Using Hybrid Functions to solve a Coupled System of Fredholm Integro-Differential Equations of the Second Kind

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1 Advances in Dynamical Systems and Applications ISSN , Volume 9, Number, pp 5 (24) Using Hybrid Functions to solve a Coupled System of Fredholm Integro-Differential Euations of the Second Kind Lina Al Ahmadieh and George M Eid Notre-Dame University Louaize Department of Mathematics and Statistics Zouk Mosbeh, Lebanon lahmadieh@nduedulb and geid@nduedulb Abstract This article introduces a numerical method that uses hybrid functions for approximating solutions of systems of Fredholm integro-differential euations of the second kind This method reduces a system of Fredholm integro-differential euations to a system of algebraic euations and is illustrated by some numerical examples AMS Subject Classifications: 34A38, 45B5, 34K2, 45J5 Keywords: Integro-differential euations, block pulse functions, hybrid functions, operational matrix Introduction Many physical phenomena may be modeled by a system of integro-differential euations Lots of work has been done on nonlinear integro-differential euations using pulse functions and Legendre polynomials, see [5, 7, 9,, 2], as well as a recent work using this techniue to solve higher dimensional problem, see [] Also different methods were used to approximate its solutions such as Chebyshev wavelets method, Galerkin method or the modified decomposition method, see [4, 8, 3, 4] In this article, we develop a method using hybrid functions ( ) 2t tk t k b km (t) b k (t)p m t k t k Received June 28, 23; Accepted October 2, 23 Communicated by Mehmet Ünal

2 2 Lina Al Ahmadieh and George Eid on the interval [, T ) defined in terms of a pulse function b k (t) k and Legendre polynomials to approximate the solution of a system of Fredholm integro-differential euations of the form u (t) + v(t) + k (t, s)u(s)ds x(t), v (t) + u(t) + k 2 (t, s)v(s)ds y(t), u() u, v() v, t [, ], where k (t, s), k 2 (t, s) L 2 ([, ] [, ]) and x(t), y(t) L 2 ([, ]) are known functions and u(t), v(t) are unknown functions The objective of using hybrid functions in this article is to show that there are important orthogonal basis functions other than those mentioned above that also yield good approximating solutions for systems of integro-differential euations by converting them into systems of linear algebraic euations Also, new proofs for some properties of hybrid functions will be given 2 Preliminaries In this section, we define block pulse and hybrid functions, and recall function approximations in L 2 [, ] Definition 2 Let {b k (t)} k be a finite set of block pulse functions [8, 5] on the interval [, T ) defined by { if t k t < t k b k (t) elsewhere, where t, t T and [t k, t k ) [, T ) for k, 2,, It follows that for t [, ), t k (k )/, t k k/ and T, we have if k t < k b k (t) elsewhere This set of block pulse functions is orthogonal [4], since { if i j, i, 2,, b i (t)b j (t) b i (t) if i j, j, 2,, and if i j b i (t), b j (t) if i j

3 Solving a Coupled System of Fredholm Integro-Differential Euations 3 Definition 22 We define the hybrid Legendre block( pulse functions (or ) simply hybrid 2t tk t k functions) on the interval [, T ) by b km (t) b k (t)p m or euivalently t k t k by ( ) 2t tk t k p m if t k t < t k b km (t) t k t k elsewhere, where k, m r and r, N, see [8] If we let t k k/, t k (k )/ and T, then ( ) 2t tk t k p m t k t k ( 2t k p m k k ( 2t k+ k p m k ) ) p m ( 2t (k ) k k (k ) p m (2t 2k + ), and hence we have the hybrid function, t [, ), p m (2t 2k + ) if k t < k b km (t) elsewhere Function Approximation in L 2 [, ] Consider the orthogonal set of hybrid functions M {b km (t) : m r, k } in the Hilbert space L 2 [, ] Then any function f L 2 [, ] may be approximated arbitrary close by a (finite or possibly infinite) linear combination of elements of M + (ie, f may be expanded to a hybrid function) Thus, f(t) f km b km (t), where f km Let f be an arbitrary function in L 2 [, ] and f(t), b km(t) b km (t), b km (t) k m Y span{b (t), b (t),, b (r ) (t), b 2 (t), b 2 (t),, b 2(r ) (t), ), b (t),, b (r ) (t)} be a finite dimensional vector space [6] Then f has the uniue best approximation f in Y in the sense that for any y Y, we have f f 2 f y 2 Since

4 4 Lina Al Ahmadieh and George Eid f Y, there exist real numbers (coefficients) c, c 2,, c (r ) such that f can be r uniuely approximated by f f c km b km (t), where the coefficients c km f, b km (t) are determined by b km (t), b km (t) 3 New Proofs f, b km (t) m k r i r i c ij b ij (t), b km (t) j c ij b ij (t), b km (t) j c km b km (t), b km (t) Now, we generalize the concept of hybrid functions [, ] and prove that they can be extended to properly handle coupled systems of Fredholm integro-differential euations of the second kind These properties are fundamental for establishing the main results of this paper This will be done by using some properties of Legendre polynomials Let B(t) ( B T (t),, B T (t) ) T be a vector function of hybrid functions on [, ), where B i (t) ( b i (t),, b i(r ) (t) ) T, i, 2,, Proposition 3 (Operational matrix of integration) The integration of the vector function B(t) may be approximated by B(s)ds P B(t), where P is an r r matrix [8, ], known as the operation matrix for hybrid functions and given by E H H H E H H P E, H E where H and E are r r matrices defined by H,

5 Solving a Coupled System of Fredholm Integro-Differential Euations 5 and 3 3 E r 3 2r 3 2r Proof Since (2m + )p m (t) p m+(t) p m (t), see [3], we get b km (s)ds k b km (s)ds }{{} k b km (s)ds + b km (s)ds k k p m (2s 2k + )ds We now discuss the different cases according to the values of t: First, we discuss the case that (k )/ t < k/ Two situations will be considered: If m, then b km (s)ds k k k b km (s)ds p m (2s 2k + )ds ( p 2 (2m + ) m+ (2s 2k + ) p m (2s 2k + ) ) ds 2 (2m + ) (p m+(2t 2k + ) p m (2t 2k + )) ( bk(m+) (t) b k(m ) (t) ) 2 (2m + ) If m, then b k (s)ds k b k (s)ds k According to [3], we have p (t) p (t) + p (t) Then p (2s 2k + )ds p (2s 2k + ) 2 (p (2s 2k + ) + p (2s 2k + ))

6 6 Lina Al Ahmadieh and George Eid and hence b k (s)ds k 2 (p (2s 2k + ) + p (2s 2k + )) 2 (p (2t 2k + ) + p (2t 2k + )) 2 (b k(t) + b k (t)) Second, we discuss the case that t k/ Again two situations will be considered: If m, then we similarly get k k t b km (s)ds b km (s)ds + b km (s)ds + b km (s) ds k k }{{} }{{} k k p m (2s 2k + )ds 2 (2m + ) k k ( p m+ (2s 2k + ) p m (2s 2k + ) ) ds Now, let u 2s 2k + Then ( b km (s)ds p (2m + )(2) 2 m+ (u) p m (u) ) du (2m + )(2) (p m+(u) 2 p m (u) ) If m, then (2m + )(2) [ 2 ( )m+ ( ( ) m )] b k (s)ds k k b k (s)ds k k ds Thus, in summary, we have if m N {}, t < k b k (t) + b k (t) k if m, t < k 2 b km ds b k(m+) (t) b k(m ) (t) if m 2(2m + ) k if m, t < if m

7 Solving a Coupled System of Fredholm Integro-Differential Euations 7 Now, consider B k (s) ( b k (s), b k (s),, b k(r ) (s) ) T Then ( B k (s)ds b k (s)ds, b k (s)ds,, b k(r ) (s)ds (,,, ) T if t < k ( bk (t) + b k (t)), b k2(t) b k (t),, b ) T k(r 2)(t) (2r ) if k t < k ( ) T b l(t),,, if l t < l, l k +,, r r B k (t) if t < k EB k (t) if k t < k HB l (t) if l t < l, l k +,, EB k (t) + HB l (t) for all t < Thus, B(s)ds lk+ B (s)ds t B 2 (s)ds t B (s)ds EB (t) + HB 2 (t) + HB 3 (t) + HB 3 (t) + + HB (t) EB 2 (t) + HB 3 (t) + + HB (t) EB (t) E H H H E H H E H B(t) H E ) T

8 8 Lina Al Ahmadieh and George Eid This concludes the proof Proposition 32 (The integration of two hybrid functions) The integration of two hybrid Legendre block pulse function vectors, see [7], is L B(t)B T (t)dt, where L diag(d, D,, D) is an r r diagonal matrix and D (, ) 3,, is an 2r r r diagonal matrix for r N Proof Consider Then B(t) (B T, B T 2, B T 3,, B T ) T, where B i (t) ( b i (t), b i (t),, b i(r ) (t) ) T B(t)B T (t) b b b b b b (r ) b b 2 b b (r ) b b b b b b (r ) b b 2 b b (r ) b (r ) b b (r ) b b (r ) b (r ) b (r ) b 2 b (r ) b (r ) b (r ) b b (r ) b b (r ) b (r ) b (r ) b 2 b (r ) b (r ) and hence B(t)B T (t)dt b (t)b (t)dt b (t)b (t)dt b (r ) (t)b (t)dt b (r ) (t)b (t)dt b (t)b (r ) (t)dt b (t)b (r ) (t)dt b (r ) (t)b (r ) (t)dt b (r ) (t)b (r ) (t)dt Thus, L B(t)B T (t)dt

9 Solving a Coupled System of Fredholm Integro-Differential Euations 9 b, b b, b (r ) b, b (r ) b, b b, b (r ) b, b (r ) b (r ), b b (r ), b (r ) b (r ), b (r ) b (r ), b b (r ), b (r ) b (r ), b (r ) Conseuently, L ( ) d ij km, where d ij km b km, b ij with k, i, 2,, and j,, 2,, r Hence, there are two cases to be considered: First, if k i, then Second, if k i, then d ij km d ij km b km, b ij Therefore, k 2 b km (t)b ij (t)dt k p m (2t 2k + )p j (2t 2k + )dt p m (s)p j (s) 2 d ij km b k (t)p m (2t 2k + )p j (2t 2k + )dt ( ) 2 δ mj, where m, j,,, r 2m + ( ) if k i or m j if k i and m j 2m + and hence L diag(d, D,, D) is an r r matrix, where b i, b i b i, b i b i, b i(r ) b i, b i b i, b i b i, b i(r ) D b i(r ), b i b i(r ), b i b i(r ), b i(r ) ( diag, ) 3,, 2r is an r r matrix for all i, 2,, This concludes the proof Now, we will clearly present the approximation of k(t, s) in L 2 ([, ] [, ]) and provide a new proof for it Let B (i) (t) denote the ith component of B(t) and B (j) (s) denote the j th component of B(s) Approximation of k(t, s) in L 2 ([, ] [, ]) Now we will approximate the function k(t, s) in L 2 ([, ] [, ]) as k(t, s) B T (t)gb(s),

10 Lina Al Ahmadieh and George Eid [], where G is an r r matrix such that g ij Proof Consider B (i) (t), k(t, s), B (j) (s), i, j, 2,, r B (i) (t), B (i) (t) B (j) (s), B (j) (s) k(t, s) l r r d k m b lk (t)b dm (s)c km ld, where c km ld is a real coefficient to be determined for all l, d, 2,, and k, m,,, r Let B (i) (t) b au (t) be the ith component of B(t) and B (j) (s) b vp (s) be the jth component of B(s), a, v, 2,, and u, p,,, r Then Now, k(t, s), B (j) (s) r r l d k m r l B (i) (t), k(t, s), B (j) (s) k b lk (t)b dm (s)c km ld, B (j) (s) c kp lv b lk(t) B (j) (s), B (j) (s) B (i) (t), B (i) (t), Thus, we conclude that the real coefficient c up av in l k r r c kp lv b lk(t) B (j) (s), B (j) (s) l k c kp lv b lk(t) B (j) (s), B (j) (s) c up av B (i) (t), B (i) (t) B (j) (s), B (j) (s) l r r d k m b lk (t)b dm (s)c km ld that is multiplied by b au (t) (the ith component B (i) (t) of B(t)) and b vp (s) (the jth component B (j) (s) of B(s)) has the form c up av B (i) (t), k(t, s), B (j) (s), i, j, 2,, r B (i) (t), B (i) (t) B (j) (s), B (j) (s) Thus, where l g ij r r d k m b lk (t)b dm (s)c km ld r i r j g ij B (i) (t)b (j) (s), B (i) (t), k(t, s), B (j) (s), i, j, 2,, r B (i) (t), B (i) (t) B (j) (s), B (j) (s)

11 Solving a Coupled System of Fredholm Integro-Differential Euations Moreover, r i r j g ij B (i) (t)b (j) (s) B T (t)gb(s), where G (g ij ), i, j, 2,, r is an r r matrix Thus k(t, s) This concludes the proof l 4 Main Results r r d k m b lk (t)b dm (s)c km ld B T (t)gb(s) In [3], a system of integro-differential euations was approximated using the modified decomposition method, and in [2], a similar system was approximated using the approximation method We now consider a system of Fredholm integro-differential euations of the form u (t) + v(t) + k (t, s)u(s)ds x(t), v (t) + u(t) + k 2 (t, s)v(s)ds y(t), u() u, v() v, t [, ], where k (t, s), k 2 (t, s) L 2 ([, ] [, ]) and x(t), y(t) L 2 ([, ]) are known functions while u(t), v(t) are unknown functions, and we propose solving it by using the following approximations u(t) (U) T B(t) B T (t)u, u (t) (U ) T B(t) B T (t)u, v(t) (V ) T B(t) B T (t)v, v (t) (V ) T B(t) B T (t)v, k (t, s) B T (t)g B(s), k 2 (t, s) B T (t)g 2 B(s) Also, by the fundamental theorem of calculus, [7], we have u(t) u (s)ds + u() Substituting u(t), u (t) and u (t) in the above euation, we get U T B(t) U T B(s)ds + U T B(t)

12 2 Lina Al Ahmadieh and George Eid U T B(s)ds + U T (t)b(t) U T P B(t) + U T B(t) (U T P + U T )B(t) Thus, U T U T P + U T and so U (P T ) (U U ) Substituting the approximated functions in the above system, we get (U ) T B(t) + (V ) T B(t) + B T (t)g B(s)B T (s)uds (X) T B(t), (V ) T B(t) + (U) T B(t) + B T (t)g 2 B(s)B T (s)v ds (Y ) T B(t) Hence, { (U ) T + (V ) T + (G LU) T (X) T, (V ) T + (U) T + (G 2 LU) T (Y ) T Therefore { U + V + G LU X, V + U + G 2 LU Y Thus { U U + P T V + p T G LU P T X, V V + P T U + P T G 2 LU P T Y, and hence { (I + P T G L)U P T X P T V + U, After some calculations, we get and V V + P T U + P T G 2 LU P T Y V ( I P T (I + P T G L) P T + P T G 2 L ) (P T Y P T (I + P T G L) U P T (I + P T G L) P T X) U (I + P T G L) (P T X P T Y + U ) Using u(t) U T B(t) and v(t) V T B(t), we get the approximated solution 5 Numerical Examples In this section, the method introduced above will be numerically applied to solve two systems

13 Solving a Coupled System of Fredholm Integro-Differential Euations 3 Example 5 Consider the system of Fredholm integro-differential euations u (t) + v(t) + (s + ) sin(t)u(s)ds ( + t) + 2 et + sin(t), v (t) + u(t) + e t s+ v(s)ds e t + e t + + t, t [, ], whose exact solution is given by u(t) + t and v(t) et For and r 2, we have the following approximations: u(t) t, v(t) t A comparison of approximate solutions versus exact solutions with an L 2 -norm of the error is given in Table 5 Table 5: Approximate versus exact solutions in L 2 -norm error for Example 5 t Approximate Exact Approximate Exact L sol of u(t) sol of 2 -norm L sol of v(t) sol of 2 -norm error u(t) error v(t), r 2 u(t), r 2 v(t) Example 52 Consider the system of Fredholm integro-differential euations u (t) + v(t) + (s + )u(s)ds t + 3, v (t) + u(t) + (s + )tv(s)ds t, t [, ], whose exact solutions are u(t) t + and v(t) t For and r 2, we have the following approximations: u(t) t t2,

14 4 Lina Al Ahmadieh and George Eid v(t) t t2 A comparison of approximate solutions versus exact solutions with an L 2 -norm error is given in Table 52 Table 52: Approximate versus exact solutions with L 2 -norm error for Example 52 t Approximate Exact Approximate Exact L sol of u(t) sol of 2 -norm L sol of v(t) sol of 2 -norm error u(t) error v(t), r 2 u(t), r 2 v(t) Conclusion This paper shows that hybrid functions may be also used to effectively approximate solutions of systems of Fredholm integro-differential euations References [] Nasser Aghazadeh and Amir Ahmad Khajehnasiri Solving nonlinear twodimensional Volterra integro-differential euations by block-pulse functions Math Sci (Springer), 7:Art 3, 6, 23 [2] M I Berenguer, A I Garralda-Guillem, and M Ruiz Galán An approximation method for solving systems of Volterra integro-differential euations Appl Numer Math, 67:26 35, 23 [3] Kanti B Datta and B M Mohan Orthogonal functions in systems and control, volume 9 of Advanced Series in Electrical and Computer Engineering World Scientific Publishing Co Inc, River Edge, NJ, 995

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