Approximate solution of linear integro-differential equations by using modified Taylor expansion method

ISSN 1 746-7233, Engl, UK World Journal of Modelling Simulation Vol 9 (213) No 4, pp 289-31 Approximate solution of linear integro-differential equations by using modified Taylor expansion method Jalil Rashidinia 1, Ali Tahmasebi 2 1 School of Mathematics, Iran University of Science & Technology, Narmak 16844, Tehran, Iran 2 School of Mathematics Computer Science, Damghan University, Damghan, Iran (Received December 5 212, Revised July 14 213, Accepted September 25 213) Abstract In this study we developed modified Taylor expansion method for approximating the solution of linear Fredholm Volterra integro-differential equations Via Taylor s expansion of the unknown function at an arbitrary point, the integro-differential equations to be solved is approximately transformed into a system of linear equations for the unknown its derivatives which can be dealt with in an easy way This method gives a simple closed form solution for a linear integro-differential equation This method also enable us to control truncation error by adjusting the step size used in the numerical scheme Some numerical examples are provided to illustrate the accuracy of our approach Keywords: linear integro-differential equations, approximate solution, taylor expansion method 1 Introduction Mathematical modeling of real-life problems usually results in functional equations, eg partial differential equations, integral integro-differential equations, stochastic equations others In particular, integro-differential equations arise in fluid dynamics, biological models chemical kinetics The analytical solutions of some integro-differential equations cannot be found, thus numerical methods are required In recent years, the numerical methods for linear integro-differential equations have been extensively studied by many authors The iterated Galerkin methods have been proposed in [1] Furthermore, mixed interpolation collocation methods for first- second-order Volterra linear integro-differential equations have been suggested in [2] Hu [3] developed the interpolation collocation method for solving Fredholm linear integro-differential equations Rashed treated a special type of integro-differential equations with derivatives appearing in integrals [4] Taylor series expansion method is a powerful technique for solving integro-differential equations The Taylor expansion approach for solving Volterra integral equations has been presented by Kanwal Liu [5] The Taylor expansion method for approximate solution of a class of linear integro-differential equations presented by Li [6] also, a Taylor series method [7 1, 22] was presented by Szer et al for differential, integral integro-differential equations Bessel or Chebyshev polynomial approach are used to solve integro-differential equations in [11 16] Sahin in [17] applied collection method for numerical solution of Volterra integral equations with variable coefficients An improved Legre method [18] is used for solution of integro-differential equations A numerical method for the solution of integro-differential-difference equation with variable coefficients has been proposed by Saadatmi [19] In this paper, the Taylors expansion method [6] is modified such that by this modification the corresponding accuracy is drastically improved Also, the approximate method described here is very easy to implement Corresponding author Tel: +98217322 E-mail address: rashidinia@iustacir Published by World Academic Press, World Academic Union

29 J Rashidinia & A Tahmasebi: Approximate solution of linear integro-differential equations 2 Volterra integro-differential equations Consider the Volterra integro-differential equations of the form x β i (x)y (i) (x) = f(x) + λ v k v (x, t)y(t)dt, x, t I = [a, b], (1) under the initial conditions η c ji y (i) (η) = µ j, j =, 1,, m 1, (2) y(x) is an unknown function to be determined, the known continuous functions β i (x), f(x) k v (x, t) are defined on the interval I Here the real coefficients λ v c ji are known constants η is a given point in the spatial domain of the problem Remark 1 The following partition is used throughout of the paper = {a = x θl < < x 1 < x = η < x 1 < < x θr = b}, is an equidistance partition on I = [a, b], h is a step size of the partition x θ = η + θh, for θ { θ r,, 1,, 1,, θ r } Theorem 1 Consider linear integro-differential Eq (1)- Eq (2) Let the functions β i (x), f(x), k v (x, t) the exact solution y(x) are sufficiently differentiable on the interval I k v (x, t) be a separable kernel Then there exist linear independent functions ϕ i (χ) constants c i for i =, 1,, α, such that y(χ) = y(χ+x θ ) is an exact solution of β i (χ + x θ )y (i) (χ) = f(χ + x θ ) + α c i ϕ i (χ) + λ v k v (χ + x θ, + x θ )y()d, (3) with initial conditions x = χ + x θ y (i) () = y (i) (x θ ), i =, 1,, m 1, (4) Proof The kernel in Eq (1) is separable, then we have k(x, t) = p u i (x)v i (t), (5) by substituting Eq (5) into Eq (3) then change of variables x = χ + x θ, t = + x θ, we have β i (χ + x θ )y (i) (χ + x θ ) = f(χ + x θ ) + d i = η x θ v i ( + x θ )y( + x θ )d p d i u i (χ + x θ ) + λ v k v (χ + x θ, + x θ )y( + x θ )d, Now, by simplifying classifying on the middle terms of RHS Eq (6), we can write (6) p d i u i (χ + x θ ) = α c i ϕ i (χ), (7) WJMS email for contribution: submit@wjmsorguk

World Journal of Modelling Simulation, Vol 9 (213) No 4, pp 289-31 291 ϕ i (χ), i =, 1,, α are known linear independent functions c i, i =, 1,, α are unknown constants Using Eq (7), by denoting y(χ) = y(χ + x θ ), we can rewrite Eq (6) Eq (2), in the form of Eq (3) Eq (4), respectively To obtain c i, by derivation on Eq (3) for α 1 times at χ =, the following algebraic system is obtained α c i ϕ i () = m g i () f(x θ ), α c i ϕ i () = m g i () f (x θ ) ρ(), α c i ϕ (α 1) i () = m g (α 1) i () f (α 1) (x θ ) ρ (α 2) (), g i (χ) = β i (χ + x θ )y (i) (χ) ρ(χ) = k v (χ + x θ, χ + x θ )y(χ) By solving algebraic system Eq (8), the unknown constants can be determined as follows c i = τ i (x θ, y(), y (),, y (m+α 1) ()), i =, 1,, α, (9) τ i are known functions y (i) () = y (i) (x θ ), i =, 1,, α For solving linear integro-differential equations Eq (1) Eq (2), by modified Taylor expansion method, we assume that y() in Eq (3) can be represented in terms of the n th -order Taylor expansion as y() = y(χ) + y (χ)( χ) + + y(n) (χ) ( χ) n + y(n+1) (ζ) n! (n + 1)! ( χ)n+1, (1) ζ is between χ It is readily shown that the Lagrange remainder y (n+1) (ζ) ( χ)n (n + 1)! is sufficiently small for a large enough n provided that y (n+1) (χ) is a uniformly bounded function for any n on the interval I, so that we neglect the remainder the truncated Taylor expansions y(χ) as y() n k= (8) y (k) (χ) ( χ) k (11) k! It is worth noting that the Lagrange remainder vanishes for a polynomial of degree equal to or less than n, implying that the above n th -order Taylor expansion is exact Also, n is chosen any positive integer such that n m Now, we substitute Taylor s expansion Eq (11) for y() into Eq (3) find k (χ)y(χ) + k 1 (χ)y (χ) + + k n (χ)y (n) (χ) = f (χ), (12) { βi (χ + x k i = θ ) λv χ k v(χ + x θ, + x θ )( χ) i d, λv χ k v(χ + x θ, + x θ )( χ) i d, i =, 1,, m i = m + 1, m + 2,, n, (13) f (χ) = f(χ + x θ ) + α c i ϕ i (χ) (14) Thus Eq (12) becomes a m th -order, linear, ordinary differential equation with variable coefficients for y(χ) its derivations up to n Instead of solving analytically ordinary differential equation, we will determine y(χ), y (χ),, y (n) (χ) by solving linear equations To this end, other n independent linear equations WJMS email for subscription: info@wjmsorguk

292 J Rashidinia & A Tahmasebi: Approximate solution of linear integro-differential equations for y(χ), y (χ),, y (n) (χ) are needed This can be achieved by integrating both sides of Eq (3) with respect to χ from to χ changing the order of the integrations Accordingly, one can get that β i ( + x θ )y (i) ()d = f ()d + λ v k v (s + x θ, + x θ )dsy()d, (15) we have replaced variable s to χ for convenience Applying integration by part, accordingly, one can get that Thus β i ( + x θ )y (i) ()d =β i ( + x θ )y (i 1) () χ β i( + x θ )y (i 2) () χ + + ( 1) i 1 β (i 1) i β i ( + x θ )y (i) ()d = ( + ( + x θ )y() χ + ( 1)i β (i) i ( + x θ )y()d ( 1) k i 1 β (k i 1) k ( + x θ ))y (i) ()) χ ( 1) k β (k) k ( + x θ)y()d, (16) Applying the Taylor expansion again substituting Eq (11) for y() into Eq (16) gives k 1i (χ) = k= k 1 (χ)y(χ) + k 11 (χ)y (χ) + + k 1n (χ)y (n) (χ) = f 1 (χ), (17) ( 1) k i 1 β (k i 1) k (χ + x θ ) + 1 λ v ( 1) k β (k) ( + x θ)( χ) i d k= k v (s + x θ, + x θ )ds( χ) i d, i =, 1,, m 1, (18) k k 1i (χ) = 1 λ v ( 1) k β (k) ( + x θ)( χ) i d k= k k v (s + x θ, + x θ )ds( χ) i d, i = m, m + 1,, n, f 1 (χ) = ( m ) ( 1) k i 1 β (k i 1) k (x θ ) y (i) () + f ()d (19) By repeating the integration on both side of Eq (15) for r times r = 2, 3,, n, one can arrive that (χ ) r 1 β i ( + x θ )y (i) ()d = By using integration by part, we can obtain (χ ) r 1 f ()d + λ v k v (s + x θ, + x θ )(χ s) r 1 dsy()d (2) WJMS email for contribution: submit@wjmsorguk

World Journal of Modelling Simulation, Vol 9 (213) No 4, pp 289-31 293 (χ ) r 1 β i ( + x θ )y (i) ()d = ( 1) k k β k ( + x θ ) (k) (χ ) r 1 k y (k) ()d k= ( m + ( 1) k i 1 k i 1 β k ( + x θ )(χ ) r 1 ) k i 1 y() χ (21) k ri (χ) = Substituting Eq (11) for y() in the Eq (21), we obtain λ v k r (χ)y(χ) + k r1 (χ)y (χ) + + k rn (χ)y (n) (χ) = f r (χ), (22) ( 1) k i 1 k i 1 β k ( + x θ )(χ ) r 1 k i 1 =χ + 1 ( 1) k k β k ( + x θ )( χ) r 1 k d k v (s + x θ, + x θ )(χ s) r 1 ds( χ) i d, r = 1,, m 1, (23) k= f r (χ) = k ri (χ) = 1 λ v ( m ( 1) k k β k ( + x θ )( χ) r 1 k d k= k v (s + x θ, + x θ )(χ s) r 1 ds( χ) i d, i = m, m + 1,, n ( 1) k i 1 k i 1 β k ( + x θ )(χ ) r 1 k i 1 ) y() =η + (χ ) r 1 f ()d (24) Therefore, Eqs (12), (17) (22) form a system of n + 1 linear equation for n + 1 unknown functions y(χ), y (χ),, y (n) (χ) For simplicity, this system can be rewritten in an alternative compact form as K nn (χ)y n (χ) = F(χ), (25) K nn (χ) is an (n + 1) (n + 1) square matrix function, Y n (χ) F n (χ) are two vectors of length n + 1, these are defined as k (χ) k 1 (χ) k n (χ) k 1 (χ) k 11 (χ) k 1n (χ) K nn (χ) =, (26) k n (χ) k n1 (χ) k nn (χ) Y n (χ) = y(χ) y (χ) y (n) (χ), F n(χ) = Therefore, by well-known Crammer s formula, we can obtain f (χ) f 1 (χ) f n (χ) (27) y (i) (χ) = Ψ i (χ, x θ, c, c 1,, c α, y(), y (),, y () ()), i =, 1,, m + α 1, (28) which is Taylor expansion of solution Eq (3) Eq (4) for y(χ) its derivations up to m + α 1 at χ =, by change of variable χ = x x θ, we have WJMS email for subscription: info@wjmsorguk

294 J Rashidinia & A Tahmasebi: Approximate solution of linear integro-differential equations y (i) (x) = Ψ i (x x θ, x θ, c, c 1,, c α, y(x θ ), y (x θ ),, y () (x θ )), i =, 1,, m + α 1, (29) which is Taylor expansion of solution Eq (1) Eq (2) for y(x) it s derivations up to m + α 1 at x = x θ Note that, in the Eq (28) Eq (29), in the any step of proposed method c i, i =, 1,, α, y (i) (), i =, 1,, m 1 are known values by using (9) 21 Algorithm of the approach In this section, we try to propose an algorithm on the basis of the above discussions suppose that we face with the linear Volterra integro-differential Eq (1) Eq (2), its kernel satisfies the conditions of Theorem 1 This algorithm is presented in two stages such as initialization step main steps Initialization step: Choose step size h > for equidistance partition on I Set c j =, j =, 1,, α y (j) () = y (j) (η), j =, 1,, m 1 Set θ = go to main steps Main steps: Step 1 Compute by Eq (29) the following approximate solution y (i) (x) = Ψ i (x x θ, x θ, c, c 1,, c α, y(), y (),, y () ()), i =, 1,, m + α 1, (3) which is Taylor expansion approximate Eq (1)- Eq (2) for y(x) it s derivations up to m+α 1 at x = x θ Go to Step 2 Step 2 Set θ = θ + 1, if θ > θ r 1, stop; Otherwise, using Eq (3) compute the approximate values y (i) () = y (i) (x θ ), i =, 1,, m + α 1, (31) which are the initial conditions Eq (1) Eq (2) at x = x θ Go to next Step Step 3 By Eq (31) compute the following approximate values c i = τ i (x θ, y(), y (),, y (m+α 1) ()), i =, 1,, α, (32) go to Step 1 Note that in the Step 2, if we replaced condition θ > θ l 1, with the condition θ > θ l + 1 also θ 1 to θ + 1, then by applying the above algorithm we obtain Taylor expansion approximate solution of Eq (1)- Eq (2) its derivations at x = x θ, θ = 1, 2,, θ l 3 Fredholm integro-differential equations In this section, our study focuses on a class of linear Fredholm integro-differential equations of the following form b β i (x)y (i) (x) = f(x) + λ f k f (x, t)y(t)dt, x, t I = [a, b], (33) subject to the initial conditions a c ji y (i) () = µ j, a η b, j =, 1,, m 1 (34) We assume that the desired solution y(t) is n + 1 times continually differentiable on the interval I, i e y C n+1 [I] Consequently, for y C n+1, y(t) can be represented in terms of the truncated Taylor series expansion as n y (k) (h) y(t) (t h) k, (35) k! k= WJMS email for contribution: submit@wjmsorguk

World Journal of Modelling Simulation, Vol 9 (213) No 4, pp 289-31 295 h is step size in the partition Substituting the approximate solution Eq (35) for y(t) into Eq (33), one can get a y(h) + a 1 y (h) + + a n y (n) (h) = f(h), (36) a i = β i (h) λ f λ f b a k f (h, t)(t h) i dt, i =, 1,, m, b a k f (h, t)(t h) i dt, i = m + 1, m + 2,, n (37) Using a procedure analogous to the previous section, we can obtain other n linear equations for y(h), y (h),, y (n) (h) as follows a ri = h h λ f a r y(h) + a r1 y (h) + + a rn y (n) (h) = f r (h), (38) ( 1) k i 1 k i 1 β k ()(h ) r 1 k i 1 =h + 1 η h η ( 1) k k (β k ()( h) r 1 ) k d k= k f (s, )(h s) r 1 ds( h) i d, r = 1,, m 1, (39) a ri = 1 h η λ f ( 1) k k (β k ()( h) r 1 ) k d k= h h η k f (s, )(h s) r 1 ds( h) i d, i = m, m + 1,, n, f r (h) = ( m ) ( 1) k i 1 k i 1 β k ()(h ) r 1 y() =η + k i 1 h η (h ) r 1 f()d (4) Therefore Eq (36) Eq (38) form a system of n + 1 linear equation for n + 1 unknown values y(h), y (h),, y (n) (h) By using Crammer s formula we can obtain is initial conditions Eq (1) Eq (2) at x = h Now let q k f = u i (x)v i (t), y i = y (i) (h), i =, 1,, q 1, q < n, (41) by using theorem 1, linear integro-differential equations Eq (1) converted to the following form with initial conditions β i (χ + x θ )y (i) (χ) = f(χ + x θ ) + y(χ) = y(χ + x θ ), x = χ + x θ q I i u i (χ + x θ ), (42) y (i) () = y (i) (x θ ), i =, 1,, m 1, (43) WJMS email for subscription: info@wjmsorguk

296 J Rashidinia & A Tahmasebi: Approximate solution of linear integro-differential equations b I i = λ f v i (t)y(t)dt a is an unknown constant From (42) the following algebraic equation cab be drived b (χ)y(χ) + b 1 (χ)y (χ) + + b n (χ)y (n) (χ) = g (χ), (44) b i (χ) = { βi (χ + x θ ), i =, 1,, m,, i = m + 1, m + 2,, n, (45) g (χ) = f(χ + x θ ) + q I i u i (χ + x θ ) (46) Also by using a procedure analogous to the section 2, we can obtain other n linear equations for y(χ), y (χ),, y (n) (χ) as follows b r (χ)y(χ) + b r1 (χ)y (χ) + + b rn (χ)y (n) (χ) = g r (χ), (47) b ri (χ) = ( 1) k i 1 k i 1 β k ( + x θ )(χ ) r 1 + 1 k i 1 =χ ( 1) k k β k ( + x θ )( χ) r 1 k d, r = 1,, m 1, (48) k= g r (χ) = b ri (χ) = 1 ( m ( 1) k k β k ( + x θ )( χ) r 1 k d, i = m, m + 1,, n, k= ( 1) k i 1 k i 1 β k ( + x θ )(χ ) r 1 k i 1 ) y() =η + (χ ) r 1 g ()d (49) Therefore Eq (44) Eq (47) form a system of n + 1 linear equation for n + 1 unknown functions y(χ), y (χ),, y (n) (χ) For simplicity, this system can be rewritten in an alternative compact form as B nn (χ)y n (χ) = F(χ), (5) B nn (χ) is an (n + 1) (n + 1) square matrix function, Y n (χ) F n (χ) are two vectors of length n + 1, these are defined as b (χ) b 1 (χ) b n (χ) b 1 (χ) b 11 (χ) b 1n (χ) B nn (χ) =, (51) b n (χ) b n1 (χ) b nn (χ) Y n (χ) = y(χ) y (χ) y (n) (χ), F n(χ) = g (χ) g 1 (χ) g n (χ) (52) WJMS email for contribution: submit@wjmsorguk

World Journal of Modelling Simulation, Vol 9 (213) No 4, pp 289-31 297 By well-known Crammer s formula, we have y (i) (χ) = Ψ i (χ, x θ, I, I 1,, I q, y(), y (),, y () ()), i =, 1,, m 1, (53) y (i) () = y (i) (x θ ), i =, 1,, m 1, {I k } q k= are unknown parameters will be determined Using Eq (41) Eq (53) the following algebraic system is yield Ψ i (, η + h, I, I 1,, I q, y(), y (),, y () ()) = y i, i =, 1,, q 1, (54) y (i) () = y (i) (x θ ), i =, 1,, m 1 By solving algebraic system Eq (54), the unknown constants can be determined as follows By substituting, the obtained values I k into Eq (53), we can obtain I k = γ k, k =, 1,, q (55) y (i) (χ) = Ψ i (χ, x θ, γ, γ 1,, γ q, y(), y (),, y () ()), i =, 1,, m 1, (56) which is Taylor expansion approximate solution Eq (42) Eq (43) for y(χ) its derivations up to m + α 1 at χ =, by change of variable χ = x x θ, we have y (i) (x) = Ψ i (x x θ, x θ, γ, γ 1,, γ q, y(), y (),, y () ()), i =, 1,, m 1, (57) which is Taylor expansion approximate solution Eq (33) Eq (34) for y(x) it s derivations up to m + α 1 at x = x θ 31 Algorithm of the approach In this section, we propose an algorithm on the basis of the above discussions This algorithm is presented in two stages such as initialization step main steps Initialization step: Choose step size h > for equidistance partition on I Set y (j) () = y (j) (), j =, 1,, m 1 Set θ = go to main steps Main steps: Step 1 By Eq (53) Eq (55), compute the following approximate solution y (i) (x) = Ψ i (x x θ, x θ, γ, γ 1,, γ q, y(), y (),, y () ()), i =, 1,, m 1, (58) which is Taylor expansion approximate Eq (33) Eq (34) for y(x) it s derivations up to m 1 at x = x θ Go to next Step Step 2 Set θ = θ + 1, if θ > θ r 1, stop; Otherwise, using Eq (58) compute the approximate values y (i) () = y (i) (x θ ), i =, 1,, m 1 (59) which are the initial conditions Eq (33) Eq (34) at x = x θ go to Step 1 Note that in the Step 2, if we replaced condition θ > θ l 1, with the condition θ > θ l + 1 also θ 1 to θ + 1, then by applying the above algorithm we obtain Taylor expansion approximate solution of Eq (33) Eq (34) its derivations at x = x θ, θ = 1, 2,, θ l WJMS email for subscription: info@wjmsorguk

298 J Rashidinia & A Tahmasebi: Approximate solution of linear integro-differential equations 4 Numerical examples To illustrate the effectiveness of the proposed method we consider several test examples to carried out the applicability of our approach Example 1 Consider the first order linear integro-differential equation [22] y = 1 + λ x y(t)dt, 1 x, t 1 (6) We consider this Example in the two different cases: one is that λ = 1, y() = corresponding exact solutions y(x) = sin x other is λ = 1, y() = 1 corresponding exact solutions y(x) = e x In the case of λ = 1, y() =, we explain the steps of our method for n = 3 h = 1 as follows By using theorem 1 linear integro-differential equations convert to the following form with initial condition y = 1 + c y()d, (61) y () = y(x θ ), (62) c = 1 y () Let n = 4, in Eq (11), by solving corresponding algebraic system Eq (25) by using Eq (28), we can obtain ( χ 9 y(χ) = 144 + c χ9 144 χ11 112 c χ11 112 + χ13 5184 + c χ13 5184 χ 15 18864 c χ15 18864 + χ8 y() 144 97 χ1 y() 324 + 59 χ12 y() 36288 χ14 y() 72576 ) ( χ 8 144 + χ1 378 + χ12 18144 + χ 16 435456 ( χ y 8 (χ) = 144 + c χ8 97 χ1 97c χ1 59 χ12 + 144 324 324 36288 ) 1 (63) 59c χ12 + 36288 χ14 72576 c χ14 72576 χ9 y() 144 + χ11 y() 112 ) ( χ13 y() χ 8 5184 144 + χ1 378 + χ12 18144 + χ 16 ) 1 (64) 435456 In the initialization step of algorithms (21), by setting c =, θ =, y() = into Eq (63), Eq (64), by using Eq (29), we have ( x 9 y(x) = 144 x11 112 + x13 5184 x 15 ) 18864 ( x 8 144 + x1 378 + x12 18144 + x 16 ) 1 (65) 435456 ( x y 8 97 x1 (x) = 144 324 ( x 8 ) 59 x12 + 36288 x14 72576 144 + x1 378 + x12 18144 + x 16 435456 ) 1, (66) WJMS email for contribution: submit@wjmsorguk

World Journal of Modelling Simulation, Vol 9 (213) No 4, pp 289-31 299 which is Taylor expansion approximation solution Eq (1) Eq (2) its derivative at x = In the next step, setting x = h, (θ = 1), into Eq (65) Eq (66), the following approximate values can be obtain y() = 998334, y () = 9954, c = 499583, y() = y(1) y () = y (1) Now again, by setting the obtained values c, y(), y () θ = 1, into Eq (63), Eq (64), by using Eq (29), Taylor approximation solution Eq (1) Eq (2) its derivative at x = h can be obtained Repeating the above procedure for θ = 2, ±3,, ±9, we can obtain Taylor approximate solution Eq (6) at x = θh in the case of λ = 1, y() = Consequently, by using similarly procedure, we can obtain the approximate solution Eq (6) for λ = 1, y() = 1 in the interval [ 1, 1] Also we employ the Taylor series method [6] to determine the three- five-order approximate solution for λ = 1, y() = 1 λ = 1, y() = 1 From numerical results in Tab 1 2, we can find that the accuracy of approximate solution by Modified Taylor series is more accurate than the solution by Taylor eries method, particularly at end points Also, Tab 1 Tab 2 show that proposed method can be used for broad intervals (67) Table 1 Absolute error of Example 1 (λ = 1, y() = ) x i Taylor series [6] n = 3 Present method n = 3 Taylor series [6] n = 5 Present method n = 5 1 434827 1 6 213995 1 12 352581 1 11 44489 1 16 2 1889 118233 1 11 188557 1 7 166533 1 15 3 321 26334 1 11 16645 1 5 277556 1 16 4 4338 117659 1 11 173967 1 4 22245 1 15 5 16527 1 11 262631 1 6 344169 1 15 6 42117 1 11 15 677236 1 15 7 297748 1 11 4675 421885 1 15 8 155588 1 11 677236 1 15 9 57558 1 11 171529 1 14 1 59699 1 11 144329 1 14 Table 2 Absolute error of Example 1 (λ = 1, y() = 1) x i Taylor series [6] n = 3 Present method n = 3 Taylor series [6] n = 5 Present method n = 5 1 169146 1 5 16737 1 11 134426 1 1 22245 1 15 2 158 762759 1 11 12321 1 6 266454 1 15 3 342535 1 1 4689 1 4 24869 1 14 4 1315 1 9 3611 99476 1 14 5 454253 1 9 255795 1 13 6 15826 1 8 26148 1 12 7 484324 1 8 14592 1 11 8 15186 1 7 49273 1 11 9 467726 1 7 142791 1 1 1 14271 1 6 469299 1 1 Example 2 Consider the second order linear integro-differential equation [6, 22] y + xy xy = e x 2 sin x + with initial conditions y() = 1, y (), which is the exact solution y(x) = e x 1 1 e t y(t) sin xdt, 1 x, t 1, (68) WJMS email for subscription: info@wjmsorguk

3 J Rashidinia & A Tahmasebi: Approximate solution of linear integro-differential equations Our method is applied in this Example with h = 1 with order of approximate n = 4 The absolute error in the grid points are given also compared the absolute error in the solution with [6] are tabulated in Tab 3 From Tab 3, we fined that numerical results based on the present method are closer to the exact solution than those in [6] Table 3 Absolute error of Example 2 x i Taylor series [6] n = 4 Present method n = 4-1 11 81745 1 4-8 42151 1 4 45573 1 4-6 12297 1 4 255 1 4-4 2981 1 5 63521 1 5-2 4569 1 7 81471 1 6 2 815147 1 6 81515 1 6 4 141856 1 4 63792 1 5 6 91994 1 4 281 1 4 8 39 47141 1 4 1 13 87295 1 4 Example 3 Consider the following high order linear Fredholm integro-differential equation 1 y (5) x 2 y (3) y xy = x 2 cos x x sin x + y(t)dt, 1 x, t 1, (69) 1 with initial condition y() = y () = y (4) =, y () = 1, y (3) = 1 exact solution y(x) = sin x We approximate the solution by n = 5, h = 2 of our method Tab 4 shows that the numerical results of our method are better than results obtained in [6] Table 4 Absolute error of Example 3 x i Taylor method [6] n = 5 Present method n = 5-1 51213 1 5 68536 1 9-8 9614 1 6 2281 1 9-6 9698 1 7 56666 1 1-4 3546 1 8 83216 1 11-2 12546 1 1 68536 1 9 2 12967 1 1 41369 1 12 4 43962 1 1 83216 1 11 6 16841 1 6 56666 1 1 8 25717 1 5 22899 1 9 1 22361 1 4 685359 1 9 Example 4 Finally, consider the following third order linear Voltera integro-differential equation y (3) xy (2) = 4 7 x9 8 5 x7 x 6 + 6x 2 6 + 4 x x 2 t 3 y(t)dt, x, t 1, (7) with the initial conditions y() = 1, y () = 2, y (2) () = We find the approximate solution by the proposed method for n = 3 is the same as y(x) = x 3 + 2x + 1, the exact solution WJMS email for contribution: submit@wjmsorguk

World Journal of Modelling Simulation, Vol 9 (213) No 4, pp 289-31 31 5 Conclusion The approximation by Taylor series method in [7 1, 22] has the appropriate accuracy at the closed neighborhood of the point of expansion, but for the points far from points the accuracy of the approximation reduced considerably, to overcome such difficultly, we modified develop Taylor expansion approximation to obtain Taylor expansion approximate at grade points x = x i on the interval [a, b] The comparison of the results obtained by the presented method conventional Taylor series method reveals that ours method is very effective convenient The efficiency of the algorithms are verified by some test problems References [1] W Volk The iterated Galerkin methods for linear integro-differential equations Journal of Computational Applied Mathematics, 1988, 21: 63 74 [2] H Brunner, A Makroglou, K Miller Mixed interpolation collocation methods for first second order Volterra integro-differential equations with periodic solution Applied numerical mathematics, 1997, 23: 381 42 [3] Q Hu Interpolation correction for collocation solutions of Fredholm integro-differential equation Mathematics of Computation of the American Mathematical Society, 1998, 223: 987 999 [4] M Rashed Numerical solution of a special type of integro-differential equations Applied mathematics computation, 23, 143: 73 88 [5] R Kanwall, K Liu A Taylor expansion approach for solving integral equations International Journal of Mathematical Education in Science Technology, 1989, 2: 411 414 [6] Y Huang, X Li Approximate solution of a class of linear integro-differential equations by Taylor expansion method International Journal of Computer Mathematics, 21, 87(6): 1277 1288 [7] M Sezer A method for the approximate solution of the second-order linear differential equations in terms of Taylor polynomials International Journal of Mathematical Education in Science Technology, 1996, 27(6) 821 834 [8] M Sezer Taylor polynomial solution of Volterra integral equations International Journal of Mathematical Education in Science Technology, 1994, 25: 625 633 [9] S Yalcinbas, M Sezer The approximate solution of high-order linear Volterra-Fredholm integro-differential equations in terms of Taylor polynomials Applied Mathematics Computation, 2, 112: 291 38 [1] S Yalcinbas Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equations Applied Mathematics Computation, 22, 127: 195 26 [11] S Yuzbasi, N Sahin, M Sezer Bessel polynomial solutions of high-order linear Volterra integro-differential equations Computers & Mathematics with Applications, 211, 62: 194 1956 [12] S Yuzbas, N Sahin, M Sezer Numerical solutions of systems of linear Fredholm integro-differential equations with Bessel polynomial bases Computers & Mathematics with Applications, 211, 61: 379 396 [13] A Akyuz-Dascioglu, M Sezer Chebyshev polynomial solutions of systems of higher-order linear Fredholm- Volterra integro-differential equations Journal of the Franklin Institute, 25, 342: 688 71 [14] A Akyuz-Dascioglu A chebyshev polynomial approach for linear Fredholm-Volterra integro-differential equations in the most general form Applied mathematics computation, 26, 181: 13 112 [15] N Sahin, S Yuzbasi, M Sezer A Bessel Polynomial approach for solving general linear Fredholm integrodifferential-difference equations International Journal of Computer Mathematics, 211, 88: 393 3111 [16] Y Gamze, Y Suayip, S Mehmet A Chebyshev Method for a class of high-order linear Fredholm integrodifferential equations Advanced Research in Applied Math, 212, 4: 49 67 [17] N Çahın, Ş Yüzbaşı, M Gülsu A collocation approach for solving systems of linear Volterra integral equations with variable coefficients Computers & Mathematics with Applications, 211, 62: 755 769 [18] S Yuzbasi, M Sezer, B Kemanci Numerical solutions of integro-differential equations application of a population model with an improved Legendre method Applied Mathematical Modelling, 213, 37(4): 286 211 [19] A Saadatmi, M Dehghan Numerical solution of the higher-order linear Fredholm integro-differentialdifference equation with variable coefficients Computers & Mathematics with Applications, 21, 59: 2996 34 [2] L Huang, X Li, Y Huang Approximate solution of Abel integral equation Computers & Mathematics with Applications, 28, 56: 1748 1757 [21] X Li, L Huang, Y Huang A new Abel inversion by means of the integrals of an input function with noise Journal of Physics A: Mathematical Theoretical, 27, 4: 347 36 [22] S Yalcinbas, M Sezer The approximate solution of high-order linear Volterra-Fredholm integro-differential equations in terms of Taylor polynomials Applied Mathematics Computation, 2, 112: 291 38 [23] M Berenguer, M Fernández Muˆnoz, et al A sequential approach for solving the Fredholm integro-differential equation Applied Numerical Mathematics, 212, 62(4): 297 34 WJMS email for subscription: info@wjmsorguk