Abstract

We present a new approach based on the formulation of the integrodifferential quadrature method (hereafter called IDQ) to handle Volterra's and Fredholm's equations. This approach is constructed and tested with some realistic numerical examples using the basic computational aspects.

1. Introduction

In the present work, we present a formulation of a new numerical approach which is based on the generalized integrodifferential quadrature method and applied to weakly singular Volterra and Fredholm integrodifferential equations in the linear case. This method studies the situation in which the unknown function is identified as the Lagrange polynomial and the interpolating points of the Tchebychev type are used. The accuracy and efficiency of solving integrodifferential equations are still an ongoing research in numerical analysis. Several different methods have also been modeled for the solutions of linear and nonlinear problem [1–27].

The central argument for the interest of the type of this problem comes naturally from its wide applications almost in any branches of science and engineering described by systems of ODEs and PDEs [28–38] which in some situations can be transformed into a set of integral equations. Much attention has been devoted to the investigation of new mathematical models and numerical approaches to evaluate the solutions of the integral equations.

New calculations are performed for the construction of the solution by a suitable choice of the interpolating points using the unified integrodifferential quadrature method. Based on this fact, this technique is a very promising and powerful methodology in successfully locating best solutions of the problem. Our main purpose is to develop a general numerical schema for the integrodifferential equations that is universally applicable.

The contents of this paper are organized as follows. In Section 2, a formulation adapted to the problem of construction of the solution of the one-dimensional weakly singular integrodifferential equation by using a summary limited to some aspects of the unified integrodifferential quadrature method is qualitatively presented. Section 3 exposes some essential examples for the construction of the unknown solution. The comments and conclusion are given in Section 4.

2. Unified Integrodifferential Quadrature Method

The description of generalized integrodifferential quadrature method is summarized as follows: the starting point of the weakly singular integrodifferential equation is written as 𝑙𝑘=1𝛾𝑘𝑑(𝑥)𝑘𝑓(𝑥)𝑑𝑥𝑘,=𝛼(𝑥)𝑓(𝑥)+𝛽(𝑥)+𝜆𝑊(𝑥,𝑧),ğ‘Žâ‰¤ğ‘¥â‰¤ğ‘‡(1) where 𝑊(𝑥,𝑧) is given by 𝑊(𝑥,𝑧)â‰¡ğ‘§ğ‘Ž(𝑥−𝑠)−𝜈𝐾(𝑥,𝑠)𝑓(𝑠)𝑑𝑠,(2) and we set 𝑊(𝑥,𝑧)=𝑊(𝑥,𝑥),forVolterratype,𝑊(𝑥,𝑇),forFredholmtype,(3) with prescribed boundary condition 𝑓(ğ‘Ž)=ğ‘“ğ‘Ž.

𝜆 and 𝜈 are parameters with 0<𝜈<1. 𝛼(𝑥),𝛽(𝑥), and 𝛾𝑘(𝑥)(𝑘=0,…,𝑚) are given functions, and 𝐾(𝑥,𝑠) is a regular kernel of the integrodifferential equation. It is assumed that the function involved in (1) is sufficiently regular. The above equations, for 𝜈=0, become regular integrodifferential equations.

In order to avoid the singularity, (1) is integrated by part and with little effort; we obtain 1𝑊(𝑥,𝑧)=(𝜈−1)(𝑥−𝑧)1−𝜈𝐾(𝑥,𝑧)𝑓(𝑧)−(ğ‘¥âˆ’ğ‘Ž)1−𝜈−1𝐾(𝑥,ğ‘Ž)𝑓(ğ‘Ž)(𝜈−1)ğ‘§ğ‘Ž(𝑥−𝑠)1−𝜈𝑑[]𝐾(𝑥,𝑠)𝑓(𝑠)𝑑𝑠𝑑𝑠.(4) Now the relation (1) becomes 𝑙𝑘=1𝛾𝑘𝑑(𝑥)𝑘𝑓(𝑥)𝑑𝑥𝑘=𝛼(𝑥)𝑓(𝑥)+𝛽𝜈+𝜆(𝑥,𝑧)(1−𝜈)ğ‘§ğ‘Ž(𝑥−𝑠)1−𝜈𝑑[]𝐾(𝑥,𝑠)𝑓(𝑠)𝑑𝑠𝑑𝑠,(5) where 𝛽𝜈(𝑥,𝑧)=𝛽(𝑥)+(𝜆/(𝜈−1))((𝑥−𝑧)1−𝜈𝐾(𝑥,𝑧)𝑓(𝑧)−(ğ‘¥âˆ’ğ‘Ž)1−𝜈𝐾(𝑥,ğ‘Ž)𝑓(ğ‘Ž)), and 𝑧=𝑥 and 𝑧=𝑇 for Volterra type and Fredholm type, respectively.

At this stage, we introduce the quadrature aspect of solutions: the derivatives and integral in expression (6) are approximated by𝑑𝑘𝑓𝑥𝑚𝑑𝑥𝑘=𝑁𝑗=0𝐷𝑘𝑚𝑗𝑓𝑥𝑗,(6)î€œğ‘¥ğ‘šğ‘Žî€·ğ‘¥ğ‘šî€¸âˆ’ğ‘ 1−𝜈𝑑𝐾𝑥𝑚𝑓,𝑠(𝑠)𝑑𝑠𝑑𝑠=𝑁𝑗=0𝐼𝑚𝑗𝑓𝑥𝑗,forVolterratype,(7)î€œğ‘‡ğ‘Žî€·ğ‘¥ğ‘šî€¸âˆ’ğ‘ 1−𝜈𝑑𝐾𝑥𝑚,𝑠𝑓(𝑠)𝑑𝑠𝑑𝑠=𝑁𝑗=0𝐽𝑚𝑗𝑓𝑥𝑗,forFredholmtype,(8) where {𝑥𝑘}𝑁𝑘=0 are interpolating points, taken as the points of Tchebychev of the form 𝑥𝑗=(1/2)𝑇[1−cos((2𝑗+1)/(2𝑁+2)𝜋)], 0≤𝑗≤𝑁. 𝐷𝑖𝑗 and 𝐼𝑚𝑗 are the weighting coefficients linked with Lagrange interpolated polynomials 𝑃𝑁,𝑗(𝑥) and are explicitly given by 𝐷𝑘𝑚𝑗=𝑑𝑘𝑃𝑁,𝑗𝑥𝑚𝑑𝑥𝑘,(9)𝐼𝑚𝑗=𝑥𝑚0𝑥𝑚−𝑠1−𝜈𝑑𝐾𝑥𝑚𝑃,𝑠𝑁,𝑗(𝑠)𝐽𝑑𝑠𝑑𝑠,𝑚𝑗=𝑇0𝑥𝑚−𝑠1−𝜈𝑑𝐾𝑥𝑚𝑃,𝑠𝑁,𝑗(𝑠)𝑑𝑠𝑑𝑠.(10)

With little effort the original problems (1) can be converted to an algebraic system and take, respectively, the following compact forms: 𝛼𝑥𝑚𝑓𝑥𝑚+𝛽𝜈𝑥𝑚,𝑥𝑚=𝑁𝑗=0𝑈𝐼𝑚𝑗𝑓𝑥𝑗,𝑚=0,…,𝑁,(11) for Volterra type, and 𝛼𝑥𝑚𝑓𝑥𝑚+𝛽𝜈𝑥𝑚=,𝑇𝑁𝑗=0𝑈𝐽𝑚𝑗𝑓𝑥𝑗,𝑚=0,…,𝑁,(12) for Fredholm type, where 𝑈𝐼𝑚𝑗=𝑙𝑘=0𝛾𝑘𝑥𝑚𝐷𝑘𝑚𝑗−𝜆𝐼(1−𝜈)𝑚𝑗,(13)𝑈𝐽𝑚𝑗=𝑙𝑘=0𝛾𝑘𝑥𝑚𝐷𝑘𝑚𝑗−𝜆𝐽(1−𝜈)𝑚𝑗.(14)

The relations (12) and (13) are real systems of 𝑀×𝑀 equations in the 𝑀×𝑀 unknown real coefficients 𝑈𝐼𝑚𝑗 and 𝑈𝐽𝑚𝑗, respectively, where 𝑀=𝑁+1. In order to avoid unnecessary calculation and to optimize the computational aspect, it is therefore more convenient to get the desired coefficients 𝐷𝑖𝑖, 𝐼𝑖𝑖, and 𝐽𝑖𝑖 in the following forms: 𝐷𝑘𝑖𝑖=−𝑁𝑗=0,𝑗≠𝑖𝐷𝑘𝑖𝑗,for𝑖=0,…,𝑁,𝑘=1,…,𝑙,(15)𝐼𝑖𝑖=î€œğ‘¥ğ‘–ğ‘Žî€·ğ‘¥ğ‘–î€¸âˆ’ğ‘ 1−𝜈𝑥𝑑𝐾𝑖,𝑠𝑑𝑠𝑑𝑠−𝑁𝑗=0,𝑗≠𝑖𝐼𝑖𝑗,for𝑖=0,…,𝑁,(16)𝐽𝑖𝑖=î€œğ‘‡ğ‘Žî€·ğ‘¥ğ‘–î€¸âˆ’ğ‘ 1−𝜈𝑥𝑑𝐾𝑖,𝑠𝑑𝑠𝑑𝑠−𝑁𝑗=0,𝑗≠𝑖𝐽𝑖𝑗,for𝑖=0,…,𝑁.(17)

Now the expressions (14), (15), (16), (17), and (18) provide the charmingly practical formulae for the weighting coefficients. Once the function values at all grid points are obtained, it is then easy to determine the function values in the overall domain in terms of polynomial approximation, such that 𝑓(𝑥)=𝑁𝑗=0𝑓𝑥𝑗𝑃𝑁,𝑗(𝑥).(18)

The practical part of this study is examined in the following section.

3. Worked Examples

We shall consider the solution of some crack situations in order to show the application and the effectiveness of the method described in Section 2. Four examples are considered for this purpose.

We write (1) in the form 𝛼𝑝𝑓𝑝+𝛽𝜈,𝑝𝑝=𝑁𝑗=0𝑈𝐼𝑝𝑗𝑓𝑗,𝑝=0,…,𝑁,(19)𝛼𝑝𝑓𝑝+𝛽𝜈,𝑝𝑇=𝑁𝑗=0𝑈𝐽𝑝𝑗𝑓𝑗,𝑝=0,…,𝑁,(20) where 𝑓𝑗 and 𝛼𝑖 stand for the values of 𝑓(𝑥𝑖) and 𝛼(𝑥𝑖), respectively. 𝛽𝜈,𝑝𝑝 and 𝛽𝜈,𝑝𝑇 denote the values of 𝛽𝜈(𝑥𝑝,𝑥𝑝) and 𝛽𝜈(𝑥𝑝,𝑇), respectively.

Evaluating (20) and (21) at 𝑥𝑝, the linear systems of algebraic equations follow, respectively, 𝐼𝐴−𝑈1𝐹=𝐵1,(21)𝐼𝐴−𝑈2𝐹=𝐵2,(22) where the vectors 𝐹 and 𝐵1(𝐵2) have components 𝑓𝑝 and 𝛽𝜈,𝑝𝑝(𝛽𝜈,𝑝𝑇), respectively, 𝑈1(𝑈2) is a matrix with the elements 𝑈1,𝑖𝑗=𝑈𝐼𝑖𝑗(𝑈2,𝑖𝑗=𝑈𝐽𝑖𝑗), and the components of 𝑈𝐼 and 𝑈𝐼 are given by (14) and (15) together with (16), (17), and (18). The equations (20) and (21) are explicit prescriptions that give the solutions in 𝑂((𝑁+1)2) operations.

The accuracy of solutions can be checked by using the error tolerance 𝐸(𝑥𝑖)=|𝑓(𝑥𝑖)−𝑓exact(𝑥𝑖)|=10−𝑗, where “𝑗” is a positive integer. We continue evaluating the solution until the above relation was satisfied.

In order to verify the efficiency of the method developed in the previous section, the following examples have been selected to provide a comparison with previously published work.

3.1. Example  1

In this example we use Volterra’s integrodifferential equation [39] given by 𝑑𝑓(𝑥)𝑑𝑥=𝑓(𝑥)+𝛽(𝑥)−𝑥0(𝑥−𝑠)−1/2𝑓(𝑠)𝑑𝑠,0≤𝑥≤1,𝑓(0)=0,(23) where √𝛽(𝑥)=(3/2)√𝑥+𝑥𝑥+(3𝜋/8)𝑥2, from the governing equation (1) and (24) subject to 𝐾(𝑥,𝑠)=1, we deduce the following: 𝑙=1,𝛾1(𝑥)=1,𝛼(𝑥)=1,𝜆=−1,𝜈=1/2,ğ‘Ž=0,𝑇=1, and 𝑓0=0. The exact solution is 𝑓(𝑥)=𝑥3/2.

Illustrated in Figure 1 is the basic application of the problem (24). The numerical result displayed in Figure 1 is qualitatively in good agreement with the exact solution in which the error tolerance, TOLV=10−5.

3.2. Example  2

We consider the numerical solution of the following Volterra’s integrodifferential equation [40]: 𝑑2𝑓(𝑥)𝑑𝑥2=𝑓(𝑥)+𝛽(𝑥)+𝑥0𝑥𝑠𝑓(𝑠)𝑑𝑠,0≤𝑥≤1,𝑓(0)=1,(24) where 𝛽(𝑥)=−𝑥(𝑥+1)+𝑥(1−𝑥)exp(𝑥)+2−𝑥5/4, and from this equation we deduce the following: 𝑙=2,𝛾1(𝑥)=0,𝛾2(𝑥)=1,𝛼(𝑥)=1,𝜆=1,𝜈=0,ğ‘Ž=0,𝑇=1,𝑓0=1, and the kernel 𝐾(𝑥,𝑠)=𝑥𝑠.

The above equation has the exact solution 𝑓(𝑥)=𝑥2+exp(𝑥).

The result is displayed in Figure 2, where the error tolerance, TOLV=10−6. With this tolerance and for the number of the interpolation points 𝑁=10, the results under consideration are very satisfactory.

3.3. Example  3

Consider the Fredholm integrodifferential equation [41]: 𝑑2𝑓(𝑥)𝑑𝑥2+𝑥𝑑𝑓(𝑥)𝑑𝑥=𝑥𝑓(𝑥)+𝛽(𝑥)+1−1𝑒−𝑠1sin(𝑥)𝑓(𝑠)𝑑𝑠,−1≤𝑥≤1,𝑓(−1)=𝑒,(25) where 𝛽(𝑥)=exp(𝑥)−2sin(𝑥), and from this equation we deduce the following: 𝑙=2,𝛾1(𝑥)=𝑥, 𝛾2(𝑥)=1, 𝛼(𝑥)=𝑥, 𝜆=1, 𝜈=0, ğ‘Ž=−1, 𝑇=1, 𝑓−1=1/𝑒, and the kernel 𝐾(𝑥,𝑠)=𝑒−𝑠sin(𝑥). The exact solution has the form 𝑓(𝑥)=exp(𝑥).

In Figure 3, the number of interpolation points is taken equal to 𝑁=10 to solve this problem with TOLV=10−8. We see from this Figure that the method provides accurate results in the full range, where the error tolerance, TOLV=10−6.

3.4. Example  4

We apply the mentioned technique to solve the fifth-order Fredholm integrodifferential equation [42]: 𝑑5𝑓(𝑥)𝑑𝑥5−𝑥2𝑑3𝑓(𝑥)𝑑𝑥3−𝑑𝑓(𝑥)𝑑𝑥=𝑥𝑓(𝑥)+𝛽(𝑥)+1−1𝑓(𝑠)𝑑𝑠,−1≤𝑥≤1,𝑓(−1)=−sin(1),(26) where 𝛽(𝑥)=𝑥2cos(𝑥)−𝑥sin(𝑥), and from this equation we deduce the following: 𝑙=5,𝛾1(𝑥)=−1,𝛾2(𝑥)=𝛾4(𝑥)=0,𝛾3(𝑥)=−𝑥2,𝛾5(𝑥)=1,𝛼(𝑥)=𝑥,𝜆=1,𝜈=0,ğ‘Ž=−1,𝑇=1,𝑓−1=−sin(1), and the kernel 𝐾(𝑥,𝑠)=1. The exact solution has the form 𝑓(𝑥)=sin(𝑥). Figure 4 presents computer simulations of this example, that is generated using the number of interpolating points 𝑁=8. The result is compared with the exact solution. From Figure 4, we see that the agreement between the numerical result of the present method and the exact result is very good, with an error tolerance TOLV=10−6.

4. Comments and Conclusions

This paper has introduced a new formulation that uses the unified integrodifferential quadrature method to handle some integrodifferential equations. The main advantage of Lagrange polynomial is that the weighting coefficients that have to be computed do not depend on the 𝑓(𝑥𝑖) and it is independent on the manner the discrete points are ordered.

The preliminary results, obtained through the use of this method, show that the resulting solutions are quite good for all examples which have been selected here as a testbed. We can note that similar patterns of convergence are seen for all four examples with a common tolerance TOLV=10−6.

This method seems to be a powerful alternative and consequently gives very accurate solutions to the problems under consideration. It would be interesting to generalize this method to the nonlinear case. This matter deserves a further work.

Acknowledgments

The authors gratefully acknowledge helpful conversations with Professor W. Cramer. This work was sponsored in part by the M.E.R.S (Ministère de l’Enseignement et de la Recherche Scientifique): under contract no. D01420060012.