The Approximate Solution of Non Linear Fredholm Weakly Singular Integro-Differential equations by Using Chebyshev polynomials of the First kind

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1 AUSTRALIA JOURAL OF BASIC AD APPLIED SCIECES ISS: EISS: Journal home page: wwwajbaswebcom The Approximate Solution of on Linear Fredholm Weakly Singular Integro-Differential equations by Using Chebyshev polynomials of the First kind 1 DrEman Ali, 2 Drabaa ajdi, 3 Wafaa Abd and 4 Ohood Fadhil 1,2,3,4 Diyala University, College of Science, Department of Mathematics, Diyala, Iraq Address For Correspondence: DrEman Ali, Diyala University, College of Science, Department of Mathematics, Diyala, Iraq A R T I C L E I F O A B S T R A C T Article history: Received 18 February 2017 Accepted 5 May 2017 Available online 10 May 2017 Keywords: Chebyshev polynomials, Integrodifferential equation, onlinear Fredholm, Trapezodial rule, Weakly singular kernel In this paper, Chebyshev polynomials of the first kind method is used to solve non linear weakly singular Fredholm integro-differential equations (LFWSIDEs) of the second kind Chebyshev polynomials are used a basis and used Trapezodial rule as an integral in this method This techniques transform the non linear Fredholm weakly singular integro-differential equations to a system of a nonlinear algebraic equations application is presented to illustrate the efficiency and accuracy of this method ITRODUCTIO Integro-differential equations (IDEs) arise in many branches of science, for example in control theory and financial mathematicsides are important, but they are hard to solve even numerically, so the progress on how to solve them is slow IDEs are equations of the form where the unknown function appears under the sign of integration and they also contain the derivatives of the unknown function Some techniques have been used for solving singular integral and integro-differential equations Therefore many researchers used several numerical methods to solve weakly singular integral equations(wsies) including A Palamora, (Product integration for Volterra integral equations of the second kind with weakly singular kernels,math Comp 65 (215) (1996) ) Z Chen, Y Xu, J Zhao, (The discrete Petrov Galerkin method for weakly singular integral equations, J Integral Equations Appl 11 (1999)1 35)And Chen,Y,Tang,T,(Spectral methods for weakly singular Volterra integral equations with smooth solutions,jcomputapplmath233,pp ,2009) As well as several numerical methods were used to solve weakly singular integro- differential equations (WSIDEs) including H Brunner, A Pedas, G Vainikko,( Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weaklysingular kernels, SIAM J umer Anal 39 (2001) ),A Pedas, E Tamme,(Spline collocation method for integro-differential equations with weakly singular kernels, J Comput Appl Math 197(2006) )and Mehrdad Lakestani, Behzad emati Saray,Mehdi Dehghan (umerical Solution for the Weakly Singular Fredholm Integro-DifferentialEquations using Legendre multiwavelets, JComputApplMath235,pp ,2011) Open Access Journal Published BY AESI Publication 2017 AESI Publisher All rights reserved This work is licensed under the Creative Commons Attribution International License (CC BY) To Cite This Article: DrEman Ali, Drabaa ajdi, Wafaa Abd &Ohood Fadhil, The Approximate Solution of on Linear Fredholm Weakly Singular Integro-Differential equations by Using Chebyshev polynomials of the First kind Aust J Basic & Appl Sci, 11(9): 37-44, 2017

2 38 DrEman Ali et al, 2017 In this paper Chebyshev polynomials of the first kind of degree n are defined in section2in the section 3 the proposed method for solving on linear Fredholm weakly singular integro-differential equations is used application is given in section 4 for confirming the efficiency of the proposed method Section 5 contains conclusions of the paper 2 Chebyshev Polynomials Of the first Kind T n (x), (Mason, Handscomb, 2003): The Chebyshev polynomials of the first kind of degree n is a set of orthogonal polynomials and it is defined by the recurrence relation: T 0 (x) = 1 T 1 (x) = x T n+1 (x) = 2xT n (x) T n 1 (x), for each n 1 (1) 21 Properties of Chebyshev polynomials T n (x): 1The Chebyshev Polynomials of the first kind T n (x), n=0,1, are aset of orthogonal polynomials over the interval [-1,1] with respect to the weight function w(x) = (1 x 2 ) 1 2, that is : 0 n m 1 π w(x)t n (x)t m (x)dx = { 1 2 n = m 0 (2) π n = m = 0 2The Chebyshev polynomials of the first kind can be defined by the trigonometric identity T n (cos(θ)) = cos (nθ)for n=0,1,2,3, 3 T n (x)has n distinct real roots x i on the interval [-1,1], these roots are defined by : x i = cos ( (2i+1)π ), i = 0,1,2,, 1 (3) 2 are called Chebyshev nodest n (x) assumes its absolute extrema at x j = cos ( jπ ) for j = 0,1,2,, (4) 4 A polynomial of degree in Chebyshev form is a polynomial p(x) = n=0 a n T n (x) (5) Where T n is the n th Chebyshev form The first few Chebyshev polynomials of the first kind for =0,1,2,3,4,5 are given in figure(1) Fig 1: The first few Chebyshev polynomials of the first kind for =0,1,2,3,4,5 22 Shifted Chebyshev polynomials: Shifted Chebyshev polynomials are also of interest when the range of the independent variable is [0,1] instead of [-1,1]The shifted Chebyshev polynomials of the first kind are defined as T n (x) = T n (2x 1), 0 x 1 (6)

3 39 DrEman Ali et al, 2017 Similarly, one can also build shifted polynomials for a generic interval [a,b] where x i = b a 2 x i + b+a 2 (7) The first few Chebyshev polynomials of the first kind for =0,1,2,3,4,5 for interval [0,1] are given in figure(2) Fig 2: The first few shifted Chebyshev polynomials of the first kind for =0,1,2,3,4,5 3 The Approximate Solution of onlinear Fredholm Weakly Singular Integro-Differential Equations: Let us consider the following first-order LFWSIDEs of the following form: b y (x) + p(x)y(x) + k(x, t, y(t))dt = f(x) x [a, b] (8) a with initial condition y(a) = β where β is a constant and k and fare given functions and y is the solution to be determined Moreover we assume that the kernel k(x, t, y(t)) = H(x,t,y(t)) x t α x, t [a, b] with where 0 < α < 1 As well as we assume that the kernel H is in L 2 [a, b]and the unknown y and the right hand side f are in L 2 [a, b] Also we suppose that y(x) satisfies in the Lipschitz condition with respect to x, y(x 1 ) y(x 2 ) L x x 1 x 2 (9) to determine an approximate solution of (9) firstly if the function y(x) defined in [ 1,1] We suppose this function may be represented by first kind CPs (Qinghua, 2014): y(x ) i=0 T i (x )b i (10) If we truncated the series (10), then we can write (10) as follows: y(x ) i=0 T i (x )b i T(x )B (11) k(x, t, y(t)) k(x, t, i=0 T i (t)b i ) k(x, t, T(t)B) (12) y (x ) ( i=0 T i (x )b i ) (T(x )B) (13) Where (x ) = [T o (x ), T 1 (x ),, T (x )], B = [b 0, b 1, b 2,, b ] T clearly T is 1 ( + 1) vectors and B is ( + 1) 1 vectors We first substitute the Chebyshev nodes,which defined by x i = cos ( (2i+1)π ), 2 i = 0,1,2,, 1 into (11) and (12) and (13) then rearrange a new matrix form to determine B: y + py + k = f (14) In which kis the nonlinear integral part of (8) and y = y (x 0) y (x 1) ( y (x )), py = p(x 0)y(x 0) p(x 1)y(x 1) ( p(x )y(x )), f = f(x 0) f(x 1) ( f(x )), k = k(x 0) k(x 1) ( k(x )) (15)

4 40 DrEman Ali et al, 2017 by substituting (11) and (12) and (13) into (14) gives nonlinear algebraic equations in (+1) unknown coefficients These equations are solved by using (Matlab R2010b) to obtain the unknown coefficients B which are then substitute into (11) to get the approximate solution of (8) Or if the function y(x) defined in [0,1]we use shifted CPs by using the transformation x = 1 2 [(b a)x + (a + b)]transforms the nodes x i in [ 1,1] into the corresponding nodes x i in [0,1] 31 The Algorithm for solving (LFWSIDEs) into [-1,1]: Input : a, b, α,, M, m, y(x), f(x), p(x), ε Output : The approximate solution of the LFWSIDEs Step 1: process : Find T i (x ), (T i (x )) (CPs) Step 2: Find roots x i, i = 0,1,, (roots of CPs) Step 3: Find roots t ij = a + j + k ± ε, k = b a, i = 0,,, j = 0,, M, x = t Step 4: Calculate (T ij (x )) m where m > 1 M Step 5: Calculate R i = k [Y 2 i0 + 2 M 1 k=1 Y ik + Y im ] (Trapezoidal rule) Where Y ij = (T ij (t))m x i t ij α, i = 0,1,,, j = 0,1,, M Step 6: Construct the system : A ik = b i ((T k (x i)) + p(x i) T k (x i)) + (b i ) m R i, B i = f(x i) i, k = 0,1,, Step 7: let b i = 0, i = 0,1,, (initial values) Step 8: Solve the non-linear system A ik = B i using the library function f solve and find the unknowns b i Step 9: Calculate the approximate function y (x ) = i=0 T i (x )b i Step 10: Calculate absolute error is the comparison between the exact andthe approximate solutions Step 11: ED of the process 32 The Algorithm for Solving (LFWSIDEs) Into[0,1]: Input: a, b, α,, M, m, y(x), f(x), p(x), ε Output : The approximate solution of the LFWSIDEs Step 1 : process: Find T i (x ) (CPs) Step 2: Find T i (x ), (T i (x )) (shifted CPs) Step 3 : Find roots x i, i = 0,, ( roots of CPs) Step 4: Find roots x i by using the transformation x i = 1 2 [(b a)x i + (b + a)], i = 0,, (roots of shifted CPs) Step 5: Find roots t ij = a + j + k ± ε, k = b a M, i = 0,, j = 0,, M, x = t Step 6: Calculate (T ij (x )) m where m > 1 i, j = 0,1,, Step 7: Calculate R i = k [Y M 1 2 i0 + 2 k=1 Y ik + Y im ] (Trapezoidal rule ) Where Y ij = (T ij, i = x i t ij α 0,1,,, j = 0,1,, M Step 8: Construct the system : A ik = b i ((T k (x i)) + p(x i) T k (x i)) + (b i ) m R i, B i = f(x i), i, k = 0,1,, Step 9: let b i = 0, i = 0,1,, (initial values) Step 10: Solve the non-linear system A ik = B i using the library function f solve and find the unknowns b i Step 11: Calculate the approximate function y (x ) = i=0 T i (x )b i Step12: Calculate absolute error is the comparison between the exact and the approximate solutions Step 13: ED of the process 4 Applications: In this section, we give applications to explain the applicability and effectiveness of the proposed method The Computations have been performed by using Matlab R2010b y 1 (x ) + p(x )y(x ) + x t 1 2 (y(t )) 2 dt = f(x ) (16) 0 with initial condition y(0) = 0 here the forcing function f is selected such that the exact solution is y(x ) = 2x, p(x ) = 2222x (t )) m

5 41 DrEman Ali et al, 2017 Table 1: illustrate the comparison between the exact and the approximate solution depending on Mean Square error (MSE) and Elapsed Time(ET) x -values Exact solution Approximate solution Absolute error MSE 2370e-003 ET Sec Results obtained and errors for example(1): =10 Table 2: illustrate the comparison between the exact and the approximate solution depending on Mean Square error (MSE) and Elapsed Time(ET) x -values Exact solution Approximate solution Absolute error MSE 2074e-003 ET Sec Results obtained and errors for example(1): =12 Table 3: illustrate the comparison between the exact and the approximate solution depending on Mean Square error (MSE) and Elapsed Time(ET) x -values Exact solution Approximate solution Absolute error

6 42 DrEman Ali et al, MSE 2006e-003 ET Sec Results obtained and errors for example(1): =14 Fig 3: A Comparison between the exact and the approximate solution using expansion method of Chebyshev polynomials of the first kind of application for =10

7 43 DrEman Ali et al, 2017 Fig 4: A Comparison between the exact and the approximate solution using expansion method of Chebyshev polynomials of the first kind of application for =12 Fig 5: A Comparison between the exact and the approximate solution using expansion method of Chebyshev polynomials of the first kind of application for =14 Conclusions: In this paper, we have submitting expansion method using Chebyshev polynomials of the first kind of degree n as basis function for approximating the solution of one weakly singular integro-differential equations: which is the LFWSIDEs In example (1) we have reduced the solution of LFWSIDEs to the system of nonlinear equations by removing the singularity using an approximate point t,and we have the following results:when the degree of expansion method of Chebyshev polynomials of the first kind is increases the

8 44 DrEman Ali et al, 2017 error is decreases Which is shown in Tables (1), (2), and (3)Aswell as the proposed method is a precise and active to solve LFWSIDEsFinally this method can be extended and applied to the system of LFWSIDEs REFERECES Palamora, A, 1996 Product integration for Volterra integral equations of the second kind with weakly singular kernels,math Comp, 65(215): Brunner, H, A Pedas, G Vainikko, 2001 Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weaklysingular kernels, SIAM J umer Anal, 39: Pedas, A, E Tamme, 2006 Spline collocation method for integro-differential equations with weakly singular kernels, J Comput Appl Math, 197: Chen, Z, Y Xu, J Zhao, 1999 The discrete Petrov Galerkin method for weakly singular integral equations, J Integral Equations Appl, 11: 1-35 Chen, Y, T Tang, 2009 Spectral methods for weakly singular Volterra integral equations with smooth solutions,jcomputapplmath, 233: Mehrdad Lakestani, Behzad emati Saray, Mehdi Dehghan, 2011, umerical Solution for the Weakly Singular Fredholm Integro-Differential Equations using Legendre multiwavelets, JComputApplMath, 235: Qinghua Wu, 2014 The Approximate Solution of Fredholm Integral Equations with Oscillatory Trigonometric kernels, ID ,7 Mason, JC, DC Handscomb, 2003 Chebyshev Polynomials, Boca Raton London ew York Washington