Haar wavelet method for nonlinear integro-differential equations

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1 Applied Mathematics and Computation 176 (6) Haar wavelet method for nonlinear integro-differential equations Ülo Lepik Institute of Applied Mathematics, University of Tartu, Liivi Str., 549 Tartu, Estonia Abstract A numerical method for solving nonlinear integral equations based on the Haar wavelets is presented. The method is applicable for Volterra integral equations and integro-differential equations; it can be used also for solving boundary value problems of ordinary differential equations. The efficiency of the proposed method is tested with the aid of four examples. High accuracy even for a small number of collocation points is stated. Ó 5 Elsevier Inc. All rights reserved. Keywords: Integro-differential equations; Numerical solution; Nonlinear; Haar wavelets; Collocation method 1. Introduction Many problems of theoretical physics and other disciplines lead to nonlinear Volterra integral equations or integro-differential equations. For solving these equations several numerical approaches have been proposed, an overview can be found in the monograph [1]. Beginning from 1991 the wavelet method has been applied for solving integral equations, a short survey on these papers can be found in []. The solutions are often quite complicated and the advantages of the wavelet method get lost, therefore any kind of simplifications are welcome. One possibility for it is to make use of the Haar wavelets, which are mathematically the most simple wavelets. The Haar wavelet method for solving linear integral equations of different type was proposed in [] and for nonlinear Fredholm integral equations in [3]. In the present article the Haar wavelets are applied for solving of nonlinear Volterra integral equations and integro-differential equations. The proposed method is based on the collocation technique; the wavelet coefficients are calculated iteratively making use of the Newton method. The main idea of the method is to double the number of collocation points at each iteration. With minor changes the proposed method is applicable also for solving two-point boundary value problems of ordinal differential equations. The method is tested by the help of four numerical examples, for which the exact solution is known. To make the article more readable in Section a short description on the Haar wavelets is added. address: ulo.lepik@ut.ee 96-33/$ - see front matter Ó 5 Elsevier Inc. All rights reserved. doi:1.116/j.amc.5.9.1

2 Ü. Lepik / Applied Mathematics and Computation 176 (6) Haar wavelet basis The Haar wavelet family is 8 >< 1 for t ½t ð1þ ; t ðþ Þ; h i ðtþ ¼ 1 for t ½t ðþ ; t ð3þ Þ; >: elsewhere. ð:1þ Here the notations t ð1þ ¼ k m ; tðþ ¼ k þ :5 m ; tð3þ ¼ k þ 1 ð:þ m are introduced. The integer m = j, j =,1,...,J indicates the level of the wavelet; k =,1,...,m 1 is the translation parameter. The integer J determines the maximal level of resolution. The index i is calculated from the formula i = m + k + 1. Here the minimal value is i = (then m =1,k = ); maximal value is i =Mwhere M = J. The index i = 1 corresponds to the scaling function 1 for 6 t 6 1; h 1 ðtþ ¼ ð:3þ elsewhere. We shall divide the interval t [,1] into M parts of equal length Dt = 1/(M); the grid points are s l ¼ðl 1ÞDt; l ¼ 1; ;...; M þ 1. ð:4þ Since in the subsequent sections the collocation method is used then we define also the collocation points t l ¼ðl :5ÞDt; l ¼ 1; ;...; M. ð:5þ Following Chen and Hsiao [4] the Haar coefficient matrix H is introduced; it is a M M matrix with the elements H(i,l)=h i (t l ). Let us integrate (.1) q i ¼ Z t h i ðtþdt. In the collocation points (.6) gets the form Q(i,l) =q i (t l ), where Q is a M M matrix. Chen and Hsiao [4] presented this matrix in the form Q = PH, where PH is interpreted as the product of the matrices P and H. Chen and Hsiao called P the operational matrix of integration; they shoved that for calculating this matrix of order l the following matrix equation holds: " P ;5l ð 1 P l ¼ lþh # ;5l ð 1 1. ð:7þ lþh ;5l It should be noted that calculations for H l, P l must be carried out only once; the obtained results are applicable for solving whatever problems. 3. Solving nonlinear integral equations by the Haar wavelet method ð:6þ Consider integro-differential equation of the type au ðxþþbuðxþ ¼ Kðx; t; uðtþ; u ðtþþdt þ f ðxþ for 6 x 6 1 and u () = u. Here a, b are constants, the functions K, f are prescribed. Satisfying (3.1) in the collocation points (.5) we obtain au ðx l Þþbuðx l Þ¼ where l =1,,...,M. l Kðx l ; t; uðtþ; u ðtþþdt þ f ðx l Þ; ð3:1þ ð3:þ

3 36 Ü. Lepik / Applied Mathematics and Computation 176 (6) The function u (t) is developed into the Haar series u ðtþ ¼ XM a i h i ðtþ. i¼1 By integrating (3.3) we obtain uðtþ ¼ XM a i q i ðtþþuðþ. i¼1 ð3:3þ ð3:4þ If (3.3) and (3.4) are substituted into (3.) and the integrations are carried out a system of M equations a i is obtained for calculating the wavelet coefficients. Since this system is nonlinear we must use some numerical approach. In the present article for this the Newton method is applied, which brings us to the equation where X M p¼1 ah p ðx l Þþbq p ðx l Þ ¼ au ðx l Þ buðx l Þþ ¼ oa p ou l l oa p dt Da p ou þ ou ¼ oa p ou oa p ou q pðtþþ ou h pðtþ. K dt þ f ðx l Þ; l ¼ 1; ;...; M; ð3:5þ The main problem is to evaluate the integrals in (3.5). This can be done in the following way. Let us denote l l uðlþ ¼ K dt; wðp; lþ ¼ dt ð3:7þ oa p and consider the subinterval t [s s,s s+1 ], s =1,,...,M, where s s is the sth grid point defined by (.4). In each subinterval ð3:6þ h p ðtþ ¼h p ðt s Þ¼const., q p ðtþ ¼q p ðs s Þþðt s s Þh p ðt s Þ. Here t s denotes the sth collocation point defined by (.5) and t s ¼ s s þ :5Dt. Since u (t)=u (t s ) = const., we obtain uðtþ ¼uðs s Þþðt s s Þu ðt s Þ; uðt s Þ¼uðs s Þþ:5u ðt s ÞDt. Next the following notations are introduced: DGðx l ; t s ; uðt s Þ; u ðt s ÞÞ ¼ D egðx l ; t l ; uðt l Þ; u ðt l ÞÞ ¼ DG u ¼ D eg u ¼ Z ssþ1 s s Z tl s l Z ssþ1 s s Z tl s l Kðx l ; t; uðtþ; u ðt s ÞÞdt; Kðx l ; t; uðtþ; u ðt l ÞÞdt; Z ssþ1 dt ¼ oa p s s ou q pðtþþ ou h pðtþ dt; oa p dt ¼ Z tl s l ou q pðtþþ ou h pðtþ dt. ð3:8þ ð3:9þ ð3:1þ ð3:11þ

4 Evaluating the integrals (3.7) for each subinterval t [s s,s s+1 ] and summing up the results we obtain uðlþ ¼ Xl 1 DG þ D eg; wðp; lþ ¼ Xl 1 DG u þ DG e u. ð3:1þ It is convenient to put our results into the matrix form. For this purpose we introduce the row vectors x =[x(l)], a =[a i ], Da =[Da i ], u =[u(t l )], u =[u (t l )], f =[f(x l )], u =[u(l)] and M M matrices H =[h p (x l )], Q =[g p (x l )]. Besides we denote S ¼ ah þ bq w; ð3:13þ F ¼ au bu þ u þ f. The matrix form of the system (3.5) is now DaS ¼ F ; ð3:14þ which has the solution Da ¼ FS 1. ð3:15þ The Volterra integral equation uðxþ ¼ Ü. Lepik / Applied Mathematics and Computation 176 (6) Kðx; t; uðtþþdt þ f ðxþ ð3:16þ is a special case of (3.5). Our results remain applicable also for the Volterra equation if out the following changes are carried out: (i) a =,b =1, ou ¼ ; (ii) Eqs. (3.3) and (3.4) are replaced with u = ah. 4. Calculation of the wavelet coefficients The wavelet coefficients a i can be calculated in the following way. Let us assume that the problem is solved for the level m = J 1, to which correspond M = J collocation points. The wavelet coefficients for this approximation are labeled as a ðmþ i ; i ¼ 1; ;...; M. Next the value of J is increased by one, thus the number of collocation points is doubled. The wavelet coefficients at the new level are estimated as ( ^a ðmþ1þ i ¼ aðmþ i for i ¼ 1; ;...; M; ð4:1þ for i ¼ M þ 1;...; M. The estimates ^u ðmþ1þ ; ð^u Þ ðmþ1þ are calculated according to (3.3) and (3.4). The values ^a ðmþ1þ i are corrected by (3.15) and we obtain a ðmþ1þ i ¼ ^a ðmþ1þ i þ Da ðmþ1þ i ; i ¼ 1; ;...; M. ð4:þ The corrected values for u (m+1),(u ) (m+1) are calculated again from (3.3) and (3.4). This cycle is repeated until the necessary exactness of the results is obtained. Assumption (4.1) is motivated by the fact that higher coefficients of the sequence a i are usually quite small. The question how to start this procedure arises. We recommend to take the starting solution in the form (u ) () = a, u() = a x + u() and satisfy (3.1) in the collocation point x =.5. This leads to the equation aa þ bð:5a þ uðþþ ¼ Z :5 from which the coefficient a can be evaluated. Kð:5; t; a t þ uðþ; a Þdt þ f ð:5þ ð4:3þ

5 38 Ü. Lepik / Applied Mathematics and Computation 176 (6) Estimates for the next step are ^a ð1þ 1 ¼ a ; ^a ð1þ ¼ and ^u ð1þ ðtþ ¼^a ð1þ 1 h 1ðtÞþ^a ð1þ h ðtþ ¼a ; ^u ð1þ ðtþ ¼^a ð1þ 1 q 1ðtÞþ^a ð1þ q ðtþþuðþ ¼a t þ uðþ. ð4:4þ These estimates are corrected by solving (3.15) for M = 1. Our calculations show that this simple method gives good results in many cases, but there may be problems for which the sequence of iterations does not converge (the convergence of the Newton method depends upon the successful choice of the initial approximation). In such cases for getting a convergent solution we could solve (3.1) directly for two or four collocation points. The other possibility is to apply the Newton method for a fixed number of collocation points more than once and only after that double the number of collocation points. 5. Error estimates With the purpose to demonstrate the applicability and efficiency of the proposed method in the following sections some tutorial examples, for which the exact solution is known, are solved. The error of the mth iteration can be estimated as eðmþ ¼ max 16i6M juðmþ ðx i Þ u ex ðx i Þj; ð5:1þ where u ex (x) is the exact solution. For determining the convergence rate of the solution the quantity qðmþ ¼ eðm 1Þ ; m ¼ ; 3; 4;... ð5:þ eðmþ is introduced. In most cases we do not know the exact solution. Here the error of the results can be estimated in the following way. We introduce the quantity SðmÞ ¼Dx ðmþ XM u ðmþ ðx i Þ i¼1 ð5:3þ where Dx (m) denotes the stepsize of the mth approximation. Since u (m) (x) is in each subinterval x [x i,x i+1 ]a linear function then S(m) has a distinct geometrical meaning: it is the area which lies in the interval x [, 1] underneath the curve ju (m) (x)j. For estimating the exactness of the solution the quantity Sðm þ 1Þ DðmÞ ¼ 1 SðmÞ ; ð5:4þ is introduced. Geometrically D(m) denotes the relative increase of the area underneath the curve ju (m) (x)j, x [, 1] for one iteration step. The convergence rate of the process can be estimated with the aid of the function Dðm 1Þ rðmþ ¼ ; m ¼ ; 3;... DðmÞ According to the criterion (5.1) the error estimate is based on a single collocation point, therefore it is a local criterion of convergence. The estimate (5.4) makes use of all collocation points therefore it is a integral criterion. 6. Examples Let us consider some test problems.

6 Example 6.1. Consider the Volterra equation 1 þ u ðtþ uðxþ ¼ ; ð6:1þ 1 þ t which has the exact solution u ex = x. The wavelet solution is sought in the form uðtþ ¼ XM i¼1 a i h i ðtþ. Since K ¼ 1þu, it follows from (3.11) and (3.1) that: 1þt uðlþ ¼ Xl 1 ½1 þ u ðt s ÞAðsÞŠ þ 1 þ u ðt l ÞaðlÞ; wðp; lþ ¼ Xl 1 uðt s Þh p ðt s ÞAðsÞþ½1þuðt l Þh p ðt l ÞŠAðlÞ; p; l ¼ 1; ;...; M; where AðsÞ ¼arctan s sþ1 arctan s s ; AðlÞ ¼arctan t l arctan s l. We start our solution with u = a = const. Satisfying (4.3) we get the equation a a arctan :5 þ 1 ¼ ; ð6:4þ which has two roots a ð1þ ¼ :6747 and a ðþ ¼ 1:481. It follows from the calculations that the value a ðþ brings to a nonconvergent iteration process and therefore we shall take a = Next two collocation points x 1 =.5 and x =.75 are taken and the estimates for a and u are ^a ¼ða ; Þ; ^u ¼ða ; a Þ. Correcting these values with the aid of (3.15) we obtain a (1) = (.53,.33) and u (1) = (.199,.846); the error function (5.1) is e (1) =.96. Results of the following iterations are shown in Table 1. Example 6.. Solve the Volterra equation of second kind Ü. Lepik / Applied Mathematics and Computation 176 (6) uðxþ ¼ 1 uðtþuðx tþdt þ 1 sin x; < x < 1. ð6:5þ The exact solution of (6.5) is u(x) =J 1 (x), where symbol J 1 denotes the Bessel function of order 1. In the present case ð6:þ ð6:3þ K ¼ 1 uðtþuðx tþ; ¼ 1 oa p ½uðx tþh pðx tþ uðtþh p ðtþš. ð6:6þ Table 1 Error estimates e, D and convergence rates q, r for Eq. (6.1) J M e q D r 1 4.7E 7.7E 8 1.6E E E E E E E E E E 5 3.9

7 33 Ü. Lepik / Applied Mathematics and Computation 176 (6) Table Error estimates e, D and convergence rates q, r for Eq. (6.5) J M e q D r E.E 8 1.E E E E E E E E E E Making use of (3.11) and (3.1) we obtain uðlþ ¼ Dt X l 1 uðt s Þuðx l t s Þ; wðp; lþ ¼ Dt X l 1 ½uðt s Þh p ðt s Þ uðx l t s Þh p ðx l t s ÞŠ. Eq. (3.14) get the form S = H, F = u + u + f, where f(l) =.5sinx l. The starting solution is taken again in the form u = a. Satisfying (4.3) we get for a the quadratic equation a 4a þ sin :5 ¼ ; which has the solutions a ð1þ ¼ :56; a ðþ ¼ 3:744. Since the value a ðþ leads to a nonconverging iteration process we take a =.56. The estimated solution for two collocation points is ^a 1 ¼ða ; Þ; ^u ¼ða ; a Þ. Correcting it with the aid of (3.15) we obtain a = (.4,.117), u = (.137,.357) with the error e =.1. Error estimates of the subsequent approximations are shown in Table. It follows from this table that the convergence rates are smaller as in the case of Example 6.1. Example 6.3. Consider the integro-differential equation u ðxþ ¼1 þ ð6:7þ uðtþu ðtþdt; 6 x 6 1; uðþ ¼. ð6:8þ The exact solution of (6.8) is p u ex ðxþ ¼ ffiffiffi x tan p ffiffi. We shall seek the wavelet solution in the form (3.3) and (3.4); from (3.11) and (3.1) we find uðlþ ¼Dt Xl 1 wðp; lþ ¼Dt Xl 1 uðt s Þu ðt s Þþ Dt uðt lþu ðt l Þ; ½uðt s Þh p ðt s Þþu ðt s Þq p ðt s ÞŠ þ Dt ½uðt lþh p ðt l Þþu ðt l Þq k ðt l ÞŠ. ð6:9þ ð6:1þ Eqs. (3.13) and (3.14) obtain the form DaS = F, where S = H w, F = u + u + E (symbol E denotes the M dimensional unit vector). For starting we take u = a, u = a x. Satisfying (4.3) for x =.5 the equation a 8a þ 8 ¼ is obtained. It has two roots from which we shall take a = 1.17 (the other root leads to a nonconverging process). Correcting the estimates ^a ¼ða ; Þ, ^u ¼ða ; a Þ, ^u ¼ :5ða ; 3a Þ with the aid of (3.15) we find a = (1.37,.19), u = (1.47,1.46), u = (.6,.88) the error of this approximation is e =.51. Results for the next approximations are presented in Table 3. This example is taken from the article [5] by Avudainayagam and Vani, who solved it by the wavelet- Galerkin method. For computing the integrals the connection coefficient method was used. This makes the solution quite complicated, besides it is applicable only in the case of quadratic nonlinearities of the type u,

8 Ü. Lepik / Applied Mathematics and Computation 176 (6) Table 3 Error estimates e, D and convergence rates q, r for Eq. (6.8) J M e q D r E 3 4.E 8 3.3E E E E E E E E E E uu or u. The same example was solved also in [6] with the aid of the AdomianÕs decomposition method. The authors of [6] denote that their solution is more simple and easy to use. We would like to note that its accuracy of the results may be insufficient. So it follows from Table 3 of the paper [6] that u = for x = 1 while the exact value is u = 1.85; consequently, the error is 9.%. 7. Boundary value problems of ODE The method of solution proposed in this article can be applied also for solving boundary value problems of ordinary differential equations. To illustrate this let us solve the two-point boundary value problem for the differential equation u ¼ Kðx; u; u Þ ð7:1þ with the boundary conditions u() = A, u(1) = B. By integrating (7.1) we get the integro-differential equation u ðxþ ¼ Kðt; uðtþ; u ðtþþdt þ u ðþ; 6 x 6 1. ð7:þ As before we seek its solution in the form (3.3) and (3.4). Since 1 for i ¼ 1; q i ð1þ ¼ ði ¼ 1;...; MÞ for i > 1; it follows from (3.4) that a 1 = B A, consequently, the first coefficient in (3.4) is specified. According to (3.3) u ðþ ¼ XM i¼1 a i h i ðþ; where 8 >< 1 if i ¼ 1; h i ðþ ¼ 1 if i ¼ J þ 1; J ¼ ; 1; ;... >: elsewhere. Due to the fact that a 1 is fixed and Da 1 = some changes in the system of Eqs. (3.14) must be executed. Let the symbol ~a denote the vector a for which the first component is deleted. In similar way the bs is a reduced matrix in which the first row and column are deleted. Instead of (3.13) (3.15) we get now D~a e S ¼ ef ; ð7:5þ where es ¼ eh w ~ ee ~ hðþ T ; ef ¼ ~u þ ~u ðþþ~u þ f ~. Here ee is a M 1 dimensional unit vector, and - denotes the Kronecker tensor product. ð7:3þ ð7:4þ

9 33 Ü. Lepik / Applied Mathematics and Computation 176 (6) The following solution proceeds according to the algorithm presented in Sections 3 and 4: we generate the vector a =(B A,a,...,a M ) and evaluate u(x), u (x) for the next approximation from (3.3) and (3.4). Example 7.1. Consider the boundary value problem u uu 1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p ffiffiffi ; uðþ ¼1; uð1þ ¼ ; 4 ð1 þ xþ 3 which has the exact solution p u ¼ ffiffiffiffiffiffiffiffiffiffiffi 1 þ x. By integrating (3.4) we obtain u ðxþ ¼ uu dt þ u ðþ 1 ð1 þ xþþ 1 p ffiffiffiffiffiffiffiffiffiffi. ð7:7þ 1 þ x p It follows from boundary conditions that a 1 ¼ uð1þ uðþ ¼ ffiffiffi 1. We start our solution with evaluating the quantities u(l) and w(p,l). Making use of (3.11) and (3.1) we obtain uðlþ ¼Dt Xl 1 ð7:6þ uðt s Þu ðt s Þþ Dt u ðt l Þ uðt l Þ Dt 4 u ðt l Þ ; ð7:8þ wðp; lþ ¼Dt Xl 1 ½h p ðt s Þuðt s Þþq p ðt s Þu ðt s ÞŠ þ Dt ðh pðt l Þuðt l Þþq p ðt l Þu ðt l ÞÞ Dt h p ðt l Þu ðt l Þ. ð7:9þ Since now a 1 is fixed we take the starting solution in the form u ðtþ ¼a 1 þ a h ðtþ; ð7:1þ uðtþ ¼a 1 t þ a q ðtþþuðþ. Replacing (7.1) into (7.7) and satisfying this equation in the point x =.5 we get the quadratic equation a þ ð5 þ p ffiffi p Þa þ ffiffiffi 8 þ p ffiffiffi 7 ¼. 6 This equation has two roots from which fits to us the root a =.7. The estimates for the first approximation are ^a ¼ða 1 ; a Þ¼ð:414; :7Þ, ^u ¼ð:4943; :3539Þ, ^u ¼ð1:14; 1:336Þ. Correcting the values with the aid of (7.5), (7.8) and (7.9) we find a (1) = (.44,.56), u (1) = (1.1,1.33), u (1) = (.48,.368) with the error e () =.9. Results of the subsequent approximations are presented in Table 4. Table 4 Error estimates e, D and convergence rates q, r for Eq. (7.7) J M e q D r E 3 6.5E 8 1.1E E E E E E E E E E 6 3.8

10 Ü. Lepik / Applied Mathematics and Computation 176 (6) Conclusions A new method for numerical solution of Volterra integral-equations and integro-differential equations, which is based on the Haar wavelets, is proposed. Its applicability and efficiency is checked on four test problems. The benefits of the Haar wavelet approach are sparse matrices of representation, fast transformation and possibility of implementation of fast algorithms. Simplicity of our solution is due to a great extent to the assumption (4.1), based on the fact that usually higher wavelet coefficients a i are small. It should be mentioned that such an approach is not applicable in the case of the conventional piecewise constant approximations method. It follows from Tables 1 4 that the accuracy of the obtained solutions is quite high even if the number of collocation points is small (as a rule a four or eight point solution guarantees satisfactory exactness). By increasing the number of collocation points the error of the solution rapidly decreases. Approximation with the Haar wavelets is equivalent with the approximation for piecewise constant functions. Therefore the convergence rate for piecewise constant functions, which is O(M ) can be transferred to Haar wavelet approach. As to the Newton method then in the case of sufficiently good initial values it has also quadratic convergence. So it could be expected that in the case of our solution by doubling the number of collocation points the error function roughly decreases four times. This theoretical estimation is in general consistent with the data in Tables 1, 3 and 4. Deviations appear only for small values of J 6 3, where due to small number of calculation points some adaption takes place. For bigger values of J the convergence rates q and r come nearer to the theoretical value 4. Exceptional is Example 6., where convergence rates are considerably smaller. Acknowledgement Financial support from the Estonian Science Foundation under Grant ETF-54 is gratefully acknowledged. References [1] H. Brunner, Collocation method for Volterra integral and related functional equations, Cambridge Monograph on Applied and Computational Mathematics, Cambridge University Press, Cambridge, MA, 4. [] Ü. Lepik, E. Tamme, Application of the Haar wavelets for solution of linear integral equations, in: Dynamical Systems and Applications, Antalaya, 4, Proceedings, 5, pp [3] Ü. Lepik, E. Tamme, Solution of nonlinear integral equations via Haar wavelet method, Int. J. Wavelets, Multiresol. Inform. Process., submitted for publication. [4] C.F. Chen, C.H. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proc. Control Theory Appl. 144 (1997) [5] A. Avudainayagam, C. Vani, Wavelet-Galerkin method for integro-differential equations, Appl. Numer. Math. 3 () [6] S.M. El-Sayed, M.R. Abdel-Azis, A comparison of AdomianÕs decomposition method and wavelet-galerkin method for solving integro-differential equations, Appl. Math. Comput. 136 (3)

Stochastic approximations of perturbed Fredholm Volterra integro-differential equation arising in mathematical neurosciences

Applied athematics and Computation 6 (27) 73 2 www.elsevier.com/locate/amc Stochastic approximations of perturbed Fredholm Volterra integro-differential equation arising in mathematical neurosciences.