On Some Volterra and Fredholm Problems via the Unified Integrodifferential Quadrature Method

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=1𝛾_𝑘(𝑥)𝑑^𝑘𝑓(𝑥)𝑑𝑥^𝑘=𝛼(𝑥)𝑓(𝑥)+𝛽(𝑥)+𝜆𝑊(𝑥,𝑧),𝑎≤𝑥≤𝑇,(1) where 𝑊(𝑥,𝑧) is given by 𝑊(𝑥,𝑧)≡^𝑧𝑎(𝑥−𝑠)^−𝜈𝐾(𝑥,𝑠)𝑓(𝑠)𝑑𝑠,(2) and we set 𝑊(𝑥,𝑧)=𝑊(𝑥,𝑥),𝑊(𝑥,𝑇),forVolterratype,forFredholmtype,(3) with prescribed boundary condition 𝑓(𝑎)=𝑓_𝑎.

𝜆 and 𝜈 are parameters with 0<𝜈<1. 𝛼(𝑥),𝛽(𝑥), and 𝛾_𝑘(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 𝑊(𝑥,𝑧)=1(𝜈−1)(𝑥−𝑧)^1−𝜈−(𝑥−𝑎)𝐾_1−𝜈(𝑥,𝑧)𝑓(𝑧)−1𝐾(𝑥,𝑎)𝑓(𝑎)(𝜈−1)^𝑧𝑎(𝑥−𝑠)^1−𝜈𝑑[𝐾(𝑥,𝑠)𝑓(𝑠)]𝑑𝑠𝑑𝑠.(4) Now the relation (1) becomes ^𝑙_𝑘=1𝛾_𝑘(𝑥)𝑑^𝑘𝑓(𝑥)𝑑𝑥^𝑘=+𝛼(𝑥)𝜆𝑓(𝑥)+𝛽_𝜈(𝑥,𝑧)(1−𝜈)^𝑧𝑎(𝑥−𝑠)^1−𝜈𝑑[𝐾(𝑥,𝑠)𝑓(𝑠)]𝑑𝑠𝑑𝑠,(5) where 𝛽_𝜈(𝑥,𝑧)=𝛽(𝑥)+(𝜆/(𝜈−1))((𝑥−𝑧)^1−𝜈𝐾(𝑥,𝑧)𝑓(𝑧)−(𝑥−𝑎)^1−𝜈𝐾(𝑥,𝑎)𝑓(𝑎)), 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 by𝑑^𝑘𝑓𝑥_𝑚𝑑𝑥^𝑘=^𝑁_𝑗=0𝐷^𝑘𝑚𝑗𝑓𝑥_𝑗,(6)^𝑥𝑚_𝑎𝑥_𝑚−𝑠^1−𝜈𝑑𝐾𝑥_𝑚,𝑠𝑓(𝑠)𝑑𝑠𝑑𝑠=^𝑁_𝑗=0𝐼_𝑚𝑗𝑓𝑥_𝑗,forVolterratype,(7)^𝑇𝑎𝑥_𝑚−𝑠^1−𝜈𝑑𝐾𝑥_𝑚,𝑠𝑓(𝑠)𝑑𝑠𝑑𝑠=^𝑁_𝑗=0𝐽_𝑚𝑗𝑓𝑥_𝑗,forFredholmtype,(8) where {𝑥_𝑘}^𝑁𝑘=0 are interpolating points, taken as the points of Tchebychev of the form 𝑥_𝑗=(1/2)𝑇[1−cos((2𝑗+1)/(2𝑁+2)𝜋)], 0≤𝑗≤𝑁. 𝐷_𝑖𝑗 and 𝐼_𝑚𝑗 are the weighting coefficients linked with Lagrange interpolated polynomials 𝑃_𝑁,𝑗(𝑥) and are explicitly given by 𝐷^𝑘𝑚𝑗=𝑑^𝑘𝑃_𝑁,𝑗𝑥_𝑚𝑑𝑥^𝑘,(9)𝐼_𝑚𝑗=^𝑥𝑚₀𝑥_𝑚−𝑠^1−𝜈𝑑𝐾𝑥_𝑚,𝑠𝑃_𝑁,𝑗(𝑠)𝐽𝑑𝑠𝑑𝑠,_𝑚𝑗=^𝑇0𝑥_𝑚−𝑠^1−𝜈𝑑𝐾𝑥_𝑚,𝑠𝑃_𝑁,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𝑥_𝑚𝑓𝑥_𝑚+𝛽_𝜈𝑥_𝑚,𝑥_𝑚=^𝑁_𝑗=0𝑈^𝐼𝑚𝑗𝑓𝑥_𝑗,𝑚=0,…,𝑁,(11) for Volterra type, and 𝛼𝑥_𝑚𝑓𝑥_𝑚+𝛽_𝜈𝑥_𝑚,𝑇=^𝑁_𝑗=0𝑈^𝐽𝑚𝑗𝑓𝑥_𝑗,𝑚=0,…,𝑁,(12) for Fredholm type, where 𝑈^𝐼𝑚𝑗=^𝑙_𝑘=0𝛾_𝑘𝑥_𝑚𝐷^𝑘𝑚𝑗−𝜆(1−𝜈)𝐼^𝑚𝑗,(13)𝑈^𝐽𝑚𝑗=^𝑙_𝑘=0𝛾_𝑘𝑥_𝑚𝐷^𝑘𝑚𝑗−𝜆(1−𝜈)𝐽^𝑚𝑗.(14)

The relations (12) and (13) are real systems of 𝑀×𝑀 equations in the 𝑀×𝑀 unknown real coefficients 𝑈^𝐼𝑚𝑗 and 𝑈^𝐽𝑚𝑗, 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 𝐷_𝑖𝑖, 𝐼_𝑖𝑖, and 𝐽_𝑖𝑖 in the following forms: 𝐷^𝑘𝑖𝑖=−^𝑁_𝑗=_{0,𝑗≠𝑖}𝐷^𝑘𝑖𝑗,for𝑖=0,…,𝑁,𝑘=1,…,𝑙,(15)𝐼_𝑖𝑖=^𝑥𝑖_𝑎𝑥_𝑖−𝑠^1−𝜈𝑑𝐾𝑥_𝑖,𝑠𝑑𝑠𝑑𝑠−^𝑁_{𝑗=0,𝑗≠𝑖}𝐼_𝑖𝑗,for𝑖=0,…,𝑁,(16)𝐽_𝑖𝑖=^𝑇𝑎𝑥_𝑖−𝑠^1−𝜈𝑑𝐾𝑥_𝑖,𝑠𝑑𝑠𝑑𝑠−^𝑁_{𝑗=0,𝑗≠𝑖}𝐽_𝑖𝑗,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𝑥_𝑗𝑃_𝑁,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 𝛼_𝑝𝑓_𝑝+𝛽_{𝜈,𝑝𝑝}=^𝑁_𝑗=0𝑈^𝐼𝑝𝑗𝑓_𝑗,𝑝=0,…,𝑁,(19)𝛼_𝑝𝑓_𝑝+𝛽_{𝜈,𝑝𝑇}=^𝑁_𝑗=0𝑈^𝐽𝑝𝑗𝑓_𝑗,𝑝=0,…,𝑁,(20) where 𝑓_𝑗 and 𝛼_𝑖 stand for the values of 𝑓(𝑥_𝑖) and 𝛼(𝑥_𝑖), respectively. 𝛽_{𝜈,𝑝𝑝} and 𝛽_{𝜈,𝑝𝑇} denote the values of 𝛽_𝜈(𝑥_𝑝,𝑥_𝑝) and 𝛽_𝜈(𝑥_𝑝,𝑇), respectively.

Evaluating (20) and (21) at 𝑥_𝑝, 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 𝑓_𝑝 and 𝛽_{𝜈,𝑝𝑝}(𝛽_{𝜈,𝑝𝑇}), respectively, 𝑈₁(𝑈₂) is a matrix with the elements 𝑈_1,𝑖𝑗=𝑈^𝐼𝑖𝑗(𝑈_2,𝑖𝑗=𝑈^𝐽𝑖𝑗), and the components of 𝑈^𝐼 and 𝑈^𝐼 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(𝑥_𝑖)=|𝑓(𝑥_𝑖)−𝑓_exact(𝑥_𝑖)|=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)−^𝑥00≤(𝑥𝑥−≤𝑠)1,𝑓^−1/2𝑓(0()𝑠=)𝑑0𝑠,,(23) where 𝛽(𝑥)=(3/2)√𝑥+𝑥√𝑥+(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,𝑎=0,𝑇=1, and 𝑓₀=0. The exact solution is 𝑓(𝑥)=𝑥^3/2.

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.

Figure 1 

Solution 𝑓(𝑥) of (23). Solid curve: exact solution: 𝑓(𝑥)=𝑥3/2, dash curve: present work.

3.2. Example  2

We consider the numerical solution of the following Volterra’s integrodifferential equation [40]: 𝑑²𝑓(𝑥)𝑑𝑥²=𝑓(𝑥)+𝛽0(𝑥≤)𝑥+≤^𝑥01𝑥,𝑓𝑠𝑓(0(𝑠))=𝑑𝑠1,,(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,𝑎=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.

Figure 2 

Solution 𝑓(𝑥) of (24). Solid line: exact solution: 𝑓(𝑥)=𝑥2+exp(𝑥), dash curve: present work.

3.3. Example  3

Consider the Fredholm integrodifferential equation [41]: 𝑑²𝑓(𝑥)𝑑𝑥²+𝑥𝑑𝑓(𝑥)𝑑𝑥=𝑥𝑓(𝑥)+𝛽(𝑥)+¹⁻¹−1≤𝑥𝑒≤^−𝑠s1in,𝑓(𝑥()−𝑓1)(𝑠=)𝑑1𝑠,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, 𝑎=−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.

Figure 3 

Solution 𝑓(𝑥) of (25). Solid curve: exact solution: 𝑓(𝑥)=exp(𝑥), dash curve: present work.

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)+¹⁻¹−1≤𝑥≤1,𝑓(−1)=−𝑓si(n𝑠)(𝑑1𝑠),,(26) where 𝛽(𝑥)=𝑥²cos(𝑥)−𝑥sin(𝑥), and from this equation we deduce the following: 𝑙1,=𝑓5₋₁,𝛾₁(𝑥)=−1,𝛾₂(𝑥)=𝛾₄(𝑥)=0,𝛾₃(𝑥)=−𝑥²,𝛾₅(𝑥)=1,𝛼(𝑥)=𝑥,𝜆=1,𝜈=0,𝑎=−1,𝑇==−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.

Figure 4 

Solution 𝑓(𝑥) of the equation (26). Solid curve: exact solution: 𝑓(𝑥)=sin(𝑥), dash curve: present work.

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(𝑥_𝑖) 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.