legendre–galerkin method for the linear fredholm

8/10/2019 LegendreGalerkin Method for the Linear Fredholm

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LegendreGalerkin method for the linear Fredholm

integro-differential equations

Mohamed Fathy a, Mohamed El-Gamel b,, M.S. El-Azab b

a Department of Engineering Physics and Mathematics, Faculty of Engineering, Helwan University, Egyptb Department of Mathematical Sciences, Faculty of Engineering, Mansoura University, Egypt

a r t i c l e i n f o

Keywords:

Galerkin

Legendre

Numerical

Gaussian quadrature

a b s t r a c t

This paper presents a study of the performance of the Galerkin method using Legendre

basis functions for solving linear Fredholm integro-differential problems. Convergence

and error estimation of the method were discussed. The method is then tested on several

examples. Numerical results are included and comparisons with other methods are made

to confirm the efficiency and accuracy of the method.

2014 Elsevier Inc. All rights reserved.

1. Introduction

The boundary value problems in terms of integro-differential equations have many practical applications such asmechanics, physics, chemistry, astronomy, biology, economics, potential theory, engineering problems[19]and electrostat-

ics[11]. The existence and uniqueness of the solutions for these problems were discussed by Agarwal[1,2].

The present work is motivated by the desire to obtain numerical solutions to boundary value problems for second-order

Fredholm integro-differential equations via LegendreGalerkin method. We will consider the numerical solution of a class of

linear Fredholm integro-differential boundary value problems in the form

Xri0

lixuix fx k

Z 11

kx; tutdt; 1:1

ru1 rr 1u1 0; 1:2

wherelix;fx; kx; t andux are continuous functions inL2 space, r 0; 1; 2 and k is a parameter.

Many numerical methods have been developed for solving the integro-differential Eq.(1.1)with Fredholm type numer-

ically. Each of these methods has its inherent advantages and disadvantages and the search for alternative, more general,

easier and more accurate methods is a continuous and ongoing process. Next, we present a selective review of the most

recent main methods. These methods include: Adomian decomposition [9], variational iteration method [5], homotopy anal-

ysis method[11], Chebyshev and Taylor collocation[22],Haar wavelet[15], sinc-collocation method[17],Tau method[10],

Taylors series expansion[8,13], hybrid Taylor and Block Pulse functions [14]and integral mean value[3].

In recent years, a lot of attention has been devoted to the study of Legendre methods to investigate various scientific mod-

els. Using these methods made it possible to solve differential equations of LaneEmden type [24], second and fourth order

http://dx.doi.org/10.1016/j.amc.2014.06.057

0096-3003/ 2014 Elsevier Inc. All rights reserved.

Corresponding author.

E-mail addresses:[email protected](M. Fathy), [email protected](M. El-Gamel).

Applied Mathematics and Computation 243 (2014) 789800

Contents lists available at ScienceDirect

Applied Mathematics and Computation

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a te / a m c
http://dx.doi.org/10.1016/j.amc.2014.06.057mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.amc.2014.06.057http://www.sciencedirect.com/science/journal/00963003http://www.elsevier.com/locate/amchttp://www.elsevier.com/locate/amchttp://www.sciencedirect.com/science/journal/00963003http://dx.doi.org/10.1016/j.amc.2014.06.057mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.amc.2014.06.057http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://crossmark.crossref.org/dialog/?doi=10.1016/j.amc.2014.06.057&domain=pdf


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equations[21], CahnHilliard equations with Neumann boundary conditions[28], Fredholm integral[12], Helmholtz equa-

tion [4], second kind Volterra integral equations [29], high-order linear Fredholm integro-differential [23] and Abels integral

equation[25].

This paper is organized as follows: in Section2,we present the preliminaries of Legendre polynomials, new lemmas and

theorems required for our subsequent development. In Section 3, convergence and error estimation of the method are

described in detail. Section4 is devoted to derivation of the discrete system. Section 5 presents appropriate techniques to

treat nonhomogeneous boundary conditions for case n 2. Section 6shows the accuracy of the proposed method using

numerical examples and making comparisons with other methods. Section 7 gives a brief conclusion. Note that we have

computed the numerical results by Matlab programming.

2. Legendre function preliminaries

Orthogonal polynomials are widely used in applications in mathematics, mathematical physics, engineering and com-

puter science. One of the most common orthogonal polynomial set is the Legendre polynomials. The Legendre polynomials

Pnxsatisfy the Legendre differential equation

1 x2y00 2xy0 nn 1y 0; 1 6 x 6 1; nP 0

with recurrence relations

P0n1x P0n1x 2n 1Pnx; 2:1

xP0nx P0n1x nPnx: 2:2

These polynomial are orthogonal on 1; 1Z 11

PmxPnxdx2

2n1; if m n;

0; if m n

( 2:3

and

Z 11

Pnxdx 2; n 0;

0; n> 0

2:4

Lemma 2.1. Let n; l and N be any two positive integer numbers such that nl 6 N and l > 0, thenZ 11

PnxP00nlxdx 0:

Proof. Integrating the left term by parts and using Eq. (2.4)can prove the above lemma. h

Lemma 2.2. Let n; m and N be any two integer numbers such that n P m and n; m 6 N, thenZ 11

PnxP0mxdx 0:

Proof. First, let n m. Integrating the left hand side in Lemma 2.2two times by parts yields

Z 11

PnxP0nxdx

1

2Pnx

2

1

1 0 )

Z 11

PnxP0mxdx 0:0

Second, letn > m. Using the recurrence Eq. (2.1) and (2.4), the left term inLemma 2.2can be written as follows:

Z 11

PnxP0mxdx

Z 11

Pnx2m 1Pm1x P0m2xdx

Z 11

PnxP0m2xdx

Z 11

PnxP0m4xdx

R11

PnxP00xdx; if m even;

R11PnxP01xdx; if m odd(

0:

790 M. Fathy et al./ Applied Mathematics and Computation 243 (2014) 789800
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Theorem 2.1. Let n; m and N be any two integer numbers such that n ; m 6 N, then

i

Z 11

P0nxPmxdx 2; if n mi;

0; if nmi or m P n

iiZ 1

1

xP0nxPmxdx

2n2n1

; if m n;

0; if n mi or m> n;

2; if nmi

8>:

iii

Z 11

xP0nxP0mxdx

0; if n m or m ni or n mi

nn 1; if m ni; m> n

mm 1; if nmi; n> m

8>:

iv

Z 11

P0nxP0mxdx

0; if n m or m ni or nmi

mm 1; if n mi; n> m

nn 1; if m ni; m> n

8>:

where i 1; 3; 5;. . .; 2k 1 6 Nm

Proof

(i) Integrating the left hand side for (i) by parts yieldsZ 11

P0nxPmxdx PnxPmx11

Z 11

PnxP0mxdx 1 1

nm1

Z 11

PnxP0mxdx: 2:5

Forn mi; i 1; 3; 5;. . . 6 Nm, by usingLemma (2.2), the integration(2.5)can be written as follows:Z 11

P0nxPmxdx 2:

As in the above case and Lemma (2.2)is considered but n mi; i 0; 2; 4;. . . 6 Nm, yields

Z 11 P

0

nxPmxdx 0:

Form P n, previous cases must be considered besidesLemma 2.2. So, the right hand side of Eq. (2.5)is equal to zero.

(ii) CombineLemma 2.2and (i) besides recurrence Eq. (2.2). To prove (ii), the proof must be divide into four cases.

First, letm n, by taking back recurrence Eq. (2.2)and recalling the orthogonality of the Legendre polynomials, the

first part of the above theorem can be drawn up asZ 11

xP0nxPmxdx n

Z 11

PnxPnxdx

Z 11

P0n1xPnxdx 2n

2n 1:

In the next case, let n mi; i 1; 3; 5; 6 Nm, the proof is as that in the previous case but n mi. The integration is

equal to zero. On the other hand, Theorem 2.1(ii) can be proved in status m > n as in the previous two cases but the inte-

gration is zero. Finally, assume n mi; i 2; 4; 6;. . . 6 Nm, by getting back to recurrence Eq. (2.2), Lemma 2.2 and

(i). Integrating by parts yields

Z 11

xP0nxPmxdx Pn1xPnix11Z 1

1Pn1xP0nixdx 2:

(iii) The left term in the above lemma can be written by using recurrence Eq.(2.2)asZ 11

xP0nxP0mxdx

Z 11

nPnxP0mxdx

Z 11

P0n1xP0mxdx 2:6

form n, recalling (i),Lemma 2.1and integrating the second term in the right hand side for Eq. (2.6), the result is zero. In

this case assume m > n andm ni, as in the first case the result is equal to nn 1. In the next status let m > n and

m ni, the integration is equal to zero. Finally, at n > m, the recurrence can be proved as in the above cases by interchang-

ingn by m andm insteadn.

(iv) The integration in the left hand side in (iv) can be written by using the recurrence (2.2)as

Z 1

1

P0nxP0mxdx Z

1

1

xP0n1xP0mxdx n 1Z

1

1

Pn1xP0mxdx 2:7

then, by using (iii) and (i) into Eq. (2.7)and substituting each case of them, (iv) is proved. h

M. Fathy et al. / Applied Mathematics and Computation 243 (2014) 789800 791


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Theorem 2.2. Let n; m and N be any two integer numbers such that n ; m 6 N, then

i

Z 11

P00nxPmxdx nn 1 mm 1; if nmi;

0; if n mi or mP n

ii

Z 11

xP00nxPmxdx nn 1 mm 1 2; if n mi;

0; if nmi or mP n

iii

Z 11

x2P00nxPmxdx

2nn12n1

; if m n;

nn 1 mm 1 4; if n mi 1;

0; if nmi 1 or m> n:

8>:

where i 1; 3; 5;. . .; 2k 1 6 Nm 1.

Proof

(i) The proof must be divided into four cases. First, let n mi; i 2; 4; 6;. . .; 2k 1 6 Nm. Integrating the left hand

side by parts twice and usingLemma 2.1produce

Z 11 P

00

nxPmxdx P

0

nxPnix

1

1Z 1

1 P

0

nxP

0

nixdx nn 1 PnxP

0

nix

1

1Z 1

1 PnxP

00

nixdx

nn 1 mm 1:

Second, letn mi; i 1; 3; 5;. . .; 2k 1 6 Nm. As in the first case, but the value ofi is odd numbers. So, the integration

is zero.

Third, letm > n. By using Eq.(2.4)and integrating the left side, the result is equal to zero.

Fourth, atm n, the value of the integration also is equal to zero by using the integration by parts.

(ii) To prove this point, Theorem 2.1(i) andTheorem 2.1(iii) are considered. Substitute each case in each of them after

integrating the left side of the above theorem by parts.

(iii) Integrating the left term in the above theorem yieldsZ 11

x2P00nxPmxdxnn 1

2 1 1

mn 2

Z 11

xP0nxPmxdx

Z 11

x2P0nxP0mxdx: 2:8

Moreover, by recalling recurrence Eq.(2.2)and using it in the last term in Eq. (2.8)then expanding the result yieldsZ 11

x2P0nxP0mxdx nm

Z 11

PnxPmxdxn

Z 11

PnxP0m1xdxm

Z 11

PmxP0n1xdx

Z 11

P0n1xP0m1xdx: 2:9

by substituting Eq.(2.9) and Theorem 2.1(ii) into Eq. (2.8), then taking back the orthogonality of the Legendre polynomial

(2.3)and usingTheorem 2.1(i) and Theorem 2.1(iv), the result produces four cases. By collecting the four cases, the above

theorem can be proved. h

3. Convergence and error estimation

3.1. Convergence of Legendre polynomial

Consider Fredholm integral equation of the second kind by puttingr 0 andl0x 1

ux fx k

Z 11

kx; tutdt: 3:1

If we define

@ : X! X; @u k

Z 11

kx; tutdt;

we can rewrite(3.1)as a form of

Ik@u f:

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Theorem 3.1 [16]. Let X be a normed space, @ : X! X a compact and Ik@ be injective. Then the inverse operatorIk@1

:X! X exist and is bounded.

By regard to the above theorem, we conclude the uniqueness of solution of Fredholm integral equation of the second kind.

Theorem 3.2 [16]. Assume @ : Y!Y is bounded, with Y a Banach space, and assume c @ : Y! Y is one to one and onto.Further assume

k@Pn@k ! 0 as n! 1;

then for all sufficiently large n, say n P N, the operatorc Pn@1 exists as a bounded operator from Y to Y. Moreover, it is uni-

formly bounded:

supnPN

kc Pn@1k< 1:

For the solution ofc Pn@xn Pny;xn 2 Y and c @x y;

xxn cc Pn@1xPnx;

jcjkc Pn@k

kxPnxk 6 kxxnk 6 jcjkc Pn@1kkxPnxk:

Lemma 3.1 ([6,16]). Let xt 2Hk1; 1(a Sobolev space) and let xnt P

ni0aiPitbe the best approximation polynomial of

xt in the L2-norm, then

kxt xntkL2 1;1 6 c0nkjjxtjjHk1;1

where

kxtkL2 1;1

Z 11

x2tdt

1=2;

kxtkHk1;1

Xki0

Z 11

jxitj2dt

!1=2:

and c0 is a positive constant, which depends on the selected norm and is independent of x tand n.By regard to theLemma 3.1we conclude that approximation rate of Legendre polynomials isnk , alsoTheorem 3.2indi-

cate thatkxxnk converge to zero at exactly the same speed as kxPnxk.

3.2. Error estimation of LegendreGalerkin method

In this section an error estimator for the LegendreGalerkin approximate solution of a Fredholm integro differential equa-

tion is obtained[10,27,26]. Let us call enx ux unxas the error function of the Legendre approximation unxto ux,

whereuxis the exact solution of(1.1) and (1.2). Hence,unx satisfies the following problem

Xri0

lixuin x k

Z 11

kx; tuntdt fx Hnx; 3:2

with boundary

run1 rr 1un1 0; 3:3

whereHn is a perturbation term associated with unx and can be obtained by substituting unxinto the equation

Hnx Xri0

lixuin x k

Z 11

kx; tuntdtfx: 3:4

We proceed to find an approximation en;Nxto theenx in the same way as we did before for the solution(1.1) and (1.2).

Subtracting(3.2) and (3.3)from(1.1) and (1.2), respectively, the error function enx satisfies the equation

Xri0

lixein x k

Z 11

kx; tentdt Hnx; 3:5

with boundary

ren1 rr 1en1 0: 3:6

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By solving this problem in the same way as Section3, we get the approximation en;Nx. It should be noted that in order to

construct the Legendre approximation en;Nxto enx.

4. The LegendreGalerkin Method

We assume the solution of(1.1)is approximated by the finite expansion of Legendre basis function

ux Xnj0

cjPjx: 4:1

The unknown coefficientscj in 4.1are determined by orthogonalizing the residual with respect to the basis functionsPrx.

This yields the discrete system

hl2xu00x; Prxi hl1xu

0x; Prxi hl0xux; Prxi hfx; Prxi kZ 1

1

kx; tutdt; Prx

: 4:2

The weighted inner product h; iis taken to be

hf;gi Z 1

1

fx gxdx:

The method of approximating the integrals in(4.2)begins by integrating by parts to transfer all derivatives from u to Pr.

Therefore, the following lemma is needed.

Lemma 4.1. The following relations hold

hl1xu0x; Prxi

Z 11

l1xPrx0uxdx; 4:3

hl2xu00x; Prxi u

0xl2xPrx1

1

Z 11

l2xPrx00uxdx; 4:4

hGx; Prxi Xmi0

Gxi 2

1 x2iP0mxi

2; 4:5

Z 1

1

Fx; tdt; Prx Xm

i0X

m

l0

xi;lFxi; tlPrxi; 4:6

where

xi;l xixl 4

1 x2i1 x2lP

0mxiP

0mxl

2:

Replacing each term of(4.2) with the approximation defined in (4.3)(4.6) respectively, we obtain the following theorem.

Theorem 4.1. If the assumed approximate solution of the boundary-value problem is (1.1), (1.2) is (4.1) then the discrete

GalerkinLegendre system for the determination of the unknown coefficients fcjgnj0

is given by

Xn

j0

cj k

Xm

i0 Xm

l0

xixlkxi; tlPjtlPrxi u0xl2xPrx

1

1

Z 11

l2xPrx00Pjxdx

Z 11

l1xPrx0Pjxdx

"

Z 11

l0xPjxPrxdx

Xmq0

xqfxqPrxqdx: 4:7

The system(4.7)takes the matrix form

Ac b; 4:8

where

A

kh0;0e0;0 v0;0 kh1;0e1;0 v1;0 khn;0en;0 vn;0kh0;1e0;1 v0;1 kh1;1e1;1 v1;1 khn;1en;1 vn;1kh0;2e0;2 v0;2 kh1;2e1;2 v1;2 khn;2en;2 vn;2kh0;3e0;3 v0;3 kh1;3e1;3 v1;3 khn;3en;3 vn;3

.

.

....

..

....

kh0;ne0;n v0;n kh1;ne1;n v1;n khn;nen;n vn;n

0BBBBBBB@

1CCCCCCCA

4:9

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and

ejr l2xP0jxPrx

1

1;

hjrXmi0

Xml0

xixlkxi; tlPjtlPrxi;

vjr

Z 11

l2xP00rxPjxdx 2

Z 11

l02xP0rxPjxdx

Z 11

l002xPrxPjxdxZ 1

1

l1xP0rxPjxdx

Z 11

l01xPrxPjxdxZ 1

1

l0xPjxPrxdx

vjrcan be evaluated from theorems and lemmas in Section2.

Now we have a linear system (4.8) ofn 1 equations forn 1 unknown coefficients. We can obtain the coefficient of the

approximate solution by solving this linear system by Q-R method. The solutionc c0;. . .; cns gives the coefficients in the

approximate LegendreGalerkin solutionux.

Algorithm

INPUT:n wheren 2 Z if the domain is a; b;x ab

2 s ab

2

evaluate A andb

solve Ac b

OUTPUT: the values ofc0; c1; c2;. . .; cn s 2

baxbaba

end

5. Treatment of boundary condition

In Section3above, the development of LegendreGalerkin technique for homogeneous boundary conditions, for n 2, if

the boundary conditions are nonhomogeneous,

u1 c; u1 e 5:1

then these conditions need be converted to homogeneous conditions via an interpolation by a known function. Applying the

transformation

yx ux 1 x

2 c

x 1

2 e;

to the problem(1.1), (1.2)yields

X2i0

lixyix fx k

Z 11

kx; tytdt

with boundary conditions

y1 0; y1 0;

where

^fx fx e c

2 l1x

e cx e c2

l0x k

Z 11

kx; t t 1e 1 tc

2

dt:

The resulting discrete system for the coefficients n 1 in the approximate LegendreGalerkin solution

ux Xnj0

cjPjx 1 x

2 c

x 1

2 e

is exactly the system in(4.8), withfin that system replaced by

^

f. Notice that ifc e 0, the problem reduces to the homo-geneous case.

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Fig. 1. (a) Comparison ofkELGk withn = 5, 10, 15, 20, 25 for example 1. (b) Legendre and exact solutions for example 1.

Table 3

Maximum absolute errors and maximum estimate error for Example 2 at different n.

N LegendreGalerkinkELGj kenk

5 5.764E05 8.700E06

10 2.530E12 1.124E12

15 1.776E15 2.288E15

20 1.332E15 2.170E15

Table 4

Comparison of maximum absolute errors for example 2.

LegendreGalerkin Sinc-Collocation[17] The rationalized

Haar wavelet[15]

2.442E15 0.4756 E06 0.0413

Fig. 2. (a) Comparison ofkELGk withn = 5, 10, 15, 20 for example 2. (b) Legendre and exact solutions for example 2.

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subject to the boundary conditions

u1 u1 0;

whose exact solution is

ux sinpx:

The computational results are summarized inTable 5. for different values ofn. Fig. 3(a) displays the maximum absolute

error and maximum estimate error at differentn andFig. 3(b) bids the Legendre and exact solutions.

Example 4 [23]. Consider the integro-differential equation

u00 xu0 xu ex 2sinx sinx

Z 11

etutdt; 1< x; t< 1;

subject to the boundary conditions

u1 e1; u1 e;

whose exact solution is

ux ex:

The computational results are summarized in Table 6 for different values ofn. Maximum absolute error is tabulated in Table 7for LegendreGalerkin together with the numerical results of S. Yalinbas [23], who used Legendre-collocation matrix

Table 5

Maximum absolute errors and maximum estimate error for Example 3.

N LegendreGalerkin kELGk jjenjj

5 6.006E01 2.919E01

10 1.008E04 8.528E05

15 3.282E08 2.836E08

20 3.752E14 4.244E14

Fig. 3. (a) Comparison ofkELGk withn = 5, 10, 15, 20 for example 3. (b) Legendre and exact solutions for example 3.

Table 6

Maximum absolute errors and maximum estimate error for Example 4 at different n.

N LegendreGalerkin kELGk kenk

5 2.136E03 6.324E03

10 1.008E09 9.187E10

15 8.104E15 1.297E15

798 M. Fathy et al. / Applied Mathematics and Computation 243 (2014) 789800


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method to obtain the numerical solution. Fig. 4(a) shows the maximum absolute error at different n and Fig. 4(b) exhibits the

Legendre and exact solutions.

7. Conclusion

This paper has discussed how the LegendreGalerkin method can be applied for obtaining solutions of integral and inte-

gro-differential equations. The formulation and implementation of the scheme are illustrated. The proposed method was

tested using some problems with results. This Paper discussed how the integro-differential equation with variable coeffi-

cients can be solved using the LegendreGalerkin method. Matlab and Mathematica had been used to obtain the approxi-

mate solution. If the problem domain is a;b, the linear transformation must be used to convert the problem domain to1; 1. The required linear transformation is s ba

2 x ba

2. Then, by using LegendreGalerkin technique, the approximate

solution is converted to linear algebraic system. By solving this system, the numerical solution is obtained. Finally, By using

inverse transformation, the final approximate solution can be produced.

Acknowledgments

The authors are grateful to the referees for their valuable comments and suggestions on the original manuscript.

References

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Table 7

Comparison of maximum absolute errors for example 4.

LegendreGalerkin Legendre-collocation

matrix method[23]

8.104E15 4.600E07

Fig. 4. (a) Comparison ofkELGk withn = 5, 10, 15 for example 4. (b) Legendre and exact solutions for example 4.

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