RESEARCH ARTICLE. Spectral Method for Solving The General Form Linear Fredholm Volterra Integro Differential Equations Based on Chebyshev Polynomials

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1 Journal of Modern Methods in Numerical Mathematics ISSN: online Vol, No, 2, c 2 Modern Science Publishers wwwm-sciencescom RESEARCH ARTICLE Spectral Method for Solving The General Form Linear Fredholm Volterra Integro Differential Equations Based on Chebyshev Polynomials H M El-Hawary and T S El-Sheshtawy Department of Mathematics, Faculty of Science, Assiut University, Assiut 756, Egypt (Received: 3 August 2, Accepted: 5 October 2) The main aim of this paper reports a pseudo spectral method based on integrated Chebyshev polynomials for numerically solving higher order linear Fredholm Volterra integro differential equations (FVIDE) in the most general form This method transforms FVIDE to matrix equations which correspond to a system of linear algebraic equations with unknown Chebyshev coefficients The present numerical results are in satisfactory agreement with the exact solutions and show the advantages of this method to some other known methods Keywords: Chebyshev collocation method; Chebyshev series approximation; FredholmVolterra integro-differential equations AMS Subject Classification: 65R Introduction The solutions of integral equations have a major role in the fields of science and engineering A physical event can be modeled by the differential equation, an integral equation or an integro-differential equation or a system of these Since few of these equations cannot be solved explicitly, it is often necessary to resort to numerical techniques which are appropriate combinations of numerical integration and interpolation [, 2] In this paper, we will consider general linear FredholmVolterra integro-differential equation in the form m P n (x)y (n) (x) = g(x) + λ under conditions F i (x, t)y (i) (t)dt + λ 2 K j (x, t)y (j) (t)dt () y (i) (c i ) = µ i ; i =,,, m, c i (2) y(x) is an unknown function, the functions g(x), P n (x), F i (x, t), K j (x, t) are defined on an interval x, t and also c i, λ, λ 2, µ i are constants For special cases λ or λ 2 is zero and without derivatives of the unknown in the integrand, some methods are given in [3, 4] and also Lagrange interpolation method [5] is given for the second order Volterra integro differential equation with derivatives of the unknown Corresponding author adresses: elhawary@aunedueg (H M El-Hawary), tareksalah@yahoocom (T S El-Sheshtawy)

2 2 2 Fundamental relations Let us write Eq () in the form D(x) = g(x) + λ S(x) + λ 2 J(x) (3) the differential part m D(x) = P n (x)y (n) (x) Fredholm integral part S(x) = F i (x, t)y (i) (t)dt and Volterra integral part J(x) = K j (x, t)y (j) (t)dt 3 Matrix Form for General Solution of The equation This is accomplished by highest-order derivative and generating approximations to the lower derivatives through successive integration of the highest-order derivative Hence assume that the function y (m) (x) and its derivatives have truncated Chebyshev series expansion of the form be written in the matrix forms y (m) (x) = T (x)a N y (m) (x) = a k T k (x) k= y () (x) = T M A + c m y (m 2) (x) = T M 2 A + c m (x + ) + c (4) Then the solution expressed by y (i) (x) = T M m i A + C i X i y(x) = T M m A + C X

3 () N+ (N+) 2 2 M = 4 4 2(N+) 2(N+m) taking M m (i, j) =, for i =,, N + m, j = N +,, N + m (N+m+) (N+m+) T (x ) T (x ) T N+m (x ) T = T (x ) T (x ) T N+m (x ), A = [ 2 a, a,, a N+m ] T, T (x N ) T (x N ) T N+m (x N ) X i = [, (x + ) (x + )m i,,! (m i )! ]T, C i = [c i, c i+,, c ] taking a k = for k > N and c, c,, c are constant compute by boundary condition Now, the matrix representation of differential part defined in (2) can be given by D(x) = = m P n (x)(t M m n A + C n X n ) (5) m P n (x)t M m n A + P n (x)c n X n for evaluating the m-fold integral of the interpolating polynomial is P N f(x) I (m) (x i ) = i P N f(x)dxdx = T M m A (6) 4 Matrix Representation for Fredholm Integral Part Let us assume that F i (x, t) is expanded to the Chebyshev series in the form N F i (x, t) = f ir (x)t r (t) r= a summation symbol with double primes denotes a sum with first and last terms halved and Chebyshev coefficients f ir (x) are determined by means of the relation f ir (x) = 2 N + N π(2j + ) F i (x, t j )T r (t j ), t j = cos( ), j =,,, N 2(N + )

4 4 Then the matrix representations of F i (x, t) become F i (x, t) = F i (x)t T (t) (7) F i (x) = [ 2 f i(x), f i (x),, f in (x)] Substituting the relations (4) and (7) in Fredholm integral part, we obtain S(x) = (F i (x)zm m i A + C i I i ) (8) so that Z = [ ] T T (t)t (t)dt = Tp T (t)t q (t)dt = z pq, p =,,, N, q =,,, N + i whose entries are given by Fox and Parker [6], and z pq = { (p+q) 2 + (p q), for even p + q; 2, for odd p + q (t + )k F i (x, t) k! N = F ir (x)t (t) = T F i (x), k =,,, m i r= F ir (x) = 2 N N F i (x, t j ) (t j + ) k T r (t j ) k! ti ik (x, t (t + )k i) = F i (x, t) dt = T MF i (x) k! I i = [ i (x, ), I() i (x, ),, I() i(m i) (x, )]T (9) 5 Matrix Form for Volterra Integral Part Similarly the previous section, suppose that the kernel functions k j (x, t) is expanded to the Chebyshev series in the form N K j (x, t) = k jr (x)t r (t) r=

5 5 k jr (x) = 2 N + N π(2i + ) K j (x, t i )T r (t i ), t i = cos( ), i =,,, N 2(N + ) Then the matrix representations of K j (x, t) become K j (x, t) = K j (x)t T (t) () K j (x) = [ 2 k j(x), k j (x),, k jn (x)] Substituting the relations (4) and () in Fredholm integral part, we obtain so that β = J(x) = (K j (x)βm m j A + C j I j2 ) () T T (t)t (t)dt = [ Tp T (t)t q (t)dt] = β pq (x), p =,,, N, q =,,, N + j whose entries are computed by Akyüz [7], we obtain 2x 2 2, for p + q = ; T p+q+(x) p+q+ Tp+q(x) p+q p+q+ + p+q + x2, for p q = β pq = 4 T p+q+(x) p+q+ + T p q+(x) p q+ + Tp q+(x) p q+ + T p+q+(x) p+q+ + 2( (p+q) 2 + (p q) 2 ), for even p + q T p+q+(x) p+q+ + T p q+(x) p q+ + Tp q+(x) p q+ + T p+q+(x) p+q+ 2( (p+q) 2 + (p q) 2 ), for even p + q (t + )k K j (x, t) k! N = K jr (x)t (t) = T K j (x), k =,,, m j r= K jr (x) = 2 N + N K j (x, t i ) (t i + ) k T r (t i ) k! I j2 = [ j (x, x), I() j (x, x),, I() j(m j) (x, x)]t

6 6 6 Matrix Form for The Conditions firstly substituting boundary conditions defined in (2) into equations (4) we obtain c, c,, c the matrix forms of constants (2) become µ c i = C i µ i C i T i MA (2) µ i T (c i )M m i µ i+ µ i =,T T (c i+ )M m i im =, T (c )M and C i = [C i, C i+,, C ] determined by boundary condition 7 Method of Solution We are now ready to construct the fundamental matrix equation corresponding to Eq () For this purpose, firstly substituting the matrix relations (5), (8) and () into Eq (3) we obtain m P n (x)t M m n A + P n (x)c n X n = g(x) + λ from equations (9), (2) we get (F i (x)zm m i A + C i I i ) + λ 2 (K j (x)βm m j A + C j I j2 ) C i I i = (C j µ j C j T j M) (j i)(x, ), i =,,, m (3) C j I j2 = (C i µ i C i T i M) (i j)2(x, x), j =,,, m (4) secondly substituting equations (3), (4) we obtain m P n (x)t M m n A + P n (x)c n X n = g(x) + λ + λ 2 C i µ i j(i j) (x, x) + λ i(j i) (x, )C j T j MA) + λ 2 j(i j) (x, x)c i T i MA) C j µ j i(j i) (x, ) (F i (x)zm m i A (K j (x)βm m j A

7 m P n (x)t M m n A λ F i (x)zm m i A λ 2 + λ = g(x) + λ K j (x)βm m j A i(j i) (x, )C j T j MA + λ 2 m P n (x)c n X n C j µ j i(j i) (x, ) + λ 2 j(i j) (x, x)c i T i MA C i µ i j(i j) (x, x) 7 For computing the Chebyshev coefficient matrix A numerically, Chebyshev collocation points defined by π(2s + ) x s = cos( ), s =,,, N 2(N + ) are substituted above relation Thereby, the fundamental matrix equation is gained m { P n T M m n λ F i ZM m i λ 2 + λ 2 + λ 2 K j βm m j + λ j(i j) (x s, x s )C i T i M}A = g(x s ) + λ C i µ i j(i j) (x s, x s ) m P n C n X n i(j i) (x s, )C j T j M C j µ j i(j i) (x s, ) (5) P n (x ) K j (x ) P n (x 2 ) P n =, K K j (x 2 ) j = P n (x N ) K j (x N ) F i (x ) β i (x ) F i (x ) F i =, β = β i (x ) F i (x N ) β i (x N ) Denoting the expression in parenthesis of Eq (5) by W, the fundamental matrix equation for Eq () is reduced to W A = G

8 8 which corresponds to a system of N + linear algebraic equations with unknown Chebyshev coefficients a, a,, a N The method can be developed for the problem defined in the domain [, ] m P n (x)y (n) (x) = g(x) + λ F i (x, t)y (i) (t)dt + λ 2 K j (x, t)y (j) (t)dt to obtain the solution in terms of shifted Chebyshev polynomials T r (x) in the form N y n (x) = a k Tk (x) k= be written in the matrix forms y (m) (x) = T (x)a y () (x) = 2 T M A + c n y (m 2) (x) = 2 2 T M 2 A + c n x + c n y (i) (x) = 2 m i T M m i A + r=i c r x r i (r i)! Then the solution expressed by y(x) = 2 m T M m A + CX X = [, x!,, x (m )! ]T, A = [ 2 a, a,, a N+m] T T r (x) = T r (2x ) It is followed the previous procedure using the collocation points defined by x s = 2 + ) ( + cos(π(2s )), s =,,, N 2(N + )

9 9 Then we obtain the fundamental matrix equation for FVIDE as m { P n T M m n λ + λ 2 + λ 2 F i Z M m i λ 2 Kj βm m j + λ j(i j) (x s, x s )C i T i M}A = g(x s) + λ C i µ i j(i j) (x s, x s ) m P n C n Xn i(j i) (x s, )C j T j M C j µ j i(j i) (x s, ) It is easily obtained Z = 2Z, β = 2β 8 Numerical Results we present some numerical results indicating the merit of the Chebyshev method for FVIDE For computational purpose, we consider four test problems Example 8 Let us first consider the second order FredholmVolterra IDE y y + x2 2 y = x [y (t) + y (t) ty]dt, y() =, y () = the exact solution y(x) = x 2 Table : Maximum absolute errors for Example 8 N Lagrange interpolation [5] Present method 2 37E-8 3 5E E-8 25E-8 5 3E-6 25E E-7 69E E-6 5E E-7 37E E-6 93E-8 28E-8 Example 82 Let us consider two examples of Volterra integral and integro-differential equations with derivatives of the unknown in the integrand (a) y xy xy = e x 2 sin x sin xe t dt, y() =, y () =

10 The exact solution to this problem y(x) = e x (b) y + x 3 y + y = 2 π sin(πx)2ex + x 2 2x [cos(πt)y (t) + 3txy (t)]dt [(x 4 t)y(t) + t 2 y (t)]dt, y() = 2 3, y ( 2 ) = The exact solution to this problem y(x) = x 2 3 Table 2: Maximum absolute errors for Example (a), (b) N Example (a) Example (b) E E E E-6 369E E E E E E E-6 847E E-6 666E-6 Example 83 Secondly, consider the FVIDE in [, ] y (t) + xe x y + y + e x+t y(t)dt = 2e x x + xe x, y() =, y () = the exact solution y(x) = e x Table 3: Maximum absolute errors for Example 83 N Lagrange interpolation [5] Present method 2 3E-3 44E E-4 55E E-6 4E E-7 75E-9 6 8E-8 E- 7 6E- 5E E- 32E E-3 53E-6 25E-4 75E-7 Example 84 Consider the following Volterra integral equation: cos(x t)y (t)dt = 2 sin(x), y() =, y () =

11 REFERENCES the exact solution y(x) = x 2 Table 4: Maximum absolute errors for Example 84 N Akyüz [8] Present method 2 26E-4 2E-4 3 8E-5 23E E-7 3E E-8 7E-8 6 8E- 76E- 7 8E- E- 8 56E-4 39E E-5 43E-5 64E-7 Conclusion In this paper, we suggest a method for numerically computation of the solution of higher order linear FredholmVolterra integro-differential equations (FVIDE) in the most general form The present method has some major advantages: Chebyshev coefficients of the solution are found very easily by using the computer programs without any computational effort and this process is very fast Acknowledgements The authors are indebted to Professor SE El-Gendi for various valuable suggestions and constructive criticism References [] C T H Baker A perspective on the numerical treatment of volterra equations J Comput Appl Math, 25:27:249, 2 [2] P Linz Analytical and numerical methods for volterra equations SIAM, Philadelphia, PA, 985 [3] Akyüz and M Sezer A chebyshev collocation method for the solution of linear integro-differential equations Int J Comput Math, 72(4):49:57, 999 [4] Ş Nas, S Yalçnbaş, and M Sezer A taylor polynomial approach for solving high-order linear fredholm integro-differential equations Int J Math Educ Sci Technol, 3(2):23:225, 2 [5] MT Rashed Lagrange interpolation to compute the numerical solutions of differential integral and integro-differential equations Appl Math Comput, 5:869:878, 24 [6] L Fox and I B Parker Chebyshev polynomials in numerical analysis Analysis, Oxford University Press, London, 968 [7] AAkyüz Chebyshev collocation method for solution of linear integro-differential equations M Sc Thesis, Dokuz Eylül University, Graduate School of Natural and Applied Sciences, 997 [8] A Akyüz and Daşcioǧlu A chebyshev polynomial approach for linearfredholmvolterra integro-differential equations in the most general form Appl Math Comput, 8:3:2, 26