Simple shooting-projection method for numerical solution of two-point Boundary Value Problems

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1 Simple shooting-projection method for numerical solution of two-point Boundary Value Problems Stefan M. Filipo Ivan D. Gospodino a Department of Programming and Computer System Application, University of Chemical Technology and Metallurgy, Kl. Ohridski 8 Blvd., Sofia 756, Bulgaria Corresponding author idg@cornell.edu

2 Keywords Ordinary differential equations; two-point boundary value problem; second-derivative preservation; simple shooting-projection method Abstract This paper presents a novel shooting algorithm for solving two-point Boundary Value Problems (BVPs) for ordinary differential equations. The algorithm includes the following steps: First, a value for the initial condition at the first boundary is guessed and a forward numerical integration of the differential equation is performed so that an Initial Value Problem (IVP) solution, called a shooting trajectory, is obtained. The shooting trajectory starts from the first boundary constraint but typically does not end at the second boundary constraint. Next, the shooting trajectory is transformed into a projection trajectory that is an approximate BVP solution. The projection trajectory satisfies both boundary constraints and has the same second derivative as the shooting trajectory. Finally, the projection trajectory is used to correct the value of the initial condition and the procedure is repeated until convergence.. Introduction This paper studies numerical solutions of two-point BVPs of second-order ordinary differential equations. Probably the most intuitive numerical method for solving two-point BVPs is the simple (single) shooting method [], []. In the simple shooting method one makes a guess for the value of the initial condition at the first boundary and performs numerical integration of the differential equation to obtain a shooting trajectory. The shooting trajectory is an IVP solution but is not a BVP solution. The end of the shooting trajectory differs from the second boundary constraint by some distance, called error. Next, one corrects the initial condition at the first boundary in an effort to decrease the absolute value of the error in the next shooting, and repeats this procedure until an initial condition that produces a zero error is reached. Thus, any simple shooting method for solving two-point BVPs is, in fact, an iterative root finding method for finding the zero of the error as a function of the initial condition. One of the most prominent root finding methods is the Newton s method [3], [4], which at each iteration evaluates the derivative of the error with respect to the initial condition and uses this derivative to produce an estimate for the zero of the error. A variant of this method is the constant-slope Newton s method [5], which evaluates the derivative only at the first iteration and then uses this value of the derivative throughout. There are also methods that do not use derivatives. Instead, they utilize the value of the error at two or more initial conditions in order to estimate the zero of the error. Prominent examples of such methods are the bisection method [6], the secant method [6], and the Muller s method [7]. The proposed simple shooting-projection method for solving two-point BVPs differs from the above-mentioned methods in the way the zero error estimate is sought. At each iteration the method starts from some initial condition, finds the corresponding IVP solution (the shooting trajectory), transforms this IVP solution into an approximate BVP solution, called a projection trajectory, and uses this projection trajectory to obtain a zero error estimate used as an initial condition in the next iteration. The projection trajectory is an approximate BVP solution because by design it has the same second derivative as the shooting trajectory and meets both boundary constraints. As it will become evident in the analysis below, this method is akin to the constant-slope Newton s method but it is derived from different premises and has important differences. The paper is organized as follows: first, the transformation of the IVP solution into an approximate BVP solution is described for functions of one independent variable; then the simple shootingprojection algorithm is formulated; then, the convergence analysis of the algorithm is performed; and finally the algorithm is tested by solving several types of two-point BVPs.

3 . Finding an approximate BVP solution using an IVP solution Let u be a real valued function of a real independent variable t [a, b]. If u satisfies the second-order ordinary differential equation u (t) = f (t, u(t), u (t)), t (a, b), () together with two initial conditions u(a) = u a, u (a) =, () then u is an IVP solution. If u satisfies () together with two boundary constraints u(a) = u a, u(b) = u b, (3) then u is a two-point BVP solution. Since a BVP solution is much harder to find than an IVP solution, we propose a simple way of finding an approximate BVP solution u using a given IVP solution u. This u satisfies the differential equation () approximately and satisfies the boundary constraints (3) exactly. As it will be shown below, this approximate BVP solution could be easily incorporated into an iterative shooting procedure for finding the exact BVP solution. A natural choice of u, which leads to simple mathematical derivations, is one that satisfies the following differential equation: u (t) = u (t), t (a, b), (4) together with the two boundary constraints u(a) = u a, u(b) = u b. (5) In (4) u is the sought unknown function, while u is the given IVP solution. Since u satisfies () we can write: u = f (t, u, u ) = f (t, u, u ) + R, t (a, b), (6) where f (t, u, u ) is expanded around u, and R stands for higher order terms. quation (6) tells us that if R is small, then u is an approximate solution to the differential eqn. (). Since, on the other hand, u satisfies the two boundary constraints (5), it follows that u is an approximate BVP solution. If f is constant or depends only on t, then u is an exact BVP solution. Since the approximate BVP solution (4), (5) is used in an iterative procedure it does not necessarily have to be a close approximation. The iterative procedure will, in general, work even if R in eqn. (6) is not small. quation (4), complemented by eqn. (5), defines a second-derivative preserving transformation of the IVP solution u into an approximate BVP solution u. It is easy to see that this transformation is a projection of u into u. Indeed, if we denote this transformation by P, then u=pu. On the other hand, since u satisfies the boundary conditions, u=pu. Thus, Pu= u=pu=p u, or, since this is true for any u, P=P, which is a definition of a projection operator. We call the function u (or the graph {(t, u), t [a, b]}) a shooting trajectory and the function u (or the graph {(t, u), t [a, b]}) a projection trajectory. An interesting property of the second-derivative preserving transformation is that u minimizes the square of the H seminorm of the difference u-u: 3

4 b S = ( u u ) dt. (7) a In (7), u is not a variable but the given IVP solution. If the function u, with two fixed boundary values (5), is to minimize (7) then the uler-lagrange equation must hold: d dt L L =, u u (8) where L=(u -u ). From where: d ( u u ) =, dt (9) which leads to eqn. (4). quation (7) tells us that the first derivative of u is minimally deviated from the first derivative of u, i.e. the two functions have similar shapes in this sense. 3. Algorithm for solving two-point BVPs using second-derivative preserving transformation of trajectories The following simple shooting-projection algorithm for solving two-point BVPs, summarized graphically in fig. below, is proposed: Step : Make a guess for the initial condition Step : Obtain a shooting trajectory We introduce v = u and numerically integrate (), using the uler s method, as a system of two first-order differential equations. We start from the first boundary constraint u a and the initial condition to obtain the following shooting trajectory (see fig. ): u i = u i- + h v i-, v i = v i- + h f(t i, u i, v i- ), i =,, N, () with initial values u = u a, and v =, where u i and v i are the values of u and t t i =a+h(i-); h =(b-a)/(n-) is the size of the integration step; and (N-) is the number of integration points. The shooting trajectory u is a numerical IVP solution that starts from the first boundary constraint (i.e. u =u a ) but does not meet the second boundary constrain (i.e. u N u b ). The distance u N -u b is denoted by and is called error. Step 3: Obtain a projection trajectory In order to obtain the projection trajectory we discretize eqn. (4) using the central difference scheme u i+ - u i + u i- = u i+ - u i + u i-, i =,, N-, () and impose the two boundary constraints (5): u = u a, u N = u b. () The solution of () together with () yields the projection trajectory (see fig. ): u i = u i - (u N - u b )(i-)/(n-), i =,, N. (3) 4

5 The projection trajectory u is an approximate BVP solution that is obtained from the shooting trajectory u, provided by Step. quation (3) is the sought second-derivative preserving projection transformation u=pu. Note, that the derivations in Step 3 are independent of the integration scheme used to obtain the IVP solution u in Step. Step 4: Obtain a new, corrected initial condition v a According to eqn. (3), the first two points of the projection trajectory are: u = u, u = u - (u N - u b )/(N-). (4) Subtracting the first equation in (4) from the second, dividing by h, and taking into account that h(n-)=b-a, a simple final expression for the new initial condition v a is obtained: v a = - ( )/k, (5) where v a =(u - u )/h, =(u - u )/h, ( )=u N ( ) - u b, and k=b-a (see fig.). Step 5: Replace in Step with v a The proposed algorithm can simply be described as: guess an initial condition; shoot to obtain an IVP solution; obtain an approximate BVP solution using this IVP solution; shoot again in the direction suggested by the approximate BVP solution, etc. Note, that eqn. (3) is used only to derive eqn. (5), which, in fact, does not contain any projection trajectory points. It contains only the endpoint u N of the shooting trajectory. Therefore, while carrying out the numerical procedure the actual construction of the projection trajectory is not necessary, i.e. Step 3 is virtual. This step was presented only to derive the mathematical equations and for conceptual clarity. Thus, the presented simple shooting-projection algorithm repeats steps, 4, and 5 until the shooting trajectory satisfies the second boundary constraint within a prescribed tolerance, i.e. u N u b. 3 5 shooting trajectory projection trajectory u N u 5 u b 5 slope u u 3 u N u a -5 u =u u u a slope t t 3 t Fig.. Second-derivative preserving transformation of the shooting trajectory u into a projection trajectory u. In the iterative algorithm is replaced by v a in the next iteration. Most shooting methods use at least two shooting trajectories per iteration in order to obtain an estimate for the zero of the error. quation (5) shows that the proposed simple shooting-projection method uses only one shooting trajectory to achieve the same. Instead of a second shooting trajectory the method makes use of a (virtual) projection trajectory that is an approximate BVP solution and therefore it is suitable for obtaining an estimate for the root of ( )=. b 5

6 Step 3 of the algorithm is equivalent to a Picard s iteration step [8], but only if u is an IVP solution. The Picard s iteration method is a relaxation method [6] that, if started from the IVP solution u, would produce the same u in the first iteration as the simple shooting-projection method, but will proceed to relax u without resorting to shooting. Thus, the Picard s method will not preserve the second derivative in the following iterations. Like other relaxation methods, and unlike the shooting methods, the Picard s method has convergence issues when the two-point BVP solution is oscillatory. In the Results section an example will be shown where the Picard s and other relaxation methods fail to converge whereas the proposed simple shooting-projection method converges. The form of eqn. (5) is similar in structure to the form of the corresponding equations for the secant, the Newton s shooting, and especially so to the constant-slope Newton s shooting methods. The properties of eqn. (5) are discussed in the next section. 4. Convergence analysis of the simple shooting-projection algorithm A plot of the error as a function of the initial condition is shown on fig. a below with a thick line. The point A in the figure corresponds to the starting initial condition. As can be seen from eqn. (5), the new initial condition v a is obtained by drawing a line with a slope k=b-a through point A. The next initial condition is obtained by drawing a line with the same slope k through point B. ach next initial condition is obtained analogously. Thus, the root of ( )= is reached after successive construction of parallel lines of slope k, shown in fig. a with thin lines. A main feature of the simple shooting-projection method is that the slope k is constant. In this sense the method is akin to the constant-slope Newton s method, where the slope is also constant, but has a different value the value of d/d evaluated at the starting initial condition. Since the starting is arbitrary, this slope will also be arbitrary and may not lead to convergence unless the starting happens to be close to the root. In contrast, the slope k does not depend on the starting. Its value b-a comes from the boundaries of the BVP. It is the most meaningful value to use under the assumptions, used to derive eqn. (5). Other shooting methods, such as the secant and the Newton s shooting methods, differ from the proposed simple shooting-projection method in that they readjust k at each iteration. Readjusting k could be an advantage, but sometimes it could be a drawback, as will be demonstrated in the Results section. Let m be the slope of ( ) at the root. Analogously to the constant-slope Newton s method, if close to the root the simple shooting-projection method converges [7] when - m/k <. (6) Otherwise the method diverges. The method converges linearly except for the trivial case m=k where the convergence is quadratic. Since k is always positive the convergence criterion above leads to the four cases, graphically shown in fig.. A A B k B k m m (a) (b) 6

7 k A m m k B A B (c) (d) Fig.. Convergence/divergence sequences of the simple shooting-projection method for the following four possible cases: (a) convergence when k > m >, (b) convergence when m/ < k < m, (c) divergence when k < m/, and (d) divergence when m <. 5. Results The simple shooting-projection algorithm for solving two-point BVPs was tested on several types of second-order ordinary differential equations. xample. Consider the following two-point BVP u = exp(u), t (, ), (7) u() =ln[(tan +)/], u() =ln[(tan (5/)+)/], with an analytical solution: u(t) =ln[(tan (t / +3/)+)/]. (8) Figure 3 traces the evolution of the shooting and the corresponding projection trajectories obtained after iterations,, 3, and 4 using the simple shooting-projection method. As discussed previously, the projection trajectories are not needed for the iteration process but they are shown in the figure to illustrate the algorithm and to demonstrate the goodness of the approximation for the BVP solution. The first shooting trajectory, shown in fig. 3a, corresponds to a starting initial condition =. The first projection trajectory predicts a negative new initial condition v a. The shooting trajectory in fig. b is obtained using this initial condition. The second projection trajectory is already very close to the analytical solution, shown by the solid curve. Convergence within a prescribed tolerance <. is reached at iteration 6. 7

8 3.5.5 shooting trajectory projection trajectory analytical solution u.5 u t t (a) (b) u.5 u t t (c) (d) Fig. 3. volution of the shooting and projection trajectories when solving eqn. (7) using the simple shooting-projection method for N=3 and starting = at: (a) iteration ; (b) iteration ; (c) iteration 3; (d) iteration 4. Figure 4 plots the graph of ( ) (which is found numerically) and the sequence of parallel lines with slope k=b-a that lead to the root of ( )=. The figure shows that the value of k is close to the value of the slope of at the root Fig. 4. Convergence sequence for the iterative solution of eqn. (7) using the simple shootingprojection method for a starting initial condition = and N=3. The root of ( )= is =

9 xample. Consider the following two-point BVP u = -3u u /t, t (, ), (9) u() = /, u() = / 5, with an analytical solution: u ( t) = t / + t. () The shape of ( ) for eqn. (9) is shown in fig. 5. This is a typical case for which the Newton s shooting method and the constant-slope Newton s shooting method will converge only if the starting initial condition is close to the root. For starting =5 the simple shooting-projection method converges within a prescribed tolerance <. in 7 iterations in the way shown in fig. 5a. For the same starting =5 the Newton s shooting method diverges in the way shown in fig. 5b. Starting from the same =5 the constant-slope Newton s shooting method oscillates around the root perpetually without being able to converge. The first few iterations are shown in fig. 5c. simple shooting-projection method Newton shooting method modified Newton shooting method (a) (b) (c) Fig. 5. Convergence/divergence sequences for the iterative solution of eqn. (9) for a starting initial condition = and N= using: (a) the simple shooting-projection method; (b) the Newton s shooting method; and (c) the constant-slope Newton s shooting method. The root of ( )= is = xample 3. Consider the following two-point BVP u = -u cosh(tu)/, t (, ), () u() =, u() =. This equation has no explicit analytical solution. The function ( ) for eqn. () has multiple roots, four of which are shown in fig. 7 below. We set out to find the solution that corresponds to root 3. Starting from = the simple shooting-projection method finds this solution in iterations within a prescribed tolerance <.. The evolution of the shooting trajectory at each iteration (numbered through ) is shown in fig. 6. 9

10 5 4 3 u t Fig. 6. volution of the shooting trajectory when solving eqn. () using the simple shootingprojection method for N= and starting = (corresponding to shooting trajectory ). Convergence is reached after iterations (corresponding to the shooting trajectory ). Figure 7 shows how root 3 of ( )= is reached. The consecutive values of at each iteration are indicated with,, 3, in the figure. They correspond to the shooting trajectories, shown in fig. 6. The function ( ) has three local extrema between root and root 3. Unlike other methods, the simple shooting-projection method does not have convergence issues around such extrema. It simply marches through them and finds root 3. In fact, the convergence of the method is not influenced by the shape of around the root. It is easy to see that the simple shooting-projection method will find root 3 for any starting that lies between root and root 4. simple shooting-projection method 3 9 root root root 3 root Fig. 7. Convergence sequence for the iterative solution of eqn. () using the simple shootingprojection method for a starting initial condition = (point ) and N=. The root 3 of ( )= is = Figure 8 demonstrates the divergence of the secant shooting method when solving eqn. () starting with the two initial conditions = (point ) and =- (point ). The figure shows that at iteration 6 the secant line becomes nearly horizontal and therefore the secant method practically fails. Although some other pairs of initial conditions may happen to find root 3 this would require a considerable trial-anderror effort.

11 secant shooting method root root root 3 root Fig. 8. Divergence sequence for the iterative solution of eqn. () using the secant shooting method for a starting initial conditions = (point ) and =- (point ), and N=. When Newton s shooting method is used to solve eqn. () for starting = it fails to find root 3. Instead, it finds root 4. The constant-slope Newton s method, started from =, finds root 3 but, as can be seen in fig. 8, if the method is started from any between root and root 3, corresponding to a negative slope of or to a positive slope of that is close (or equal) to zero, the method will fail to find root 3. Figure 6 shows that the solution of the two-point BVP () is oscillatory. As mentioned previously, the relaxation methods, like the Jacobi [9], the Picard s, or the Newton s relaxation [6] methods, have convergence issues with oscillatory solutions, while the shooting methods, in general, do not. All three relaxation methods were tried for solving eqn. (). The relaxation methods require a starting trajectory guess that does not necessarily have to be an IVP solution. In order to meaningfully compare these methods to the simple shooting-projection method, as a starting trajectory guess we used the shooting trajectory, corresponding to = and N= (corresponding to trajectory in fig. 6 and to point in fig. 7). Then all three relaxation methods failed to converge. Figures 4, 5a, and 7 show that the value of k is close to the value of the slope of at the root for all examples, considered in this work. From the presented results one can see that the majority of the well-known existing shooting and relaxation methods for solving two-point BVPs fail to find the solution of eqn. () corresponding to root 3. As discussed, the proposed simple shooting-projection method will find this solution, regardless of the starting, as long as it lies between root and root Conclusion This paper described a novel simple shooting method for numerical solution of two-point BVPs. The method starts with a guess for the initial condition, obtains a shooting trajectory, and then transforms this shooting trajectory into a projection trajectory that is an approximate two-point BVP solution. From the projection trajectory a new initial condition is obtained, which is used for the next iteration. This idea leads to a simple, easy to analyze iteration formula for correcting the initial condition. It was demonstrated that at certain cases the method outperforms prominent shooting and relaxation methods. The proposed simple shooting-projection method has three important features: (i) it uses only one shooting trajectory per iteration in order to correct the initial condition; (ii) the slope k in the iterative correction formula (5) is a constant with a value that comes from the boundaries and thus it does not depend on the value of the initial condition ; and (iii) the projection step is a relaxation step.

12 A recommendation for future work is to perform an analysis that shows how the value of the slope k=b-a relates to the value of the slope of at the root of ( )=. This would shed additional light on the convergence properties of the method. Acknowledgements We would like to thank Dr. Stephen Vavasis, Dr. Grigor Georgiev, Dr. Stefka Dimova, Dr. Geno Nikolov, Dr. Jordanka Angelova, and Ana Avdzieva for the valuable discussions. References [] Herbert B. Keller. (968) Numerical Methods For Two-Point Boundary-Value Problems. Blaisdell. [] Herbert B. Keller. (976) Numerical Solution of Two Point Boundary Value Problems. SIAM. [3] Newton, I. (664-67) Methodus fluxionum et serierum infinitarum. [4] Raphson, J. (69) Analysis aequationum universalis. London. [5] Steward. G. W. (996) Afternotes on Numerical Anlysis. SIAM. [6] Press, W. H., Flannery, B. P., Teukolsky, S. A., Vetterling, W. T. (99) Numerical Recipes in FORTRAN: The Art of Scientific Computing, nd ed. Cambridge, ngland: Cambridge University Press. [7] Muller, David. (956) A Method for Solving Algebraic quations Using an Automatic Computer, Mathematical Tables and Other Aids to Computation,, 8-5. [8] Picard. (893) Sur lapplication des metodes dapproximations succesives a letude de certains equations differentielles ordinaires, Journal de Mathmatiques Pures et Appliques, 9, 7 7. [9] Bronshtein, I. N. and Semendyayev, K. A. (997) Handbook of Mathematics, 3rd ed. New York: Springer-Verlag.