carefully in the computer programmation are as follows: (3.1) the choice of the values s j (j =1, 2,..., m 1) in the first iteration, (3.2) the numerical integration to solve the Cauchy problems (4) (5) and (16) (17), (3.3) the numerical method to solve the system of linear equations (11), (3.4) the numerical method to determine λ in the formula (18) of the modified Newton method. 6

7 The choice of the values s j (j =1, 2,..., m 1) in the first iteration step, and the computation of these values in the following iteration steps, have crucial importance not only for the convergence of the method. In some practical problems the partial derivative 2 f is unbounded on the regions [t j, t j+1 ] R n, and in such cases the solutions of the initial value problems (4) (5) and (16) (17) for certain values of s j can only be defined in a neighborhood of t j, and not on the whole intervals [t j, t j+1 ]. So the method would break down in these cases. Moreover, even if the initial value problems (4) (5) and (16) (17) can be solved in principle, in practice if the solutions depend very sensitively on the initial values, there can be considerable inaccuracies at the end of the intervals [t j, t j+1 ]. The computation even to full machine accuracy and by quasi-exact initial value problem solvers (codes) does not guarantee that the computed solutions can be determined accurately. These problems can be demonstrated by the well-known estimate (3.5) ξ j (t; s 1 j) ξ j (t; s 2 j) s 1 j s 2 j e L j t t j, where ξ j (t; s 1 j ) and ξ j(t; s 2 j ) are the values at t of the solutions of the initial value problem (4) (5) with s j := s 1 j and s2 j, respectively; and L j is a Lipschitz constant for the function f with respect to its second variable on the region [t j, t j+1 ] R n. The estimate (3.5) shows that the influence of inaccurate initial values s j can be reduced arbitrarily small if we define sufficiently close points in the intervals [t j, t j+1 ], t j =: t j1 < t j2 <... < t jl := t j+1, such that the values e L j t jk+1 t jk are sufficiently small. We can consider these intermediate points as if were given points (together with the endpoints t j and t j+1 ) for the MBVP problem (1) (2) (3). Simple continuity conditions have to be given for these newly defined points, such as (3.6) x(t jk +) x(t jk ) = 0, thus the boundary conditions (3) have to be completed by the mapping (3.7) g jk : (R n ) 2 R n, (u, v) u v. for the newly generated points t jk. 7

8 The function r in the boundary condition (2) has to be also modified. Its domain has to be completed corresponding to the newly defined intermediate points, but the mapping really does not change. Since we do not know the Lipschitz constants L j, in each iteration step we generate new t jk points if the magnitude of the solution of the initial value problems (4) (5) or (16) (17) at some point t becomes greater than a user prescribed value. While integrating the differential equations (4) and (16) on the intervals [t j, t j+1 ], we examine whether the values of ξ j (t; s j ) and η j,k (t) become too large to s j and 1, respectively. If (3.8) ξ j ( t; s j ) > γ ( s j + 1 ) or η j,k (t) > γ, then we terminate the integration and generate the point t as a new intermediate point t jk. The parameter γ has to be be given by the user. To continue the integrations on the interval [t jk, t j+1 ] the initial values in (5) are computed by linear interpolation from the output trajectory of the previous iteration or from the starting trajectory in the first iteration. It is important to mention that (i) (ii) (iii) for numerical reasons, the program uses greater γ in the second and in the following iterations than in the first iteration, sometimes, especially when large derivatives are expected for the trajectories, it is recommended to put intermediate points as given points of the multipoint boundary conditions (2) (3), the given points t j together with the program generated intermediate points t jk will be called shooting points in the following. The first estimations for the solution at the given points, i.e. the values of s j (j = 1, 2,..., m 1), have to be given by the user. This can easily be done when the qualitative behavior of the solution is known by the phisycal meaning of the problem, so at least a rough approximation can be found. Howewer, in complicated cases, the first approximation must be treated very carefully. In many cases the so-called continuation 8

9 or homotopy method is the most effective method, where we solve the problem gradually. Starting with a neighboring simpler problem, we go toward the solution of the actually posed problem, step by step, using the output result of the previous step as input data for the actually executed step. The homotopy procedure can easily be programmed by the following possibilities to give the first approximations. (3.9) The first approximative trajectory for the differential equation (1) is the constant zero function, thus, all components of s j equal zero in the first iteration (j = 1, 2,..., m 1). In this case the user does not have to prepare data before entering the program. (3.10) The first estimations of s j for the initial value problems (4) (5) are computed by interpolation from a data file which has to be given by the user. Data must be written in matrix form, the rows should contain points τ k (at which approximations are given for the trajectory) and the estimated components of ξ(τ k ). At least two points, t b and t e have to be given (together with the corresponding estimations of the trajectory) in such a way that the interval [t b, t e ] includes all given points t j. (3.11) The first estimations of s j for the Cauchy problems (4) (5) are interpolated from the results at the shooting points of the previous run, so this can be chosen only when the program run previously. The user does not have to give estimations, since the program automatically creates the necessary input data file from the results of the previous run. (3.12) The first approximations of the trajectory have to be found in a data file like in case (3.10). The set of given points will be enlarged so that the shooting points of the previous run will be embedded in the set of the actually given points. The estimations at these points are interpolated from the input data file, which is recommended to be the output results of the previous run. 9

10 (3.13) The program creates the given points t j together with the approximations s j automatically so that they will be exactly the same as they were in the last iteration of the previous run. The user does not have to give approximations, since the program utilizes the output of the previous run as input for the actual run. We emphasize that in the cases (3.9), (3.11) and (3.13) the program automatically creates the necessary input data file (by the constant zero function in (3.9) or by the output results at the shooting points of the previous run in (3.11) and (3.13) ), while in the cases (3.10) and (3.12) the input data file has to be created by the user. When the user applies the homotopy method, we propose to use the case (3.13). However, if the solutions of the differential equation (4) depend very sensitively on the initial values s j, the case (3.12) can be used the most efficiently (by more detailed approximations in the data file). For the integration of the differential equations (4) and (16) different one-step methods are implemented in the program package, such as (i) third, fourth, fifth-sixth and seventh-eighth order Runge-Kutta methods ([4],[6],[7]), (ii) fourth order Runge-Kutta-Fehlberg methods ([4]), (iii) fourth, fifth and sixth order ROW methods for stiff ODE s ([8],[9]). Each code is developed to work with automatic stepsize control (two algorithms are implemented for each code), and forward or backward direction can be chosed for the integrations. 10

11 Reference: [ 1 ] G.Varjú,K.Károlyi: Calculating screening effect of a metal cable sheath with nonlinearity, Int.Symp.EMC. Wroclaw,1990. ( ) [ 2 ] F.Jonas,G.Varjú: Gen.Model & Num.Method for Multicond.Systems Frequ.Dom., IEEE/KTH Pow.Tech.Conf.Stockholm,1995. ( ) [ 3 ] K.Károlyi: An interactive code to solve MBVP s, Int.Conference on Diff.Equations, Barcelona,1991. ( ) [ 4 ] J.Stoer,R.Bulirsch: Introduction to Numerical Analysis, Springer-Verlag [ 5 ] R.E.Bank,D.J.Rose: Glob.Appr.Newton Meth. Num.Math ( ) [ 6 ] P.J.Prince,J.R.Dormand: High order embedded Runge-Kutta formulae, J.Comp.Appl.Math.vol.7.no (67-75) [ 7 ] J.H.Verner: Explicit Runge-Kutta Methods with Estimates of the Local Error, Report 92.Univ.Auckland, New Zealand,1976. [ 8 ] P.Kaps,P.Rentrop: Generalized Runge-Kutta Meth.of Ord.Four with Steps Contr. for Stiff ODE s, Numer.Math.vol.33,1979 (55-68) [ 9 ] P.Kaps,G.Wanner: A Study of Rosenbrock-Type Methods of High Order, Numer.Math.vol.38,1981 ( ) 11