An Efficient Method for Solving Multipoint Equation Boundary Value Problems

Transcription

1 Vol:7, No:3, 013 An Efficient Method for Solving Multipoint Equation Boundary Value Prolems Ampon Dhamacharoen and Kanittha Chompuvised Open Science Index, Mathematical and Computational Sciences Vol:7, No:3, 013 waset.org/pulication/7803 Astract In this wor, we solve multipoint oundary value prolems where the oundary value conditions are equations using the NewtonBroyden Shooting method (NBSM).The proposed method is tested upon several prolems from the literature and the results are compared with the availale exact solution. The experiments are given to illustrate the efficiency and implementation of the method. Keywords Boundary value prolem; Multipoint equation oundary value prolems, Shooting Method, NewtonBroyden method. I. INTRODUCTION OUNDARY value prolems of ordinary differential Bequations appear in a large domain of sciences and many practically important prolems lead to Multipoint Boundary Value Prolems (MBVP's).They are used to descrie a large numer of physical, iological and chemical phenomena. The wors of Prescott [10] and Timosheno [16] on elasticity, the monographs y Mansfield [5] and the wor of Dulácsa [4] on the effects of soil settlement are rich sources of such applications. Let us consider the oundary value prolem: Differential equation: f(y'(t), t) = 0 Boundary value condition: g(y'(t 0 ), Y'(t 1 ),..., Y'(t )) = 0 (1a) (1) where t R, Y'(t) = (y (n) (t), y (n1) (t),..., y'(t), y(t)) R m (n+1), f R m, y(t) R m, and g R m n. If = 0 the prolems ecome the initial value prolem. The prolems are called a twopoint oundary value prolem if = 1, and a multipoint oundary value prolem if > 1. Under certain conditions on the differential equations and the oundary conditions, systems (1a), (1) have solutions (or a unique solution). The proof of existence, and uniqueness, for some oundary value prolems is possile [6,0,1], ut for the A. Dhamacharoen is with the Department of mathematics, Faculty of Science, Burapha University, Bangsaen, Chonuri 0131, Thailand; ( ampon@uu.ac.th). Kanittha Chompuvised is with the Department of mathematics, Faculty of Science, and Technology, Nahon Ratchasima Raahat University, Nahon Ratchasima, Thailand; ( oein@hotmail.com). maority of prolems, providing proof is difficult, and so for this reason they remain unproved. Also, solving the prolems y analytical methods are rare since most of the prolems encountered were difficult, with complicate differential equations or complicate oundary conditions. On the contrary, various numerical methods were developed to approximately solve the prolems. Several techniques, such as shooting method [15,19] Adomian decomposition method (ADM) [,1], differential transform method (DTM) [18], variational iteration method (VIM) [9], successive iteration [13], splines [17], homotopy perturation method (HPM) [8,11], homotopy analysis method (HAM) [14] have een used to handle these types of prolems. In this paper, we concentrate on method of solving the multipoint oundary value prolem where the oundary conditions are equations, which we shall call the multipoint equation oundary value prolem. We adapted a method which was success in finding the approximate solution of twopoint BVPs (see []). The method maes use of the shooting method, ut has also adapted the technique of updating the initial values. Several examples are given to illustrate the efficiency and implementation of the method. II. THE NEWTON AND BROYDEN SHOOTING METHOD The process of shooting method is as follows: Firstly, guess the initial value z = Y'(t 0 ) and solve the differential equation (1a) with this initial value to get Y'(t i ), i = 1,,..., m; chec whether the condition (1) is satisfied; if not, adust the initial z and repeat until the condition (1) is met. More precisely, for the +1point oundary value prolems: Differential equation: f(y'(t), t) = 0 (a) Boundary value condition: g(y'(t 0 ), Y'(t 1 ),..., Y'(t )) = 0 () Let z = Y'(t 0 ) and F(z) = g(z, Y'(t 1 ),..., Y'(t ))), where Y'(t) and Y'(t ), = 1,,...,, are considered to e functions of z implicitly through the equation (a). The equation () can e written in the form F(z) = 0 (3) To solve (3) numerically, the Newton method has the scheme: z i+1 = z i [F'(z i )] 1 F(z i ), i = 1,, 3,... (4) International Scholarly and Scientific Research & Innovation 7(3) ISNI:

2 Vol:7, No:3, 013 Open Science Index, Mathematical and Computational Sciences Vol:7, No:3, 013 waset.org/pulication/7803 where F'(z) is the Jacoian matrix of the function F. The method is called Newton Shooting Method if (3) is solved using the Newton scheme (4). The most difficult tas in this prolem is to find the value F'(z i ) in each iteration, since the function F has implicit form, and its value is derived from solving the differential equations (1a). Let S(t, z) = Y'(t), we have F(z) = g(z, S(t 1, z),..., S(t, z)). Therefore: F'(z) = g where, and, = 1,,...,, are Jacoian matrices with appropriate sizes. The values and will e otained directly from (). The value i can e computed y =, = 1,,..., n. t= t From (1a) differentiating oth sides with respect to z gives: [ f (Y'(t), t)] = 0 Let p'(t) = = Y'(t), then can e otained from z z solving the system of linear differential equations f [ (Y'(t), t)] p'(t) = 0 (6) for p(t) and then set = p(t ), = 1,,..., n. Note that computing the value of F'(z) in each iteration requires so much wor, therefore, any method which could e utilized to avoid this wor, and which also wors satisfactorily well, would e preferred. To avoid the difficult calculation of F'(z), the Broyden scheme z i+1 = z i D 1 i F(zi ), i = 1,, 3,... (7) can e used to replace (3), that is, replacing F'(z) with a nonsingular matrix D i. With the appropriate initial z 0 and D 0, and with the Broyden updating D i+1 = D i + a T 1 where a = F(zi+1 T ), and = z i+1 z i, the scheme (7) will produce the sequence that hopefully converges to the solutions. In solving the system of ordinary equations, the Broyden method performs well, provided that a good initial point z and D were chosen. In case of convergence, the Broyden method has qsuper linearly order of convergence [7]. So, compared to the Newton method, which has quadratic convergence, ut involves a large amount of computation, the Broyden method may sometimes e preferale. Unfortunately, it is sometimes difficult to come up with a good initial guess, and thus the (5) solutions cannot e found. For the oundary value prolems with simple twopoint oundary conditions, experiments have showed that the Broyden wors well most of the time, even for nonlinear differential equations [3]. If the oundary conditions are equations, the Broyden still wors well if the differential equations are linear. However, many times, it can not reach a satisfactory solution if the differential equations are nonlinear. III. THE NEWTONBROYDEN SHOOTING METHOD Now, let's consider the function of the form The Newton scheme of this prolem is F(u(x), x) = 0 (8) x i+1 = x i [F'(u(x i ), x i )] 1 F(u(x i ), x i ), i = 0, 1,,... (9) where F'(u(x i ), x i ) = F u (u(x i ), x i )u'(x i ) + F x (u(x i ), x i ) Replace u'(x i ) y D i, gives x i+1 = x i [F u (u(x i ), x i )D i + F x (u(x i ), x i )] 1 F(u(x i ), x i ), i = 0, 1,,... (10) with updating: 1 D i+1 = D i + T (u(x i+1 ) u(x i ) D i )T (11) where = x i+1 x i. This method retains the good part of Newton method, and replaces the difficult part of Newton's y Broyden's. With a good initial guess for z 0 and D 0, the NewtonBroyden scheme (10) will produce the sequence that converges to a solution of (8), with qsuper linearly order of convergence [1]. Consider the function of the form: F(u 1 (x), u (x),, u (x), x) = 0 (1) where x, u (x) R m, = 1,,...,. The derivative is: F'(u 1 (x), u (x),, u (x), x) = Fu ( u 1( x),..., u( x), x) u '(x) 1 = + F x (u 1 (x), u (x),, u (x), x) The Newton scheme of this prolem is x i+1 = x i [ Fu ( u( x i), xi) u '(x i ) + F x (u(x i ), x i )] 1 F(u(x i ), x i ), 1 = i = 0, 1,, (13) where u(x) = (u 1 (x), u (x),, u (x)). Replace u '(x i ) y D i, gives x i+1 = x i [ Fu ( u( x i), xi) D i + F x (u(x i ), x i )] 1 F(u(x i ), x i ), 1 = i = 0, 1,, (14) with updating: International Scholarly and Scientific Research & Innovation 7(3) ISNI:

3 Vol:7, No:3, 013 Open Science Index, Mathematical and Computational Sciences Vol:7, No:3, 013 waset.org/pulication/7803 D i+1 = D i + where = x i+1 x i. 1 (u T (x i+1 ) u (x i ) D i ) T (15) Consider the equation (5) in the form of (13), the derivatives tae the forms: (F u1 (u(x i ), x i ), F u (u(x i ), x i ),..., F u (u(x i ), x i )) =, u '(x i ) =, = 1,,...,, and F x (u(x i ), x i ) = g. In using the scheme (14), the matrix at the ith iteration was replaced y D i, and the Broyden updating technique was used to otain D i+1. The full process (which should e called the NewtonBroyden Shooting method (NBSM)) will e as follows: S 1. Let Y'(t 0 ) = z 0 and a matrix D 0 (same size of ) e given. Solve the differential equation (a) with the initial values z 0 to otain Y'(t i ), i = 1,,...,. Compute F = F(z) = g(y'(t 1 ), Y'(t ),...,Y'(t ), z). Let u i = Y'(t ), = 1,,...,.. For i = 0, 1,,... do the following: g Compute, =, = 1,,...,, and H = + g D 1 = i Solve the system of linear equations H = F for. Let z i+1 = z i +. Solve the differential equation (a) with the new initial values z i+1, to otain Y'(t ), = 1,,...,. Compute u i+1 = Y'(t ) and F = F(z i+1 ) = g(y'(t 1 ), Y'(t ),...,Y'(t ), z i+1 ). Compute T 1 and a = (ui+1 T u i D i ), and update D i y D i+1 = D i + a T 3. Stop the iteration if i n, a prescried positive integer, or F < ε, or < ε, a given small numer. In the algorithm, the matrix was replaced y D i, so that sipping the step which involves solving the system of linear differential equations can save a large amount of wor, in terms of the numer of computation and implementation. The success of the method for solving oundary value prolems generally depends on two phases, which are namely the method of solving the initial value prolems, and the method of solving systems of equations. In this proposed method, the first part use the standard technique of solving the initial value prolems, such as the Taylor's method or the RungeKutta's method; the second part use the new technique t for solving the system of nonlinear equations derived y A. Dhamacharoen [1]. IV. EXPERIMENTS Various systems of oundary value prolems, with ordinary differential equations and multipoint equation oundary values conditions, have een solved using the scheme in section III. The prolems are as follow: Prolem 1: We consider the following thirdorder linear ordinary differential equation: 3 t x = tx + (t t 5t 3)e 1 x ( 0.5) x ( 0.5) x (1) + x(0) + x (0)x (1) = e e x (0)x(1) + x ( 0.5) x (0) + 4x (1) x (1) = 0 x (0)x ( 0.5) x ( 0.5) + x(1) x(0)x ( 0.5) = 0 t The exact solution of this prolem is x(t) = (t t )e Define the vectors x = (x(0), x (0), x (0)) ; u 1 (x) = x 0.5,x 0.5,x 0.5 and u (x) = ( ( ) ( ) ( )) ( x1,x ( ) ( 1,x ) ( 1) ), the function F(u 1 (x),u (x), x) is the vector in the oundary condition. The value of u can e otained y solving the differential equation with the initial value x, so that the function F is well defined. The Jacoian u ( x ) could e computed y solving the system of linear differential equations. However, use the approximating matrix D instead. Prolem : We consider the following thirdorder nonlinear ordinary differential equation: t x = e x 1 x( 0) + x ( 0.5) x(1) x (1) x(0.5) = 1 e x(0) x(0)x(1) + x( 0.5x(0.5) ) + x(1) = 1+ e x (0)x () 1 x () 1 + x (0.5) + x(0) x ( 0) x (0.5) = 0 t The exact solution of this prolem is, x(t) = e Prolem 3: We consider the following fourthorder nonlinear ordinary differential equation: (4) 7 x + xx 4t 4= 0 x (0) + x (0)x (0.5) + x (0.5) x (1) + x (1) 4x (0.5) = 3 x(0) + x (0)x (1) x (1) + x (0.5)x (0.5) 16x(0.5) + x(0.5) = 6 x (0) + x (0) + x (1) x (0.5) + x (0.5) x (0.5) = 5 x (0) + x (1)x (0.5) x (0.5) + x(0.5) x (0.5) = 6 4 The exact solution of this prolem is, x(t) = t. Define the vectors x = (x(0), x (0), x (0), x (0)) ; x 0.5, x 0.5, x 0.5, x (0.5) ; u 1 (x) = ( ( ) ( ) ( ) ) International Scholarly and Scientific Research & Innovation 7(3) ISNI:

4 Vol:7, No:3, 013 Open Science Index, Mathematical and Computational Sciences Vol:7, No:3, 013 waset.org/pulication/7803 u (x) = ( x ( 0.5 ), x ( 0.5 ), x ( 0.5 ), x (0.5)) u 3 (x) = ( x1,x ( ) ( 1,x ) ( 1,x(1) ) ) and, the function F(u 1 (x), u (x), u 3 (x), x) is the vector in the oundary condition. The value of u can e otained y solving the differential equation with the initial value x, so that the function F is well defined. Prolem 4: We consider the following nonlinear system of secondorder ordinary differential equation: y + tx + xy = t t + t + t t+ 3 x ty + x = t t + 6t x (0)y(0.5) + x (0)x (1) + y(1) = x (1) + y (0.5) + y (1) x(0.5)y(0) = 1 4x (0.5) y (0) + x(0) + y (1)x (1) = = y(0.5) x (0.5) x(0.5) y (0) x (0) y(1) 0 The exact solution of this prolem is, y(t) = t t. 3 x(t) = t t and Define the vectors x = (x(0), x (0), y(0), y (0)) ; u 1 (x) = x 0.5, x 0.5, y 0.5, y (0.5) and u (x) = ( ( ) ( ) ( ) ) ( x1,x ( ) ( 1,y1,y(1) ) ( ) ), the function F(u 1 (x), u (x), x) is the vector in the oundary condition. The value of u can e otained y solving the differential equation with the initial value x, so that the function F is well defined. The aove prolems were solved using three methods for comparison; the Broyden Shooting Method (BSM), the Newton Shooting Method (NSM) and the NewtonBroyden Shooting Method (NBSM). The computations for each prolem were made y using the same initial guess for all methods. The suitale initial value, and the numer of iterations (N) are shown in Tale I. The numer of operations were counted, and the saving cost of the proposed methods, which is the percentage of saving numer of operations relative to the Newton method, is shown in Tale II. The results show that the NBSM can compete with NSM. Although the numer of iterations of the NBSM to reach the solution is more than that of the NSM, overall the numer of operations is less than that of the NSM. TABLE I COMPARISON OF VARIOUS ITERATIVE METHODS Initial guess z 0 Solution Suitale Initial values for the solutions BSM NSM NBSM Prolem 1 N = 14 N = 10 N = 9 (, 0, 0.5) (0, 1, 0) (3.6E08, , 1E09) (E09, , 1.6E08) (1.E08, , 9E09) Prolem N = 64 N = 13 N = 7 (1.5, 1.5, 1.5) (1, 1, 1) ( , , ) ( , , ) ( , , ) Prolem 3 N = 11 N = 16 (1, 1, 1, 1) (0, 0, 0, 0) Diverge (6.870E09,.718E09,.4E10, 1.1E10) (1.077E08,.794E09,.44E10, 7.31E09) Prolem 4 N = 18 N = 18 (1, 1, 1, 0) (0, 1,0, 1) Diverge (9.90E09, , 6.994E09, ) (7.680E09, , 7.659E09, ) TABLE II THE NUMBER OF OPERATIONS AND THE SAVING COST OF ITERATIONS Initial guess Flops Saving(%) z 0 BSM NSM NBSM NBSM Prolem 1 (,0,0.5) Prolem (1.5,1.5,1.5) Prolem 3 (1,1,1,1) Diverge Prolem 4 (1,1,1,0) Diverge A. Figures and Tales Comparisons of the solution from the NBSM with the exact values are given in tale IIIVI elow. We oserved from the tale that the values otained y the NBSM are reasonaly closed to the exact values, and the NBSM still remain efficient, even for higher order BVP and nonlinear systems of BVP. TABLE III RESULT OF PROBLEM 1 t Exact solution NBSM solution Error E E E E E E E E E E E E E E09 International Scholarly and Scientific Research & Innovation 7(3) ISNI:

5 Vol:7, No:3, 013 Open Science Index, Mathematical and Computational Sciences Vol:7, No:3, 013 waset.org/pulication/7803 TABLE IV RESULT OF PROBLEM t Exact solution NBSM solution Error E E E E E E E E E E E09 TABLE V RESULT OF PROBLEM 3 t Exact NBSM solution Error solution E E E E E E E E E E E11 TABLE VI RESULT OF PROBLEM 4 t Exact solution x(t) NBSM solution Error E E E E E E E E E E E09 V. CONCLUSION The multipoint equation oundary value prolem, can e solved using the NewtonBroyden Shooting method. The experiments have shown that the method wored well, as expected, which is to say that the good initial guesses were easily found. ACKNOWLEDGMENT The author would lie to than the referees for their comments. This research was supported y the Research Fund from the National Research Council of Thailand). [3] A. Prachanuru, Broyden method in solving equations oundary value prolems, Master's thesis, Department of Mathematics, Graduate School, Burapha University (007). [4] E. Dulácsa, Soil Settlement Effects on Buildings,Developments Geotechn. Engrg., vol. 69, Elsevier, Amsterdam,199. [5] E. H. Mansfield, The Bending and Stretching of Plates Internat. Ser. Monogr. Aeronautics and Astronautics, vol. 6 Pergamon, New Yor, [6] H. B. Keller, Existence theory for two point oundary value prolems, Bull. Amer. Math. Soc., 7 (1966), [7] J. E. Dennis, and R. B. Schnael, Numerical Methods forunconstrained Optimization and Nonlinear Equations, SIAM,1966. [8] J. H. He, Homotopy perturation method for solving oundary value prolems, Phys. Lett. A. 350 (006), [9] J. H. He, Variational approach to the sixthorder oundaryvalue prolems, Appl. Math. Comput. 143 (3) (003), [10] J. Prescott, Applied Elasticity, Dover, New Yor, [11] M. A. Noor, S.T. MohyudDin, An efficient algorithm for solving fifthorder oundary value prolems, Math. Comput. Model. 45 (78) (007), [1] M. Tatari, M. Dehgan, The use of the Adomian decomposition method for solving multipoint oundary value prolems, Phys. Scr. 73 (006), [13] Q. Yao, Successive iteration and positive solution for nonlinear secondorder threepoint oundary value prolem, Comput. Math. Appl. 50 (005), [14] S. J. Liao, Beyond Perturation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC, Florida, 004. [15] S. N. Ha, A nonlinear shooting method for twopoint oundary value prolems, Comput. Math. Appl., 4 (1011) (001), [16] S. P. Timosheno, Theory of Elastic Staility, McGraw Hill, New Yor, [17] SiraUlIslam, I.A. Tirmizi, FazaliHaq, M.A. han, Nonpolynomial splines approach to the solution of sixthorder oundaryvalue prolems, Appl. Math. Comput. 195 (008), [18] SiraUlIslam, SiraulHaq, Javed Ali, Numerical solution of special 1thorder oundary value prolems using differential transform method, Comm. Nonl. Sci. Numeric. Simul. 14 (4) (009), [19] U. M. Ascher, R. M. M. Matthei, R. D. Russell, Numerical solution of oundary value prolems for ordinary differential equations, Englewood Cliffs: Prentice Halll (1988). [0] W. Jiang, M. Cui, Constructive proof for existence of nonlinear twopoint oundary value prolems., Appl. Math. Comput., 15 (5) (009), [1] Y. Tao, G. Gao, Existence, uniqueness and approximation of classical solutions to nonlinear twopoint oundary value prolems., Nonlinear Anal., 50 (7) (00), [] K. Chompuvised, A. Dhamacharoen, Solving oundary value prolems of ordinary differential equations with nonseparatedoundary conditions, Applied Mathematics and Computation,Volume 17, Issue 4, 15 August 011, Pages REFERENCES [1] A. Dhamacharoen, TwoStage Iteration Methods for Solving Systems of Nonlinear Equations, In The 15th Annual Meeting in Mathematics., Bango., Annu. 010, 68. [] A. M. Wazwaz, The numerical solution of sixthorder oundary value prolems y the modified decomposition method, Appl. Math. Comput. 118 (001), International Scholarly and Scientific Research & Innovation 7(3) ISNI: