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Numerical Solutions of Fredholm Integral Equation of Second Kind Using Piecewise Bernoulli Polynomials

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Numerical Solutions of Fredholm Integral Equation of Second Kind Using Piecewise Bernoulli Polynomials Numerical Solutions of Fredholm Integral Equation of Second Kind Using Piecewise Bernoulli Polynomials Afroza Shirin1, Md. Shafiqul Islam*2 1Institute of Natural Sciences, United International University, Dhaka-1209, Bangladesh. Email: [email protected]. 2Department of Mathematics, University of Dhaka, Dhaka – 1000, Bangladesh *Corresponding author, Email: [email protected] Abstract: The aim of this paper is to solve the integral equations numerically using piecewise Bernoulli polynomials. The Bernoulli polynomials are derived explicitly over the unit interval. A matrix formulation for a non-singular linear Fredholm integral equation of the second kind is derived by the technique of Galerkin method. In the Galerkin method, the Bernoulli polynomials are exploited as the linear combination in the approximation as basis functions. Numerical examples are considered to verify the effectiveness of the proposed derivations. Keywords: Fredholm integral equation, Galerkin method, Bernoulli polynomials, Numerical solutions. I. Introduction In the survey of solutions of integral equations, a large number of analytical but a few approximate methods are available for solving numerically various classes of integral equations [1, 2, 7, 8 ]. Since the piecewise polynomials are differentiable and integrable, the Bernstein polynomials [5 – 8] have been used for solving differential and integral equations numerically. Recently, integral equations have been solved by the well known variational iteration method [9]. In the literature [7], Mandal and Bhattacharya have attempted to solve integral equations numerically using Bernstein polynomials, but they obtained the results in terms of finite series solutions In contrast to this, we solve the linear Fredholm integral equation of the second kind by exploiting very well known Galerkin method, and Bernoulli polynomials [4] are used as trial functions. For this, we give a short introduction of Bernoulli polynomials first. Then we derive a matrix formulation by the technique of Galerkin method. To verify our formulation we consider three examples, in which we obtain exact solutions for two examples even using a few and lower order polynomials. On the other hand, the last example shows an excellent agreement of accuracy compared to exact solution, which confirms the convergence. All the computations are performed using MATHEMATICA. 1 II. Bernoulli Polynomials: The Bernoulli polynomials [Atkinson, 4] upto degree n can be defined over the interval [0,1] implicitly by n n nk Bn (x) bk x k0 k where bk are Bernoulli numbers given by 1 b 1 and b B (x)dx . 0 k k k 1 0 These Bernoulli polynomials may be defined explicitly as B0 (x) 1 m n m n 1 k n m 1 k n m (1) Bm (x) (1) (x k) (1) k , m 1 n0 n 1k0 k n0 n 1 k0 k The first 6 Bernoulli polynomials (n 5)are given bellow for using in this paper: 2 2 3 4 B0 (x) 1 B2 (x) x x B4 (x) x 2x x x 3x2 x 5x3 5x4 B (x) x B (x) x3 B (x) x5 1 3 2 2 5 6 3 2 Note that Bernoulli polynomials have a special property at x 0 and x 1, respectively, Brn (0) 0, n 1 and Bn (1) 0, n 2. III. Formulation of Integral Equation in Matrix Form Consider a general linear Fredholm integral equation (FIE) of second kind [1, 2] is given by b a(x)(x) k(t, x)(t)dt f (x) , a x b (2) a where a(x) and f (x) are given functions, k(t, x) is the kernel, and (x) is the unknown function or exact solution of (2), which is to be determined. Now we use the technique of Galerkin method [Lewis, 3] to find an approximate solution ~ (x) of (2). For this, we assume that ~ n (x) ai Bi (x) (3) i0 where Bi (x) are Bernoulli polynomials (basis) of degree i defined in eqn. (1), and ai are unknown parameters, to be determined. Substituting (3) into (2), we obtain 2 n b n a(x) a B (x) k(t, x) a B (t) dt f (x) i i i i i0 a i0 n n b or, a a(x)B (x) a k(t, x)B (t) dt f (x) i i i i i0 i0 a n b or, a a(x) B (x) k(t, x)B (t)dt f (x) (4) i i i i0 a Then the Galerkin equations [Lewis, 3] are obtained by multiplying both sides of (3) by B j (x) and then integrating with respect to x from a to b, we have b n b b a a(x) B (x) k(t, x)B (t)dt B (x)dx B (x) f (x)dx i i i j j a i0 a a or, n b b b a a(x) B (x) k(t, x) B (t)dt B (x)dx B (x) f (x)dx, j 0,1,,n (5a) i i i j j i0 a a a In each equation, there are three integrals. The inner integrand of the left side is a function of x and t and is integrated with respect to t from a to b. As a result the outer integrand becomes a function of x only and integration with respect to x yields a constant. Thus for each j ( j 0,1,,n ) we have a linear equation with n 1 unknowns ai (i 0,1,,n ) . Finally (5a) represents the system of linear equations in unknowns. Equivalently, n aiCi, j Fj , j 0,1,2,,n, (5b) i0 where b b C a(x) B (x) k(t, x) B (t)dt B (x) dx i, j 0,1,2,,n. (5c) i, j i i j a a b F B (x) f (x)dx, j 0,1,2,,n (5d) j i a Now the unknown parameters ai are determined by solving the system of equations (5), and ~ substituting these values of parameters in (3), we get the approximate solution (x) of the integral equation (2). The absolute error E for this formulation is defined by (x) ~(x) E . (x) 3 IV. Numerical Examples In this section, we explain three integral equations which are available in the existing literatures [2, 7]. For each example we find the approximate solutions using Bernoulli polynomials. Example 1: We consider the FIE of 2nd kind given by [7] 1 (x) (xt x2t 2 )(t)dt 1, 1 x 1, (6) 1 10 having the exact solution (x) 1 x2 . 9 Using the formulation described in the previous section, the equations (5) lead us, respectively, 1 1 1 C B (x)B (x) dx (xt x 2t 2 ) B (t)dt B (x) dx , i, j 0,1,2,,n (7a) i, j i j i j 1 11 1 F B (x) dx , j 0,1,2,,n (7b) j j 1 Solving the system (7) for n 3, the values of the parameters are: 10 10 a 1, a , a , a 0 0 1 9 2 9 3 and the approximate solution is ~ 10 (x) 1 x2 9 which is the exact solution. Example 2: Now we consider another FIE of 2nd kind given by [Mandal, 7] 1 (x) (x4 t 4 )(t)dt x , 1 x 1 (8) 1 having the exact solution (x) x Proceeding as the example 1, the system of equations becomes as where, 1 1 1 C B (x)B (x)dx (x4 t 4 )B (t)dt B (x)dx i, j 0,1,2,,n, (9a) i, j i j i j 1 11 4 1 F x B (x)dx j 0,1,2,,n, (9b) j j 1 For n 3, solving system (9), the values of the parameters ( ai ) are: a0 0, a1 1, a2 0, a3 0, ~ and the approximate solution is (x) x which is the exact solution. Example 3: Consider another FIE of 2nd kind given by [pp 213 (1), with 1, Jerry, 1] 1 (x) (tx 2 xt 2 )(t)dt x , 0 x 1 (10) 0 180 80 having the exact solution (x) x x 2 119 119 Proceeding as the previous examples, the system of equations becomes as n aiCi, j Fj , j 0,1,2,,n, (11a) i0 where, 1 1 1 C B (x)B (x)dx (tx 2 xt 2 )B (t)dt B (x)dx (11b) i, j i j i j 0 0 0 1 F x B (x)dx (11c) j j 0 For solving system (11), the values of the parameters ( ) are: 260 80 a 0, a , a , a 0, 0 1 119 2 119 3 and the approximate solution is ~ 180 80 (x) x x 2 119 119 which is the exact solution. Example 4: Consider another FIE of 2nd kind given by [pp 124 (iv), with , Shanti, 2] 1 (x) 2e xet(t)dt e x , 0 x 1, (12) 0 e x having the exact solution (x) . 2 e2 Since the equations (5b) and (5c) are of the form i, j 0,1,2,,n, 5 1 1 1 C B (x)B (x)dx 2e xet B (t)dt B (x)dx , i, j 0,1,2,,n (13a) i, j i j i j 0 0 0 1 F e x B (x) dx , j 0,1,2,,n, (13b) j j 0 Solving the system (5a) using (13a) and (13b) instead of (5b) and (5c) respectively, we have the following results: For n 3, the approximate solution is ~ (x) 0.185387 0.188957 x 0.078167 x2 0.051702 x3 For n 4 , the approximate solution is ~ (x) 0.185571 0.185273 x 0.0947437 x2 0.025916 x3 0.012893 x4 For n 5, the approximate solution is ~ (x) 0.185561 0.18558 x 0.0925986 x 2 0.0316362 x3 0.00645779 x 4 0.00257408 x5 For n 6, the approximate solution is ~ (x) 0.185561 0.18556 x 0.0925932 x 2 0.0308581x3 0.00791665 x 4 0.0012903 x5 0.000427923x6 Plot of relative error, E is depicted in Fig.
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