Numerical Solution of Fredholm Integro-differential Equations By Using Hybrid Function Operational Matrix of Differentiation

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1 Available online at Int. J. Industrial Mathematics (ISS 8-56) Vol. 9, o. 4, 7 Article ID IJIM-8, pages Research Article umerical Solution of Fredholm Integro-differential Equations By Using Hybrid Function Operational Matrix of Differentiation R. Jafari, R. Ezzati, K. Maleknejad Received Date: 6-3- Revised Date: 6-8- Accepted Date: 7-5- Abstract In this paper, first, a numerical method is presented for solving a class of linear Fredholm integrodifferential equation. The operational matrix of derivative is obtained by introducing hybrid third kind Chebyshev polynomials and Block-pulse functions. The application of the proposed operational matrix with tau method is then utilized to transform the integro-differential equations to the algebraic equations. Finally, show the efficiency of the proposed method is indicated by some numerical examples. Keywords : Fredholm integro-differential equation; Hybrid function; Chebyshev polynomial; Blockpulse function; Operational matrix of derivative. Introduction n recent years, there has been a growing interest in the integro-differential equations, which I provide an important tool for modeling numerous real world problem in engineering, mechanics, physics, chemistry, astronomy, biology, economics, potential theory and electrostatics. The kinds of equations are usually difficult to solve analytically, so it is required to obtain an efficient approximate solution. Therefore, many different methods are used to obtain the solution of the linear and nonlinear Integro-differential equations such as: the successive approximations, Adomian decomposition, Homotopy perturbation method, Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran. Corresponding author. ezati@ kiau.ac.ir, Tel: +98(9)36858 Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran. Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran. Chebyshev and taylor collocation, Hybrid function, Cas and Haar wavelet, Tau and Walsh series methods []-[8]. In this paper, a numerical method using hybrid of third kind Chebyshev polynomials and Blockpulse functions (HTKCPBPF) is presented for the following linear Fredholm integro-differential equations of the type: u (x) f(x) + u(x) + λ K(x, t)u(t)dt, u() a, (.) λ, a, are constants, f(x)) and K(x, t) are Known and u(t) is the unknown function to be determined. This method reduces the integral equation to a set of algebraic equations by expanding u(x) as (HTKCPBPF) with unknown coefficients. The paper is organized as follows: In Section, we review briefly about Block-pulse functions and third kind Chebyshev polynomials and hybrid of them. Section 3 is devoted to function approximation. In Sections 4, we construct 349

2 35 R. Jafari et al. /IJIM Vol. 9, o. 4 (7) the operational matrices of derivative based on the (HTKCPBPF). Convergence analysis of the proposed method is done in Section 5. In Section 6 and 7, we show the validity and efficiency of the proposed method, we present some numerical examples. Finally, Section 8 concludes the paper. Function and hybrid function. Block-pulse functions A set of Block-pulse functions b i (x), i,,...,, on the interval [, ) are defined as [9]: i, x < i b i (x), otherwise, (.) These functions satisfy in the following properties: i- disjointness b i (x), fori j b i (x)b j (x), fori j. Third kind of Chebyshev polynomials The third kind of Chebyshev polynomial V n (x) is a polynomial of degree n in x defined by [9] : V n (x) cos(n + )θ cos( )θ, (.3) x cos θ. clearly from., fundamental recurrence relation as follows: V n (x) xv n (x) V n (x), n, 3,..., V (x), V (x) x, These polynomials are orthogonal on [, ] with +x respect to the weight function ω(x) x, that is V i (x)v j (x)w(x)dx δ ij..3 Hybrid functions For n,..., and m,..., M, the HTKCPBPF are defined as follows [3]: φ nm (x) ii- orthogonality V m(x n + ), n x < n b i (x)b j (x)dt δ ij, otherwise, (.4) i, j,,...,, and δ ij is the Kronecker delta, iii- completeness for every f L [, ) when m approach to the infinity, parsevals identity hold: f (x)dx f i f(x)b i(x)dx. (fi b i (x) ), with the following weight function ω n (x) ω(x n + ). 3 Function approximation A function f(x) L [, ) may be expanded as: f(x) n m c nm φ nm (x), (3.5) c nm f(x), φ nm(x) φ nm (x), φ nm (x) ω n (x)φ nm (x)f(t)dx. (3.6)

3 R. Jafari et al. /IJIM Vol. 9, o. 4 (7) In 3.6,.,. L ω [,) denotes the inner product in L ω[, ), with weight function w n (x). If the infinite series in 3.5 is truncated, then equation 3.5 can be written as: f(x) M n m c nm φ nm (x) C T φ(x), (3.7) C and φ(x) are M matrices given by: C [c, c, c,..., c,m, c,..., c,m ] T, (3.8) φ(x) [φ (x), φ (x),..., φ,m (x), φ (x) (3.9)..., φ,m (x)] T. ob- The differentiation of vector φ(x) can be tained by: dφ(x) dx Dφ(x). (3.) We derive the matrix D in the following section for some particular values of and M. 4 Operational matrix of derivative In this section, we figure out the precise derivative of the HTKCPBPF with and M 3. In this case, the six basis functions are given by : φ φ (x), φ φ (x) 8x 3, φ 3 φ (x) 64x 4x + 5, (4.) for t [, ), and φ 4 φ (x), φ 5 φ (x) 8x 7, φ 6 φ (x) 64x 4x + 4, (4.) for t [, ). Let φ 6(t) (φ (t) φ (t) φ (t) φ (t) φ (t) φ (t)). By differentiation 4., 4.3 from to t, and representing them in the matrix form, we obtain dφ dx, dφ dx 8 8φ, dφ 3 dx 8x 4 6φ + 8φ, dφ 4 dx, dφ 5 dx 8 8φ, dφ 6 dx 8x 4 6φ + 8φ. Thus, we have Where dφ(x) dx D 6 6φ(x). (4.3) 4 D The matrix D 6 6 can be written as [ ] F3 3 D F 3 3 F In general, for M 4, we have dφ(x) Dφ(x), (4.4) dx φ(x) is given in 3. and D is a M M matrix given by F F D F, F F a (ij) is M M matrices, whose the elements are given explicitly by: (i + j ), i > j, (i + j)odd, a ij (i j), i > j, (i + j)even,, otherwise. (4.5)

4 35 R. Jafari et al. /IJIM Vol. 9, o. 4 (7) For example if M 7 as follows: F Using the above procedure, the operational matrix of nth derivative can be derived as: d n φ(x) dx n D n φ(x), (4.6) The integration of two HTKCPBPF vectors is obtained as E φ(t)φ(t) T dt, (4.7) E is a M M symmetric matrix. For example if and M 6 as follows: Out[3] E Exact Presented method, Figure : The exact and Presented method solution of Example 7. 5 Convergence analysis The following theorem gives the convergence and accuracy estimation of HTKCPBPF. 5 Out[8] 5 Exact Presented method Figure : The exact and Presented method solution of Example Out[36].3... Exact Presented method Figure 3: The exact and Presented method solution of Example 7.4 Theorem 5. Let f(x) be a second-order derivative square-integrable function defined on [, ) with bounded second-order derivative, say f (x) A for some constant A, then (i) f(x) can be expanded as an infinite sum of the HTKCPBPF and the series converges to f(x) uniformly, that is f(x) n m c nm φ nm (t), c nm f(x), φ nm (x) L ω [,). (ii) [ β f,n,m A 8 f(x) n m n+ mm β f,n,m M n 5 (m ) 4, c nm φ nm (x) ω n (x)dx].

5 R. Jafari et al. /IJIM Vol. 9, o. 4 (7) Table : shows some valuse of the solutions and absolute errors x Hybrid function Exact solution Absolute error Table : shows some valuse of the solutions and absolute errors x Hybrid function Exact solution Absolute error Table 3: shows some valuse of the solutions and absolute errors x Hybrid function Exact solution Absolute error Proof. To prove (i), we have: (i)c nm f(x), φ nm (x) L ω [,) n n ω n (x)φ nm (x)f(x)dx f(x) V m(x n + ) ω(x n + )dx. Let t (x n + ) then dt dx. Clearly, we have c nm f( t + n + t )V m (t) t dt. By letting t cosθ and the definition of the HTKCPBPF, it follows that c nm cosθ + n f + f( (cos mθ + cos(m + )θ)dθ [ cosθ + n fcos mθ cosθ + n )cos(m + )θdθ]. Using the integration by parts, we have c nm 4 [ f cosθ + n m (sin mθsinθ)dθ+

6 354 R. Jafari et al. /IJIM Vol. 9, o. 4 (7) and f ( m + cosθ + n ) (sin(m + )θsinθ)dθ] I m f ( (sin mθsinθ)dθ, I m + cosθ + n ) f ( (sin(m + )θsinθ)dθ. cosθ + n ) 4 [I + I ], (5.8) ow, we estimate I and I, respectively. A simple computation shows that I f cosθ + n m and [cos(m )θ cos(m + )θ]dθ f cosθ + n m [cos(m )θdθ f cosθ + n m cos(m + )θ]dθ I I, I [ f cosθ + n m cos(m )θdθ, I f cosθ + n m cos(m + )θdθ. By using the integration by parts, and for m >, we get I I f cosθ + n 4m(m ) [sin(m )θsinθ]dθ f cosθ + n 8m(m ) [cos(m )θdθ cos mθ]dθ, f cosθ + n 4m(m + ) [sin(m + )θsinθ]dθ 8m(m + ) f ( cosθ + n ) [cos mθ cos(m + )θ]dθ. Thus, for m >,we conclude that and hence I f cosθ + n 8m cos(m )θ cos mθ [ (m ) cos mθ cos(m + )θ ]dθ, (m + ) I f cosθ + n 8m cos(m )θ cos mθ [ (m ) cos mθ cos(m + )θ ]dθ (m + ) 64m f cosθ + n cos(m )θ cos mθ [ (m ) cos mθ cos(m + )θ ]dθ. (m + )

7 R. Jafari et al. /IJIM Vol. 9, o. 4 (7) By the fact that f (x) A and Schwartz inequality, it follows that I A 64m (m ) (m + ) (m + )cos(m )θ + mcos mθ+ [ (m )cos(m + )θ dθ A 64m (m ) (m + ) (m + ) cos (m )θdθ+ 4m cos mθdθ+ (m ) cos (m + )θdθ] A 64m (m ) (m + ) [ (m + ) + 4m + (m ) ] For m >, we obtain A (3m + ) 64m (m ) (m + ) I In a similar way, we will have I A 4 (m ) 4. A (m ) (5.9) A (m ) (5.) Therefore, for m >, we conclude that c nm 4 4 [I + I ] A (m ) A n 3 (m ) (5.) ote that f (x) is bounded on [, ) due to the fact that f (x) A, indeed, by the Differential Mean Value Theorem and for any t (, ), there exists some γ x (, x) such that So f (x) f () f (γ x )x, f (x) f () +A, for x (, ). Thus f (x) is bounded on [, ),say f (x) Ã for some constant Ã. Hence, by 5.8, we have c n, 4 [ f cosθ + n dθ + f cosθ + n dθ] and c n, [ f ( cosθ + n ) dθ 3Ã 3Ã 4 4 n (5.) f cosθ + n dθ f cosθ + n dθ] f ( cosθ + n ) dθ 5Ã 5Ã 4 6 n (5.3) Relations show that the series n m c nm is absolutely convergent. For m and according to the definition of φ n, (x), the series n c n,φ n, (x) is convergent. Therefore, the series n m c nmφ nm (x) converges to f(x) uniformly.

8 356 R. Jafari et al. /IJIM Vol. 9, o. 4 (7) (ii) β f,n,m f(x) ( M n m n+ mm n+ mm n ) n y nm c nm φ nm (x) ω n (x)dx y nm φ nm (x) ω n (x)dx V m (x n + ) + (x n + ) (x n + ) dx. Let t x n + then dt ndx. Therefore βf,n,m n+ mm we have c nm + t Vm(t) t dt, + t Vm(t) dt, t the last equality follows due to the orthogonality of φ nm (x). Together with 5. we get β f,n,m A 8 n+ mm n 5 (m ) 4. 6 Solution of Fredholm integrodifferential equation Consider the linear Fredholm integro-differential equation given by.. We approximate u(x),k(x, t),f(x) by the way mentioned in section as: u(x) C T φ(x), K(x, t) φ T (x)kφ(t), f(x) F T φ(x), (6.4) by using 4.4 we have u (x) C T Dφ(x). (6.5) With substituting in. we have C T Dφ(x) F T φ(x) + C T φ(x)+ φ T (x)kφ(t)φ T (t)cdt. (6.6) Applying 4.6, the residual R(x) for. can be written as: R(x) C T Dφ(x) F T φ(x) C T φ(x) φ T (x)kec. (6.7) As in a typical Tau method, we generate M linear equations by applying ω(x)r(x)φ i (x)dx, ı,,..., M. (6.8) Also, by substituting initial conditions. we have u() C T φ() a, (6.9) Eqs generate M set of linear equations. These linear equations can be solved for unknown coefficients of the vector C. 7 umerical Examples In this Section, linear Fredholm integrodifferential equation have been solved using the proposed method. Example 7. Consider the integro-differential equation u (x) 3 x + xtu(t)dt, u(). (7.3) with the exact solution u(x) x. We apply the method that was explained in Section 6 for, M 3. After performing some manipulations, the components of the vector C are given by Thus u(x) C T φ(x). c 3 4, c 4, c. u(x) c φ (x) + c φ (x) + c φ (x) (7.3) ) ( (4x 3) (6x x + 5) which is the exact solution. x, (7.3)

9 R. Jafari et al. /IJIM Vol. 9, o. 4 (7) Example 7. Consider the integro-differential equation u (x) xe x + e x x + xu(t)dt, u(). (7.33) with the exact solution u(x) xe x. We apply the method that was explained in Section 6 for, M 4. After performing some manipulations, the components of the vector C are given by Thus c.4557, c.56498, c.6836, c 3.4, u(x) x x x 3. (7.34) Table shows some values of the solutions and absolute errors at some x, and plot of the exact and approximate solutions are shown in Figure. u (x) u(x) x + u(), (ln) +x x +t y(t)dt, ln( + x)+ (7.37) with the exact solution u(x) ln( + x). We apply the method that was explained in Section 6 for, M 5. After performing some manipulations, the components of the vector C are given by Thus c.3875, c.789, c.94358, c 3.578, c u(x) x.45577x +.76x x 4. (7.38) Table 3 shows some values of the solutions and absolute errors at some x, and plot of the exact and approximate solutions are shown in Figure 3. Example 7.3 Consider the integro-differential equation u (x) 3e 3x 3 (e3 + )x + 3xtu(t)dt, u(). (7.35) with the exact solution u(x) e 3x. We apply the method that was explained in Section 6 for, M 5. After performing some manipulations, the components of the vector C are given by Thus c 8.745, c 4.69, c.34, c 3.378, c u(x) x + 5.3x 7.95x x 4. (7.36) Table shows some values of the solutions and absolute errors at some x, and plot of the exact and approximate solutions are shown in Figure. Example 7.4 Consider the integro-differential equation 8 Conclusion In this paper, we constructed operational matrix of derivative of hybrid the third kind Chebyshev polynomials and Block-pulse functions. Also, we applied these matrices to convert Fredholm integro-differential equations to system of linear algebraic equations. As to validity and efficiency of the proposed method, we presented some numerical examples. References [] S. H. Behiry, Solution of nonlinear Fredholm integro-differential equations using a hybrid of Block pulse functins and normalized Bernstein polynomials, Comput. Appl. Math. 6 (4)58-65 [] H. Danfu, Sh. Xufeng, umerical solution of integro-differential equations by using CAS wavelet operational matrix of integration, Appl. Math. Comput. 94 (7) [3] M. Dehghan, F. Shakeri, Approimate soluation of differential equation arising in astrophysics using the variation method, ew Astronomy 3 (8) 53-59

10 358 R. Jafari et al. /IJIM Vol. 9, o. 4 (7) [4] P. Darania, A. Ebadian, A method for the numerical solution of the integrodifferential equations, Appl. Math. Comput. (6), [5] R. Ezzati, S. ajafalizadeh, umerical solution of nonlinear Volterra-Fredholm integral equation by using Chebyshev polynomials, Appl. Math. Comput. (3) [6] A. Golbabai, M. Javidi, Application of He s homotopy perturbation method for nthorder integro-differential equations, Appl. Math. Comput. 9 (7) [7] J. Hou, Ch. Yang, umerical method in solving Fredholm integro-differential equations by using hybrid functin operational matrix of derivative, Journal of Information and Computational Science (3) [8] H. Jaradat, O. Alsayyed, S. Al-Shara, umerical solution of linear integro-differential equations, Journal of Mathematics and Statistics (8) [9] Z. H. Jiang, W. Schaufelberger, 99, Blockpulse Functions and their Applications in Contorl Systems, Springer-Verlag Berlin Heidelberg ew York 37. [] A. Karamete, M. Sezer, A Taylor collocation method for the solution of linear integrodifferential equations, Int. J. Comput. Math. 79 () 987- [] S. J. Liao, An explicit analytic solution to the Thomas-Fermi equation, Appl. Math. Comput. 44 (3) [] K. Maleknejad, F. Mirzaee, umerical solution of integro-differential equations by using rationalized Haar functions method, Kybernetes, Int. J. Syst. Math. 35 (6) [3] K. Maleknejad, M. Tavassoli Kajani, A hybrid collocation metod based on combining the third kind Chebyshev polynomials and Block-pulse functions for solving higer-order initial value problems, Kuwait journal of Science (6). [5] S. Yeganeh, Y. Ordokhani, A. Saadatmandi, A Sinc-collocation method for secondorder boundary value problems of nonlinear integro-differential equation, J. Inf. Comput. Sci. 7 () 5-6. Reza Jafari was born in Miandoab, Iran, in 976. He obtained his M.Sc degree in Applied Mathematics from Islamic Azad University Karaj Branch, Iran, in 7 and he is currently Ph.D. Student in IAU- Karaj branch. Also he is one of the researcher in this university. R. Ezzati received his PhD degree in applied mathematics from IAU- Science and Research Branch, Tehran, Iran in 6. He is an professor in the Department of Mathematics at Islamic Azad University, Karaj Branch, (Iran) from 5. He has published more than papers in international journals, and he also is the associate editor of Mathematical Sciences (a Springer Open Journal). His current interests include numerical solution of differential and integral equations, fuzzy mathematics, especially, on solution of fuzzy systems, fuzzy integral equations, and fuzzy interpolation. K. Maleknejad received his PhD degree in Applied Mathematics in umerical Analysis area from the University of Wales, Aberystwyth, UK in 98. He has been a professor since, at IUST. His research interests include numerical in solving ill-posed problems and solving Fredholm and Volterra integral equations. He has authored as the editor-in-chief of the International Journal of Mathematical Sciences, which publishers by Springer. [4] J. C. Mason, D. C. Handscomb, Chebyshev polynomials () ISB: