Two new algorithms for discrete boundary value problems

International Journal of Stochastic Analysis, Jul 2018

We propose two new methods of constructing the solutions of linear multi-point discrete boundary value problems. These methods are applied to solve some continuous two-point boundary value problems which are known to be numerically unstable.

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Two new algorithms for discrete boundary value problems

Journal of Applied Mathematics and Stochastic Analysis Two New Algorithms for Discrete Boundary Value Problems* Ravi P. Agarwal and Tara R. Nanda Ravi P. Agarwal 0 Tara R. Nanda 0 0 AMS Subject Classification: 65Q05 , 65L10 We propose two new methods of constructing the solutions of linear multi-point discrete boundary value problems. These methods are applied to solve some continuous two-point boundary value problems which are known to be numerically unstable. The purpose of this paper is to provide two new algorithms to compute the solution of the linear discrete system u(k + 1) = A(k)u(k) + b(k), k e N(k1, kr) r where A(k) is a given nonsingular n x n matrix with elements aqp(k), 1 q, p< n; b(k) is a given n x 1 vector with components bq(k), 1 <_ q n; u(k) is an finknown n x 1 vector with components uq(k), 1 < q _< n; 0 _< k 1 < k2 < ...< kr (r > 2 ) where each k is a positive integer, N(k1, kr) is the discrete interval [k1, kl+ 1,...,kr]; Oqp ,tq, 1 <q,p<n, 1 _< i <_ r are given constants. Adjoint Identities; Discrete Systems; Multi-point Boundary Value Problems - 1. INTRODUCTION satisfying the multi-point boundary conditions (1.1) (1.2) 1 _< i _< r _< (2 _< r _< n), Motivated by the work of Angel and Kalaba [ 4 ] on two-point boundary value problems for difference equations, recently in [ 1, 3, 7, 12 ], we have discussed several new methods of constructing the solutions of linear as well as nonlinear multi-point discrete boundary value problems. In this paper we shall develop adjoint identifies which are in terms of solutions of (1.1) and its adjoint system. These identifies lead to the backwardforward and forward-backward methods, which seem to be new processes for computing the solutions of (1.1) and (1.2). However, the application of adjoint equations to solve discrete problems is not new, e.g., Clenshaw [6] used adjoint difference equations to sum the Chebyshev series. To demonstrate the usefulness of the proposed methods we solve some continuous two-point boundary value problems which are known to be unstable. BACKWARD-FORWARD AND FORWARD-BACKWARD METHODS The adjoint system of the difference system (1.1) is defined as v(k) = AT(k) v(k + I), k N(kI, kr) (2.1) where AT(k) is an n x n matrix with elements aqp(k), 1 _< p, q < n and v(k) is an n x 1 vector with components vq(k), 1 < q _< n. obtain We multiply the qth equation of (1.1) by vq(k + 1) and sum over all n equations to n n n n Z uq(k + l)vq(k + I) = Z vq(k + I) Z aqp(k)up(k) + Z bq(k)vq(k + I). q=l q=l p=l q=l Next we multiply the qth equation of (2.1) by uq(k) and sum over all n equations to get n ql n ql n 11 uq(k)vq(k)= uq(k) aqp(k)vp(k+ 1). On subtracting (2.3) from (2.2), we find Let k0 e N(k1, kr) be fixed. In (2.4) letting k = t and summing from k0 to k 1 results in n n Z [uq(k + 1)vq(k + 1) uq(k)vq(k)] = Z bq(k)vq(k + 1). q=l q=l n k n Z [uq(k)vq(k)- uq(kOlVq(kO)] = Y. Z Vq(t) bq(t- 11, q=l t=k0+l q=l for all k0 <_ k N(kl, kr) (2.2) (2.3) (2.4) N(k1, kr) (2.5/ and, similarly n Z [uq(k)vq(k)- uq(k0)vq(k0)] ql n X X Vq(t) bq(t t=k+1 q=l I), (2.6) for all k0>_k N(kl, kr). p(i) (ki) Vq q Equations (2.5) and (2.6) will be referred to as adjoint identities. We compute backward solutions once for each uq(ki), 2 _< i _< r appearing in (1.2) with the conditions into the adjoint identity (2.5) with k0 = k1, we obtain where v(i) (ki) is the qth component at k for the pth backward solution. Substituting (2.7) n 10;tN uq(ki). n q=l vqp(i) (kl) uq(kl) = ki t=kl+1 n ql vpq(i) (t)bq(t 1), 2 < i _< r. (2.8) 2 _<i_< r, 1 < p, q _< n (2.7) Summing (r 1) equations (2.8) and making use of (1.2), we get n r r ki X [xt + X V(i) (kl)]uq(kl)=tp- X X q=l i=2 i=2 t=kl+l q=l vPq(i)(t)bq(t 1), l_<p_<n. (2.9) r i--2 ki Z t=k+l If the matrix [:ilN + v(i) (kl)] is nonsingular, then the system (2.9) provides the unknowns uq(kl), 1 _< q < n. The solution of the problem (1.1) and (1.2) is obtained by computing the solution of (1.1) with these values of uq(kl), 1 < q < n. However, to evaluate the summation term in (2.9) we need to store the solutions of (2.1). This can be avoided at the cost of solving another (r 1) system. For this, we denote Wp(i)(k) = Zv(i)(t)bq(t 1); l<p<n, 2<i<r, q=l n Wp(i)(k)=- E v.,t.tj)(k + 1) bq(k) + Wp(i)(k + 1) q=l Wp(i)(ki) = 0; 1 _< p n, 2 i _< r. (2.10) (2.11) (2.12) Thus, at the point ki, 2 < i _< r, we solve a system of order 2n given by (2.1) and (2.10) subject to the conditions (2.7) and (2.11). With this adjustment system (2.9) takes the form X [orq + X vqi) (kl)] uq(k1) = tp + X Wp(i)(kl), 1 <_ p _< n. n n r q=l i=2 i=2 This method of constructing the solution of (1.1) and (1.2) is called the backwardforward process and requires (r 1)n backward solutions of the adjoint system (2.1) satisfying (2.7), (r 1) backward solutions of (2.10) satisfying (2.11), and 1 forward solution of (1.1) with the obtained values of u(k 1), 1 _< q _< n, from the system (2.12), i.e., a total of (r 1) (n + 1) + 1 solutions of ntfi order systems. In particular, if r=2 then we need (n + 2) solutions. Similar to the backward-forward process we have the forwardbackward process. For this we solve (2.1) forward once for each uq(ki), 1 < i < r 1, appearing in (1.2) with the conditions v i) (ki) = lXl l<i<r-l,l<p,q_<n, (2.13) where v)P. (ki) is t_he qth component at ki for the pth forward solution. Substituting (2.13) into the adjoint identity (2.6) with k0 = kr, we obtain n Y. O;pq uq(ki)q=l n q=l ,,ti) vt? (kr) uq(kr) = kr t=ki+l q=l vP_q(i) (t)bq(t 1), Summing (r- 1) equations (2.14) and making use of (1.2), we get l_<i<r- 1. (2.14) n q=l (2.15) We introduce W.p(i)(k) = t=ki+ 1 q=l v (t)bq(t 1); 1 _< p _< n,1 i _< r 1, n q=l W.p(i)(k) = vqp(i) (k + 1) bq(k) + W.p(i)(k + 1) W.p(i)(ki) = O; 1 < p <_ n, 1 <_ i _< r 1. (2.16) (2.17) Thus, the system (2.15) is the same as n E [a q+ r-1 r-1 Z v.p) (kr)] uq(kr) = tp + Z W-p(i)(kr), 1 <_ p < n. i=l i=l The solution of the problem (1.1) and (1.2) is obtained by solving backward the system (1.1) with the obtained values of uq(kr), 1 < q < n from the system (2.18). Next we shall consider the system (1.1) together with the implicit separated conditions (1.3). We compute (n I]1) solutions of (2.1) backward with the conditions where Viq(Si)(ki) is the qth component at k for the sith backward solution. Substituting (2.19) into (2.5) with k0 = k1 and using (1.3), we obtain vSi)(ki) = cti(si),q; 2 < i < r, 1 < s _< [I 1 < q < n, (2.19) n i(si) Z *q q=l (kl)Uq(kl) = ti,i(si) ki Z t=kl +I n X viq(Si) (t)bq(t 1)" 1 2_<i <r, 1 < si< i. (2.20) We introduce ki Wi(si)(k) =- Z t=k+l n "i(si) Z "q q=l (t)bq(t I); 2<i_<r, 1 si_< i Thus, the system (2.20) can be written as n viq(Si) (kl)Uq(kl) = 1.i,i(si) + Wi(si)(kl); 2 i <_ r, 1 < s _< i. System (2.23) together with (1.3) for i = 1, i.e., n O;l(Sl), q uq(kl) = tl,l(s1), 1 _< s < i, forms a system of n equations in n unknowns uq(kl), 1 _< q <_ n. The solution of (1.1) and (1.3) is obtained by solving forward the system (1.1) with these values of uq(kl), l_<q_<n. In practice we couple the adjoint system (2.1) with the equation (2.21) and solve this system of (n + 1) equations from the point ki, 2 < i _< r to k1 with the conditions (2.19) and (2.22). Similarly, in the forward-backward process for (1.1) and (1.3), the unknowns uq(kr), 1 _< q _< n are computed from the system n qE=l vi(qSi) (kr)uq(kr) = ti,i(si) + w-i(si)(kr); 1 _< i < r- 1 1 < s < i, n Z (XrCsr) q uq(kr) = tr,rCsr), 1 < Sr< r, ql where vi,Si,t (k) is the qth component of the sith forward solution from the point k of the adjoint system (2.1) satisfying i(si) (ki) = Oq(si), q; 1 < i _< r 1, 1 V_q s <_ li, 1 < q _< n and w.i(si)(k) is the forward solution of the initial value problem w.i(si)(k) = n vi(Si) (k + 1)bq(k) + w.i(si)(k + 1) w.i(si)(ki) = O; 1 < i _< r 1, 1 s _< i. (2.21) (2.22) (2.23) (2.24) (2.25) (2.26) (2.27) (2.28) The solution of (1.1) and (1.3) is obtained by solving backward the system (1.1) with the obtained values of uq(kr), 1 _< q < r. We shall construct an appproximate solution of the continuous boundary value problem 3. NUMERICAL EXAMPLES y"= f(t)y + g(t) y(cc) = A, Y(I) = B by employing the discrete variable method due to Nomerov. Let h =K+i? tk = cc + kh, k N(O,K+I), fk = f(tk), gk = g(tk), and u(k) be the approximation to the true solution of y(t) at tk, satisfies the second order difference equation, (- 1 + 1-h2fk. 1)u(k- 1) + (2 + -1i0h.2fk)u(k) + (- 1 + l_h2fk+ 1)u(k+ 1) = l?h22(gk-1+ 10gk + gk+l), k N(1,K) together with the boundary conditions u(0) = A, u(K+ 1) = B. Theorem 3.1. (see [ 12 ])The discrete boundary value problem (3.2) and (3.3) has a unique solution provided (a) f(t) >_ 0 on [ct,13] and 1-h22fk < 1, k N(0,K+I); or 4sin 2K+I max If(t)l < 1 a-<t-<l] [ 12 ] if l_h22 maax_f<(tt) < 1 then in system form the problem (3.2) and (3.3) can be written as c2(k) 1 2 u 1 (k + 1) = u2(k) Ul (0) = A, u2(K) = B (3.1) (3.3) (3.4) (3.5) d(k) -- (k) x Applying the backward-forward process for the boundary value problem (3.4) and (3.5), we note that (2.1), (2.21), (2.19) and (2.22) reduce to Further, the system (2.23) and (2.24) takes the form which easily determines v22(k) = v21(k + 1)+ c (k) c2(k) v22(k + l) w2(k) =- v22(k + 1) d(k) + w2(k + 1) v(K) = O, v(K)= 1, w2(K)= 0. v21(0)u1(0) + v22(0)u2(0) = B + w2(0) u(0) =A, UI(0) = A, u2(0) = B + w2(0) v](0)A v(0) The solution of (3.4) and (3.5) is obtained by recursing forward the system (S.4) with the initial values (S.8). Similarly, applying the forward-backward process we f?md that (2.1), (2.28), (2.27) and (2.29) reduce to v.: (k + 1)= v.21(k)+c- v-ll (k) Cl(k] 1 * + c2(k) V-l W.l(k + 1) = W.l(k) + v.2(k + 1)d(k) Further, the system (2.25) and (2.26) becomes v.11 (K)Ul (K) + v.2I(K)u2(K) = A + w. I(K) u2(K) = B (3.6) (3.7) (3.8) (3.9) (3.10) Two New Algorithms for Discrete Boundary Value Problms: Agarwal & Nanda which gives u 1 (K) = A + W.l(K)- v.](K)B vI (K) u2(K) = B. (3.1 I) The solution of (3.4) and (3.5) is obtained by recursing backward the system (3.4) with the final values (3.11). 2. y(t) y,, = 2 t both the methods discussed in this section work equally well. The errors obtained, as 1 with h = 25? are presented in Table 1. calculated from the exact solution y(t) = 6 (19t- 5t2 0 ) and approximate solution U l(k) Example 3.2. For the discrete analogue (3.4) and (3.5) of the boundary value problem y" = 400y; y(0) = 1, y(5) = e-100, we apply both the methods. The errors obtained, as calculated from the exact solution 5 y(t) = e-20t and approximate solution Ul(k) with h = 1024? are presented in Table 2. 0.3125 0.6250 0.9375 1.2500 1.5625 1.8750 2.1875 2.5000 2.1825 3.1250 3.4375 3.7500 ..........4,.0625 .,..,,, 4.6875 Example 3.3. The boundary value problem y"= (2m + 1 + t2)y; y(0) = 13, y(oo) = 0, (3.12) where rn >_ 0 and 13 are known constants, is known as Holt?s problem. This problem is a typical example wherein usual shooting methods fail [ 2,5,7,9-11 ]. Replacing the boundary condition y(,,o) = 0 by y(T) = 0 (T finite) Holt [8] used f?mite dif,ference methods (however, for rn = 0, 13 = 1, T = 12; m = 1, 13 = -1/2, T = 8; rn = 2, 13 = T = 8, the results are unsatisfactory [ 8,11 ]), whereas Osborne [9] used a multiple shooting method and Roberts and Shipman [ 10 ] used a multi-point approach. In [ 2 ] we have formulated a new shooting method which gives accurate solutions of (3.12) for several different values of rn and 13 up to T = 18. (This value of T has been chosen in view of restricted computer capabilities.) For the same and several other values of rn and 13 accurate solutions of (3.12) up to T = 18 have also been obtained in [ 5 ]. Here the error estimates in the solution of (3.12) when approximating y(.o) = 0 by an appropriate boundary condition at T are also available. For ,1 the discrete analogue (3.4) and (3.5) of (3.12) (replacing y(,o) = 0 by y(18) = 0) with rn = 0, = 1 and h = we apply both the methods of this section. The numerical solution u 1 (k) is shown in Table 3. 7.0 $.0 iO.?O ?ii?.0? i?. 13.0 14.0 15.0 16.0 17.0 18.0 Example 3.4. For the discrete analogue (3.4) and (3.5) of the boundary value problem y" = (sin2t)y + cos2t; y(- 1) = y(1) = 0, 1 we apply both the methods. The numerical solution Ul(k) for h = 5--4-- is shown in Table 4. t Advances in ns Research Hindawi Publishing Corporation ht p:/ www.hindawi.com Advances in Hindawi Publishing Corporation ht p:/ www.hindawi.com bability and Statistics Hindawi Publishing Corporation ht p:/ www.hindawi.com Hindawi Publishing Corporation ht p:/ www.hindawi.com Hindawi Publishing Corporation ht p:/ www.hindawi.com The Scientiifc World Journal Hindawi Publishing Corporation ht p:/ www.hindawi.com International Journal of Combinatorics Hindawi Publishing Corporation ht p:/ www.hindawi.com Submit your manuscr ipts Journal of Mathematics gineering Discrete Nature and Society International Journal of Mathematics and Mathematical Sciences Journal of Discrete Mathematics ht p:/ w w.hindawi.com Journal of [ 1] R.P. Agarwal , Initial-value Methodsfor Discrete Boundary Value Problems , J. Math. Anal. Appl . 100 ( 1984 ), 513 - 529 . [2] R.P. Agarwal and R.C. Gupta , On the Solution ofHolts Problem, BIT 24 ( 1984 ), 342 - 346 . [3] R.P. Agarwal , Computational Methodsfor Discrete Boundary Value Problems, Appl. Math. Comp. 18 ( 1986 ), 15 - 41 . [4] E. Angel and R. Kalaba , A One-sweep Numerical Methodfor Vector-matrix Difference Equations with Two-point Boundary Conch'tions , J. Optimization Theory and Appl . 6 ( 1970 ), 345 - 355 . [5] K. Balla and M. Vicsek , On the Reduction ofHolt's Problem to a Finite Interval, Numer . Math. 51 ( 1987 ), 291 - 302 . [6] C.W. Clenshaw , A Note on the Summation of Chebyshev Series , MTAC, 9 ( 1955 ), 118 - 120 . [7] R.C. Gupta and R.P. Agarwal , A New Shooting Methodfor Multi-point Discrete Boundary Value Problems , J. Math. Anal. Appl . 112 ( 1985 ), 210 - 220 . [8] J.F. Holt , Numerical Solution ofNonlinear Two-point Boundary-value Problems by Finite-difference Methods , Commun. ACM 7 ( 1964 ), 366 - 373 . [9] M.R. Osborne , On Shooting Methodsfor Boundary Value Problems, J. Math. Anal. Appl . 27 ( 1969 ), 417 - 433 . [10] S.M. Roberts and J.S. Shipman , Multipoint Solution of Two-point Boundary-value Problems , J. Optimization Theory and Appl . 7 ( 1971 ), 301 - 318 . [ 11] S.M. Roberts and J.S. Shipman , Two-point Boundary Value Problems: Shooting Methods , Elsevier, New York ( 1972 ). [12] R.A. Usmani and R.P. Agarwal , On the Numerical Solution of Two Point Discrete Boundary Value Problems , Appl. Math. Comp. 25 ( 1988 ), 247 - 264 . Volume 2014 Volume 2014 Volume 2014 Hindawi Publishing Corporation ht p:/ www .hindawi.com


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Ravi P. Agarwal, Tara R. Nanda. Two new algorithms for discrete boundary value problems, International Journal of Stochastic Analysis, DOI: 10.1155/S1048953390000028