Research Journal of Recent Sciences _________________________________________________ ISSN 2277-2502 Vol. 3(12), 75-77, December (2014) Res.J.Recent Sci. International Science Congress Association 75 Solving Integral equations on Semi-Infinite Intervals via Rational third kind Chebyshev functions Karimi Dizicheh A., Davari A. and Tavassoli Kajani M.Department of Mathematics, Mobarakeh Branch, Islamic Azad University, Mobarakeh, IRAN Department of Mathematics, Faculty of Sciences, University of Isfahan, Isfahan, IRAN Department of Mathematics, Khorasgan Branch, Islamic Azad University, Isfahan, IRAN Available online at: www.isca.in , www.isca.me Received 12th July 2013, revised 4th January 2014, accepted 28th May 2014Abstract In this paper, we employ the rational third kind Chebyshev functions on the interval [0, ¥ , to solve the linear integral equations of the second kind over infinite intervals. The properties of the rational third kind Chebyshev functions together with the Galerkin method are applied to reduce the integral equation to a system of linear algebraic equations. Using two numerical examples, we show that our estimates have a good degree of accuracy. Keywords: Integral equation, Rational third kind Chebyshev functions, Semi-infinite interval, Galerkin method. IntroductionIn recent years, many different basic functions have been used to estimate the solution of integral equations, such as wavelets1-, orthonormal bases4,5 and combination of Block-Pulse functions6,7. Besides many different method have been used to estimete the solution of mathematics equations, see8,9. In this paper we are going to use an efficient base that is rational third kind Chebyshev functions on [0, ¥ , which is called RTC functions. Properties of RTC functions In this section, we present some properties of RTC functions. RTC functions: The third kind Chebyshev polynomials are orthogonal in the interval 1,1] - with respect to the weight function 1+x (x)= 1-x and we find that satisfies the recurrence relation10 V(x)=1,V(x)=2x-1,01 V(x)=2xV(x)-V(x), n³2. n-1n-2 (1) The RTC functions are defined by x-L R(x)=V, nn x+L    thus RTC functions satisfy x-LR(x)=1,R(x)=2-1,01x+Lx-L R(x)=2R(x)-R(x), n³2. n-1n-2x+L (2) Function approximation: Let 2Lx w(x)= 2 (x+L) denotes a non-negative, integrable, realvalued function over the interval [0, +¥ . We define { } 2 L(I)=y:I®R | y is measurable and y¥, (3) where () 1 2¥ 2 y=|y(x)|w(x)dx, (4) is the norm induced by the scalar product ¥ ,&#xy32.;Í¥z=y(x)z(x)w(x)dx. (5) Thus )} denote a system which are mutually orthogonal under Eq. (5), i.e., ¥ R(x)R(x)w(x)dx=  0 nmnm (6) where nm d is the Kronecker delta function11,12. This system is complete in ; as a result, any function can be expanded as follows: Research Journal of Recent Sciences _____________________________________________________________ ISSN 2277-2502Vol. 3(12), 75-77, December (2014) Res.J.Recent Sci. International Science Congress Association 76 ¥ y(x)=aR(x), kkk=0 (7) with 1 a=y,R&#x-6.2;ã–’. w kk(8) The 's are the expansion coefficients associated with the family )}. If the infinite series in Eq. (7) is truncated, then it can be written as y(x)»aR(x)=AR(x), kkk=0 (9) where T A=[a,a,…,a], N 01 T R(x)=[R(x),R(x),…,R(x)]. N 01 We can also approximate the function in L(I×I) as follows k(x,t)»k(x,t)=R(x)KR(t), M (10) where K is an M M ´ matrix that 1 K=R(x),k(x,t),R(t)&#x-3.5;é¡&#x-3.5;é¡, i,j=0,1,…,M. wwijij  Product integration of the RTC functions: We also define the matrix as follows P=R(t)R(t)dt. 0 a (11) To illustrate the calculation we choose , we obtain 83 11-4ln29-12ln217-24ln2-40ln2L 5391 1-4ln29-8ln29-16ln2-28ln2-44ln2L 335967559 9-12ln29-16ln2-24ln2-36ln2-52ln2L 33155967559623 17-24ln2-28ln2-36ln2-48ln2-64ln2L 3315158391559623-40ln2-44ln2-52ln2331515 . 6217 -64ln2-80ln2L 105 MMMMMO \n  Second kind integral equations over semi-infinite interval: In this phase, at first we consider the following second kind integral equation, y(x)=f(x)+k(x,t)y(t)dt, xÎI, (12) where and . Then we approximate , y and using (9) and (10) as follows y(x)»YR(x),f(x)»FR(x),k(x,t)»R(x)KR(t).With substituting in (12) we have TTTT R(x)Y=R(x)F+R(x)KR(t)R(t)Ydt TTT =R(x)F+R(x)KR(t)R(t)dtY =R(x)(F+KPY), then one can conclude that (I-KP)Y=F, N+1a(13) where is the identity matrix. By solving this linear system of algebraic equations we can find the vector Y . Numerical examples With best of our knowledge this is the first time that the following examples are solved. Example 1.: Consider the integral equation 14y(t) y(x)=-+dt, xÎI, x+1(x+1)(t+1) (14) with the exact solution 1 y(x)= x+1 . In order to solve this example using the present method, we choose 1 = L and therefore we have 1- 1-1 14 F=, K=. 4 1-11 \n \n \r \rSo by solving the linear system (I-KP)Y=F 21 we obtain T 11 Y= - 44 \n \r , thus 111 T y(x)=YR(x)=R(x)-R(x)=, 01 44x+1 which is the exact solution. Example 2.: Consider the integral equation Research Journal of Recent Sciences _____________________________________________________________ ISSN 2277-2502Vol. 3(12), 75-77, December (2014) Res.J.Recent Sci. International Science Congress Association 77 -x-2-x-t y(x)=e+2ey(t)dt, xÎI, (15) with the exact solution -x y(x)=e . In Table 1, a comparison is made between the values of y obtained using the proposed method with N=9 14 and the exact solution. Table-1 Numerical results of for Example 2 x 14Exact 0 1.01004 1.00109 1.00000 1 0.36957 0.36791 0.36788 2 0.13467 0.13540 0.13534 3 0.05060 0.04972 0.04979 4 0.01924 0.01836 0.01832 5 0.00712 0.00681 0.00674 6 0.00206 0.00248 0.00248 7 0.00042 0.00085 0.00091 8 0.00050 0.00026 0.00034 9 0.00020 0.00006 0.00012 10 0.00009 0.00001 0.00005 ConclusionThe fundamental goal of this paper has been to construct an approximation to the solution of the second kind integral equations in a semi-infinite interval. In the above discussion, the Galerkin method with RTC functions, which have the property of orthogonality, were employed to achieve this goal. The contribution of this paper is that we do not reform the problem to a finite domain and with an small value of accurate results are obtained. There is a good agreement between obtained results and exact values that demonstrates the validity of the present method for this type of problems and gives the method a wider applicability. 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