We study Legendre polynomials and develop new operational matrix of integration. Based on the operational matrix, we develop a new method to solve a coupled system of Fredholm integral equations of the form U(x)+λ11∫01K11(x,t)U(t)dt+λ12∫01K12(x,t)V(t)dt=f(x), V(x)+λ21∫01K21(x,t)U(t)dt+λ22∫01K22(x,t)V(t)dt=g(x), where λ11, λ12, λ21, and λ22 are real constants and f,g∈C([0,1]). The method reduces the coupled system to a system of easily solvable algebraic equations without discretizing the original system. As an application, we provide examples and numerical simulations demonstrating that the results obtained using the new technique match very well with the exact solutions of the problems. To show the efficiency of the method, we compare our results with some of the results already studied with other available methods in the literature.

1. Introduction

Fredholm integral equations are frequently encountered in many physical processes such as dynamic stiffness of rigid rectangular foundations [1], soil mechanics and rock mechanics [2], diffraction of waves by randomly rough surface in two dimensions [3], thermoelasticity [4], and scattering problem [5], to name a few. For systems of such equations, various techniques such as extrapolation method, Galerkin discretization, collocation methods, and quadrature, iterative, spline, orthogonal polynomial, and multiple grid methods have been proposed to determine desired solutions (see, e.g., [6–9] and the references quoted there). These methods include approximate analytical and numerical approaches.

Recently, approximate solutions to system of integral equations have attracted the attention of many authors and they obtained solutions using various available techniques in the literature. For example, system of integral equations has been studied with wavelets techniques in [10, 11], with Adomian decomposition method in [12, 13], with Tau method in [14], with chebesheve polynomial and block pulse function in [15, 16], and with Taylor expansion and some modified methods based on taylor series expansion in [17–25].

In this paper, we use shifted Legendre polynomials and develop a new operational matrix of integration. Based on the operational matrix of integration, we develop a simple method to find solutions of the coupled system of Fredholm integral equations. The method reduces the coupled system to a system of easily solvable algebraic equations without discretizing the original system of equations. Besides simplicity, the method yields accurate results even for small value of M resulting in the reduction of the system to small system of algebraic equations. It is verified by examples and their numerical simulations demonstrating that the results obtained using the new technique match very well with the exact solutions of the problems. To show the efficiency of the method over some of the well-known techniques, we compare our results with some of the results already studied with other available methods such as Taylor series approximation method [19] and block pulse method [16]. We find that the new techniques provide highly accurate solutions as compared to Taylor series approximation method and block pulse method.

2. Main Results: New Operational Matrix of Integrations

The Legendre polynomials defined on [-1,1] are given by the following recurrence relation:
(1)Łi+1(z)=2i+1i+1zŁi(z)-ii+1Łi-1(z),i=1,2,…,whereŁ0(z)=0,Ł1(z)=z.
The transformation x=(z+1)/2 transforms the interval [-1,1] to [0,1] and the polynomials transformed to the so called shifted Legendre polynomials given as [26] follows:
(2)Pi(x)=∑k=0i(-1)i+k(i+k)!(i-k)!xk(k!)2,i=0,1,2,3,…,
where Pi(0)=(-1)i, Pi(1)=1. The orthogonality condition is
(3)∫01Pi(x)Pj(x)dx={12i+1,ifi=j;0,ifi≠j.

Consequently, any f(x)∈C[0,1] can be approximated by shifted Legendre polynomial as follows:
(4)f(x)≈∑a=0mcaPa(x),whereca=〈f(x),Pa(x)〉=(2a+1)∫01f(x)Pa(x)dx.
In vector notation, we write
(5)f(x)=KMTP^M,
where M=m+1, K is the coefficient vector, and P^ is M terms vector function. In case of function of two variables, that is, f∈C([0,1]×[0,1]), we write
(6)f(x,t)≈∑i=0m∑j=0mcijPi(x)Pj(t),wherecij=(2i+1)(2j+1)∬01f(x,t)Pi(x)Pj(t)dxdt.
The orthogonality condition of Pi(x)Pj(t) is found to be
(7)∬01Pi(x)Pj(t)Pa(x)Pb(t)dxdt={1(2i+1)(2j+1),ifa=i,b=j;0,otherwise.
In vector notation, (6) can be written as
(8)f(x,t)≈(P^M(x))TC(M×M)P^M(t),
where P^M(x) and P^M(t) are column vectors containing Legendre polynomial and C is the coefficient matrix whose entries are obtained by using (6).

2.1. Error Analysis

For sufficiently smooth function f(x,y) on [0,1]×[0,1], the error of the approximation is given by
(9)∥f(x,y)-Pn(x,y)∥2≤(C1+C2+C31MM+1)1MM+1,
where
(10)C1=14max(x,y)∈[0,1]×[0,1]|∂M+1∂xM+1f(x,y)|,C2=14max(x,y)∈[0,1]×[0,1]|∂M+1∂yM+1f(x,y)|,C3=116max(x,y)∈[0,1]×[0,1]|∂2M+2∂xM+1∂yM+1f(x,y)|.
We refer the reader to [27] for the proof of the above result.

Lemma 1.

Let f(x,t)∈C([0,1]×[0,1]) and g(t)∈C([0,1]); then
(11)∫01f(x,t)g(t)dt≈KMGM×MP^(x),
where KM is the Legendre coefficient vector of g(t) and the matrix G=[qji], where qji=(1/(2j+1))cij.

Proof.

In view of (5) and (6), we have
(12)f(x,t)≈∑i=0m∑j=0mcijPi(x)Pj(t),wherecij=(2j+1)(2i+1)∬01f(x,t)Pi(x)Pj(t)dxdt,g(t)≈∑a=0mdaPa(t),whereda=(2a+1)∫01g(t)Pa(t)dt.
Using (12), we obtain
(13)∫01f(x,t)g(t)dt≈∫01(∑i=0m∑j=0mcijPi(x)Pj(t))(∑a=0mdaPa(t))dt,
which implies that
(14)∫01f(x,t)g(t)dt≈∑i=0m∑j=0m∑a=0mdacijPi(x)∫01Pj(t)Pa(t)dt.
Using the orthogonality relation, we get
(15)∫01f(x,t)g(t)dt≈∑i=0m∑j=0mdjcijPi(x)(12j+1)=∑j=0m∑i=0mdjqjiPi(x),
where qji=(1/(2j+1))cij. In matrix form, we have
(16)∫01f(x,t)g(t)dt≈KMGM×MP^(x).

3. System of Fredholm Integral Equations

Consider the following coupled system of Fredholm integral equations:
(17)U(x)+λ11∫01K11(x,t)U(t)dtU(x)+λ12∫01K12(x,t)V(t)dt=f(x),V(x)+λ21∫01K21(x,t)U(t)dtV(x)+λ22∫01K22(x,t)V(t)dt=g(x),
where λ11, λ12, λ21, and λ22 are real constants, f,g∈C([0,1]), K11,K12,K21,K22∈C([0,1]×[0,1]), and U(x), V(x) are unknown functions to be determined. Approximating U(x) and V(x) in terms of Legendre polynomials, we obtain
(18)U(x)≈HMTP^(x),V(x)≈NMTP^(x).
Using Lemma 1, we have the following approximations:
(19)∫01K11(x,t)U(t)dt≈HMTG11P^(x),∫01K12(x,t)V(t)dt≈NMTG12P^(x),∫01K21(x,t)U(t)dt≈HMTG21P^(x),∫01K22(x,t)V(t)dt≈NMTG22P^(x).
Using (18) and (19) in the coupled system (17), we obtain the following system of algebraic equations
(20)HMTP^(x)+λ11HMTG11P^(x)+λ12NMTG12P^(x)=F1P^(x),NMTP^(x)+λ21HMTG21P^(x)+λ22NMTG22P^(x)=F2P^(x),
which can be written as
(21)(HMTP^(x)NMTP^(x))+(λ11HMTG11P^(x)λ22NMTG22P^(x))+(λ12NMTG12P^(x)λ21HMTG21P^(x))=(F1P^(x)F2P^(x)).
The transpose of the above system is given by
(22)(HMTP^(x)NMTP^(x))+(λ11HMTG11P^(x)λ22NMTG22P^(x))+(λ12NMTG12P^(x)λ21HMTG21P^(x))=(F1P^(x)F2P^(x))
which can further be written as
(23)(HMTNMT)A+(HMTNMT)(λ11G1100λ22G22)A+(HMTNMT)(0λ21G21λ12G120)A=(F1F2)A,
where
(24)A=(P^(x)00P^(x)).
Hence it follows that
(25)(HMTNMT)+(HMTNMT)(λ11G11λ21G21λ12G12λ22G22)-(F1F2)=0,
which is a generalized Sylvester type equation and can easily be solved for the unknown HM and NM by any computational software.

4. Illustrative ExamplesExample 1.

Consider the following system of Fredholm integral equation:
(26)U(x)-13∫01(x+t)U(t)dt-13∫01(x+t)V(t)dt=x18+1736,V(x)-∫01(xt)U(t)dt-∫01(xt)V(t)dt=x2-1912x+1.
The exact solutions of the system are U(x)=1+x and V(x)=x2. The solutions (U(x), V(x)) obtained via our technique for M=3 (small enough) are compared with the exact solutions of the problem in Figure 1, where dots represent the exact solutions and the curves are for the solutions obtained via the new method. From Figure 1, it follows that our solutions matchs very well with the exact solution of the problem even for small value M, which shows the effectiveness of our technique.

Comparison between the exact solutions and the solutions obtained via the new method for M=3. Dots represent the exact solution and the approximate solutions are represented by curved lines.

Example 2.

For comparison purposes, consider the following coupled system of Fredholm integral equations:
(27)U(x)+∫01e(x-t)U(t)dt+∫01e(xt+2t)V(t)dt=2ex+1(x+1)(e(x+1)-1),V(x)+∫01e(xt)U(t)dt+∫01e(x+t)V(t)dt=ex+e-x+1(x+1)(e(x+1)-1).
The exact solutions of the system are U(x)=ex and V(x)=e-x. We obtain the approximate solutions of the system for different values of M and compare the results with the exact solutions of the system. For M=2 and M=3, the comparison is shown in Figure 2, where dots represent the exact solutions of the system and doted curves (red and yellow) represent the approximate solution (U(x) and V(x)) obtained via our technique for M=2 while Blue and orange dots represent the approximate solutions (U(x) and V(x)) obtained via our technique for M=3. It is clear that the approximate solutions approach rapidly the exact solutions as the values of M increase. It also shows that the approximate solutions are very close to the exact ones for M=3. For example, error of approximation in both U(x) (red doted curve) and V(x) (blue doted curve) is less than 10-6 for M=5 as shown in Figure 3, which is much more acceptable number and demonstrates high accuracy of the new technique. Further, we compare our results with some other available results in the literature. We compare the absolute errors (red line) with the absolute error obtained in [19] using Taylor series approximation and also with absolute error obtained in [16] using numerical solution with block pulses. The results are shown in Figures 4 and 5. From these analyses, it is clear that the absolute error in our method even for small value of M=4 is much smaller than those obtained in [16, 19] even for much larger values of m such as m=16,32. It is a clear indication that the new techniques provide highly accurate solutions as compared to Taylor series approximation method and block pulse method.

Comparing exact solutions with the solutions obtained by our method at different values of M.

Error analysis in U(x) and V(x) for M=5.

Comparing the error estimates in U(x) by our method with error found with Taylor series approximation method (purple dots) and block pulse method (green and blue dots).

Comparing the error estimates in V(x) by our method with error found with taylor series approximation method (green dots) and block pulse method (orange and purple dots).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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