Numerical Solution of Second order Integro- Differential Equations(Ides) with Different Four Polynomials Bases Functions

Numerical Solution of Second order Integro- Differential Equations(Ides) with Different Four Polynomials Bases Functions

1Taiwo O. A , 2Raji M. T

1Department of Mathematics, University of Ilorin

2Department of Mathematics and Statistics, The Poly., Ibadan

Abstract: – In this paper, a method based on the collocation methods with some bases functions are developed to find the numerical solution of Fredholm Integro-Differential Equations; four different polynomial bases functions used were: Legendary, Leguerre, Hermite and Fibonacci polynomial bases functions. The differential part appearing in the integro-differential equation is re-defined and used to generate each of the polynomial bases functions. Some numerical results are given to demonstrate the superior performance of the various collocation methods, particularly, the table of error with the various value of N.

Keywords: Collocation method, Integro-differential equations (IDEs), Chebyshev, Hermite, Fibonacci

and Leguerre Polynomial.

1. INTRODUCTION

   Integro-differential equations (IDEs) plays an important role in many branches of linear and Non-linear functional analysis. In addition to their occurrence in the field of mechanics and mathematical physics, Integro- differential equations have found wide applications in the theory of engineering, chemistry, astronomy, biology, economics potential theory and electro-static.

   This paper concerns the developments of the various polynomial bases function (see Ortiz [15] and Ortiz and Samara [17]) with Legendre, Leguerre, Fibonacci and Hermite polynomials for the numerical solution of integro-

   differential equations (IDEs). The polynomials has found extensive application in recent years presented in a series of papers, for example, in [2-8] for the case of numerical solution of ordinary differential equations (ODEs) and in [4,9,10] for the case of numerical solution of partial differential equations (PDEs). Application of the Chebyshev and Legendre polynomials and their numerical merits in solving ODEs and PDEs numerically have been discussed in a series of papers (for example, [2-13]). We are, therefore, motivated to work in this direction of extending to various polynomials proposed in the literature to handle IDEs numerically [13-17].

   Yalcinbas and Sezer [19] proposed an approximating solution in terms of Taylor polynomials which we believe it is a particular case of the method presented in this paper. Also, this paper is organized as follows. In section 2, the formulation of the various polynomials bases are developed (such as: Fibonacci, Hermite, Legurre and Legendry); the matrix representations of each part of IDE and its supplementary conditions are obtained. In section 3, as efficient Tau error estimator is introduced. In section 4, preliminary steps towards construction of various polynomial bases functions were considered and taking Chebyshev polynomial as an example. Finally, in section 5, some numerical results are provided to demonstrate the efficiency of using various polynomials bases and compared with those of [19].

2. CONVERTING INTEGRO-DIFFERENTIAL EQUATION TO A SYSTEM OF LINEAR ALGEBRAIC EQUATIONS

   v d n

   dx

   L : gn (x) n ,

   n0

   2.1

   We shall write for

   gn (x)

   gn x: gn j

   j 0

   x j ,

   2.2

   Where n is the degree of

   gn (x) and

   g g ,…g ,0,0,…, X 1, x, x2 ,…T .

   n n0 n

   Unless otherwise stated, x will always be the independent variable of the functions which appear throughout this paper and will be defined in a finite interval.

   Let

   y(x) be the exact solution of the integro-differential equation,

   Lyx b mx,tytdt

   a

   f x,

   x a,b

   2.3

   With

   v

   c1 ym1 a c2 ym1 b d ,

   j 1,…, v,

   jm jm j

   m1

   2.4

   Where

   f xand mx,t are given continuous functions and , a,b, c1 , c2 and d some given constants.

   jm jm j

   Matrix representation for the different parts

   Let V : v0 x, v1 x,… be a polynomial basis by V : V X , where V is a non-singular lower triangular matrix

   and degree v x i, for i 0,1,2,…. . Also for any matrix P, P

   VPV 1 .

   i v

   Now we convert the Eqs. (2.3) and (2.4) to the corresponding linear algebraic equations in three parts; (a), (b) and (c).

   1. Matrix representation for Dy(x):

      Ortiz and Samara proposed in [17] an alternative for the Tau technique which they called the operational approach as it reduces differential problems to linear algebraic problems. The effect of differentiation, shifting and integration on the coefficients vector

      a~ : (a~ , a~ ,…., a~ ,0,0,…)

      n 0 1 n

      u (x) a~ X

      a~ by the matrices , and i respectively,

      Of a polynomial n n is the same as that of post-multiplication of n

      dun (x) a~ X ,

      u (x) a~ X

      x u (t)dt a~ i X

      dx n n n

      and 0 n n

      where

      0

      0 …

      …

      0 1

      ……

      0 1

      … …

      1 0

      0 …

      1

      , 0

      0 1 …, i 0 0 2 ….

      0

      …

      2

      …

      0

      …

      …

      …

      …

      …

      ……

      …

      …

      … …

      We recall now the following theorem given by Ortiz and Samara [17].

      Theorem 2.1 For any linear differential operator L defined by (2.1) and any series

      y(x) : aV ,

      a : (a0 , a1 , a2 ,…),

      we have

      Ly(x) a~ X avV ,

      2.5

      where

      g (),

      v

      i

      i

      i0

      and

      v

      V V 1.

   2. Matrix representation for the integral term:

   Let us assume that

   n n

   m(x, t) mij vi (x)v j (t),

   i0 j 0

   and

   y(x) ai vi (x) aV .

   i0

   Then, we can write

   b n n b

   a m(x, t) y(t)dt mij ai vi (x)a v j (t)vi (t)dt aMV ,

   10 i0 j 0

   2.6

   where

   n m

   m .	0 .	0 .	…
   m .	0 .	0 .	… …

   … n

   j 0 0 j jo

   M .

   j 0

   nj jo

   n m … n

   j 0 0 j jn

   .

   j 0 nj jn

   b

   with

   ji

   a v j

   (t)v1 (t),

   for j, l 0,…, n.

   (c) Matrix representation for the supplementary conditions:

   y(x) a v (x) in the left hand side of (2.4), it can be written as

   Replacing

   i0 i i

   v

   v

   c(1) y (m1) (a) c(2) y (m1) (b)

   c(1) vm1 (a) c(2) v(m1) (b) aB ,

   jm

   m1

   jm

   i0 m1

   jm i

   jm i j

   2.7

   where for j = 1, ,v,

   c(1) v

   1. c(2) v

   jm 0 jm 0

   2

   c(1) v m1 (a) c(2) v(m1) (b)

   B j

   m1

   jm 1

   jm 1

   2.8

   v

   c(1) v(m1) (a) c(2) v(m1) (b)

   m1

   jm v1

   jm v1

   We refer to B as the matrix representation of the supplementary conditions and for computing the elements of the matrix B can be deduced from (2.7):

   B j as its jth column. Thefollowing relations

   b v

   c(1) v(m1) (a) c(2) v(m1) (b),

   for i, j 1,2,…, v,

   ij m1

   jm v1

   jm v1

   2.9

   and

   b

   v

   c(1) v(m1) (a) c(2) v(m1) (b),

   for i v 1, v 2,…, j 1,2,…, v.

   ij m1

   jkm

   v1

   jm v1

   2.10

   We introduce

   d (d1 , d2 ,…, dv ) , the vector that contains right hand sides of conditions. Then the supplementary

   conditions take the form

   aB d.

   2.11

   It follows from (2.5) and (2.6) that

   b

   Ly(x) a m(x,t) y(t)d a( v M )V.

   2.12

   Let

   M : M

   and

   M stands for its ith column and let

   f (x) n

   f v (x) f V

   with

   v

   v

   vi

   i0

   i

   i

   f ( f0 ,…, fn ,0,0,…).. Then the coefficient of exact solution infinite algebraic system:

   y aV of problem (2.3) and (2.4) satisfies the following

   aM vi

   fi ;

   i 0,…, n,

   aM

   vi

   0;

   i n 1,

   j

   aB

   di ;

   j 1,2,…, v.

   2.13

   setting

   G (B1 ,…, Bv , M v0 , M v1 ,…),

   and

   g (d1 ,…, dv , f0 , f1 ,…),

   we can write instead of (2.13)

   aG g.

   2.14

   Definition 2.2. The polynomial

   n

   y a V

   n

   will be called an approximate solution of (2.3) and (2.4), if the vector

   n

   a (a0 ,…, an )

   is the solution of the linear algebraic equations.

   n

   a Gn gn .

   Where Gn is the matrix defined by restriction of G to its first (n + 1) rows and columns.

   2.15

   Remark 2.3. For v 0 and G0 (x) 1, Eq. (2.3) is transformed into a Fredholm integral equation of second kind and for

   0 , it is transformed into a Differential Equation.

3. ERROR ESTIMATION

   In this section an error estimator for the approximate solution of (2.3) and (2.4) is obtained. Let us call

   en (x) y(x) yn (x) as the error function of the approximate solution

   yn (x)

   to y(x) , where,

   y(x)

   is the exact

   solution of (2.3) and (2.4). Hence

   yn (x) satisfies the following problem:

   Lyn

   (x)

   a m(x,t) yn

   (t)dt

   f (x) H n

   (x),

   x a,b

   b

   3.1

   with

   v

   c(1) y (m1) (a) c(2) y (m1) (b) d ,

   j 1,…, v.

   jm jm j

   m1

   3.2

   The perturbation term H n (x) can be obtained by substituting the computed solution yn (x) into the equation

   b

   H n (x) Lyn (x) a m(x,t) yn (t)dt f (x).

   3.3

   We proceed to find an approximation en ,N (x) to the error function en (x) in the way as we did before for the solution

   b

   of problem (2.3), (2.4). Subtracting (3.1) and (3.2) from (2.3) and (2.4) respectively, the error function problem

   en (x) satisfies the

   Len

   (x)

   a m(x,t)en

   (t)dt H n

   (x),

   x a,b

   3.4

   with the homogeneous conditions

   v

   c(1) e(m1) (a) c(2) e(m1) (b) 0,

   j 1,…, v.

   m1

   jm n

   jm n

   3.5

   It should be noted that in order to construct the approximant en ,N (x) to en (x) , only the right hand side of system (2.15) needs to be recomputed; the structure of the coefficient matrix Gn remains the same.

4. CONSTRUCTION OF VARIOUS POLYNOMIAL BASES FUNCTION

   In section 2, V : v0 (x), v1 (x),… was considered as a polynomial basis given by V : V X , where V is a non-

   singular lower triangular matrix and degree

   (vi (x)) i, 0,1,2,…. It was used to convert (2.3) and (2.4) into a system of

   linear equations. The various polynomials are very interesting polynomial basis with a matrix V of the same structure. We argue the application of the Tau method for the case of Chebyshev polynomials, though the case of Lengendre, Leguerre, Hermite and Fibonacci may also be similar. But different recursive formulae will be employed.

   For instance, the shifted chebyshev polynomials are defined as

   T (*) (x) 1,

   T (*) (x) 2x (b a) ,

   x a,b

   0

   and for i 1

   0 b a

   T (*) (x)

   2x (b a)

   (*)

   T (*) (x),

   x a,b

   i1

   2 b a Ti

   (x)

   i1

   In this case the functions m(x,t), f(x) and the approximate solution

   yn (x) are written as

   n n

   m(x,t) n n m T (*) (x)T (*) (x),

   i0

   i0

   ij i j

   n

   i

   i

   f (x) n f T (*) (x),

   and

   n

   n

   i

   i

   y (x) a T (*) (x),

   Where mi j and fi are computed by the following relations and a0 , a1 ,…, an are obtained from (2.11),

   m 4 n '' n ''

   b a

   b a

   b a

   b a

   it

   jl

   i j n2

   m 2

   xt 2 , 2

   x1

   2 cos n

   cos n ,

   f 2

   t 0

   n

   ''

   l 0

   b a

   b a

   n

   i

   m0

   m xt ,

   2

   2

   For i, j 0,1,…, n and

   .

   x cos i

   i n

   A summation symbol with double primes denotes a sum first and last terms halved. Also are computed easily in this case, since

   aijs

   (see part (b) section 2),

   a b T (*) (t)T (*) (t)dt

   b T 2t (b a)

   2t (b a) dt

   jl a j l

   a j

   b a

   Tl

   b a

   1

   b a

   2

   1Tj

   (t)Tl (t)dt,

   Where Tj and Tl

   are Chebyshev polynomial of degree j and l respectively. But these polynomials are even for even

   degree and odd for odd degree, hence

   a jl

   0; if

   b a

   j 1

   1

   odd

   1

   ; if j l

   even

   2

   1 ( j l)2 1 ( j l)2

   All other elements are computed as in Section 2. The same application will definitely adopted for the other Polynomials when the recursive formulae were introduced to arrive at the approximate solution of the various polynomial bases function.

5. NUMERICAL EXAMPLES

   In this section, we consider some examples demonstrating the accuracy of the method and effectiveness of the Chebyshev polynomial basis function compared with the other various polynomial bases functions. Hence, error of examples 1 and 2 are also compared with [19]

   Example 1. This example was considered by Yalcinbas and Sezer [19] for the method of solution in terms of Taylor polynomials.

   y'' (x) xy ' (x) xy(x) ex 2sin(x) 1 sin(x)et y(t)dt,

   1

   1 x 1

   y(0) 1

   y' (0) 1

   The exact solution is

   y(x) ex .

   Table 1: Tables of errors when N = 4

   td>

   3.97E-02

   X	CHEBYSHEV	LEGENDARY	LAGUERRE	HERMITE	FIBONACCI	Yalcinbas et al. [19]
   -1.00	0.36787942	3.42E-01	3.51E-01	6.65E-01	0	0.367879
   -0.80	0.44932896	3.54E-01	9.96E-02	1.80E-02	8.89E-05	0.449328
   -0.60	0.54881164	2.54E-01	4.27E-01	1.75E-01	6.75E-04	0.548811
   -0.40	0.67032004	1.36E-01	6.68E-01	4.63E-01	2.15E-03	0.67032
   -0.20	0.81873075	7.76E-01	9.50E-02	4.80E-03	0.81873
   0	1	0.00E+00	0.00E+00	7.00E-10	8.75E-03	1
   0.20	1.22140276	3.55E-04	1.16E+00	5.48E-02	1.40E-02	1.2214
   0.40	1.49182469	2.56E-02	1.39E+00	3.01E-01	2.03E-02	1.49182
   0.60	1.82211880	1.16E-01	1.60E+00	5.52E-01	2.75E-02	1.82211
   0.80	2.22554092	3.21E-01	1.78E+00	2.06E-01	3.48E-02	2.22554
   1.00	2.71828180	6.92E-01	1.90E+00	1.75E+00	4.15E-02	2.71828

   Table 2: Tables of errors when N = 5

   X	CHEBYSHEV	LEGENDARY	LAGUERRE	HERMITE	FIBONACCI	Yalcinbas et al. [19]
   -1	2.12E-08	1.54E-03	1.42E-03	5.18E-03	0	4.41E-07
   -0.8	4.10E-09	2.43E-03	6.29E-04	8.86E-04	5.90E-07	9.64E-07
   -0.6	3.90E-09	3.61E-03	7.42E-03	9.95E-03	3.57E-06	6.36E-07
   -0.4	6.00E-09	4.75E-03	8.33E-03	2.96E-03	4.83E-06	4.60E-08
   -0.2	3.10E-09	7.04E-04	8.55E-03	9.82E-04	2.01E-06	7.53E-07
   0	0.00E+00	0.00E+00	0.00E+00	0.00E+00	4.94E-06	0.00E+00
   0.2	2.00E-09	4.77E-06	5.08E-02	4.85E-04	4.93E-05	2.76E-06
   0.4	8.00E-09	5.95E-04	6.30E-02	1.15E-03	3.46E-05	4.70E-06
   0.6	0.00E+00	4.95E-03	1.53E-02	1.70E-03	4.58E-05	8.80E-06
   0.8	8.00E-09	5.51E-03	6.64E-02	5.78E-03	7.65E-05	9.28E-07
   1	2.80E-08	4.82E-03	7.82E-02	3.41E-03	5.13E-05	1.83E-06

   From the numerical results and Table of error (see Tables 1, 2). In Table 2, it is evident that better performance provided by the various method here proposed compared with the results of Yalcinbas and Sezer [19].Hence, it was observed that the Legendre and Hermite solutions have same errors over the interval. The error of the Fibonacci solution shows a tendency to increase rapidly, as N increases.

   Example 2. (see [19])

   y '' (x)

   2 xty(t)dt x sin(x),

   0

   y(0) 0

   0 x

   2

   y' (0) 1.

   The exact solution is

   y(x) sin(x) .

   For numerical results see table 3 and 4

   Table 3: Tables of errors when n = 4

   X	CHEBYSHEV	LEGENDARY	LAGUERRE	HERMITE	FIBONACCI	Yalcinbas et al.
   0	8.79E-05	3.75E-05	5.13E-04	2.00E-04	5.00E-03	4.41E-04
   0.1	4.32E-04	9.98E-02	5.41E-03	9.81E-02	4.56E-03	9.64E-04
   0.2	2.90E-02	1.99E-01	3.95E-02	1.96E-01	8.56E-02	3.60E-03
   0.3	8.26E-02	2.90E-01	6.54E-02	2.92E-01	7.45E-02	5.46E-03
   0.4	3.49E-02	3.82E-01	5.63E-02	3.84E-01	4.60E-02	7.53E-03
   0.5	3.40E-02	4.69E-01	8.23E-02	4.73E-01	3.10E-02	0.00E+00
   0.6	5.68E-02	5.52E-01	8.12E-02	5.58E-01	9.71E-02	2.76E-03
   0.7	5.17E-02	6.28E-01	8.54E-02	6.36E-01	4.55E-02	4.70E-03
   0.8	7.60E-02	6.98E-01	3.48E-01	7.08E-01	4.84E-02	8.85E-03
   0.9	1.57E-02	7.60E-01	1.23E-01	7.73E-01	7.28E-02	9.28E-03
   1	6.68E-02	8.14E-01	3.75E-01	8.30E-01	5.89E-02	1.83E-03
   /2	9.22E-02	9.38E-01	9.34E-01	9.77E-01	1.68E-02	1.12E-03

   Table 2: Tables of errors when n = 5

   X	CHEBYSHEV	LEGENDARY	LAGUERRE	HERMITE	FIBONACCI	Yalcinbas et al.[18]
   0	1.93E-06	0.00E+00	2.12E-04	0.00E+00	9.96E-05	5.44E-06
   0.1	5.41E-06	1.10E-05	4.10E-04	6.62E-05	2.46E-05	1.96E-06
   0.2	5.39E-05	3.30E-05	3.90E-05	6.93E-05	8.86E-05	6.36E-05
   0.3	5.60E-05	2.50E-04	6.00E-05	2.53E-04	6.75E-04	5.46E-06
   0.4	3.31E-06	1.50E-04	3.10E-05	4.22E-05	5.46E-04	4.75E-06
   0.5	7.53E-05	8.60E-05	2.00E-05	8.63E-05	4.31E-04	9.80E-05
   0.6	9.82E-07	4.80E-04	8.12E-05	1.52E-04	5.97E-04	2.28E-05
   0.7	6.88E-06	1.12E-04	8.00E-05	1.17E-04	7.45E-04	5.47E-05
   0.8	7.65E-05	9.66E-04	3.48E-04	9.63E-04	4.80E-04	5.88E-05
   0.9	7.58E-06	5.85E-04	2.12E-04	3.92E-06	7.24E-04	4.93E-06
   1	6.28E-05	4.72E-04	7.38E-05	1.28E-04	6.59E-04	9.18E-05
   /2	1.37E-06	8.26E-04	7.93E-04	1.37E-03	2.16E-04	7.11E-06

   The table shows that the results of the polynomial bases are numerically stable

6. CONCLUSIONS

Our results indicate that the tau method with various polynomial bases can be regarded as a structurally simple algorithm that is conventionally applicable to the numerical solution of IDEs. In addition, although we have restricted our attention to linear Fredholm IDEs, we expect the method to be easily extended to more general IDEs.

Despite the relatively low degrees used the numerical results show the superior performance of the Tau method, particularly, with the Chebyshev and Legendre bases. Nevertheless, the error of the Tau solution shows a tendency to increase rapidly, as N increases. This behaviour also indicated in other polynomial/methods.

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