On-line version ISSN 1807-0302
Comput. Appl. Math. vol.31 no.1 São Carlos 2012
Zakieh Avazzadeh*; Mohammad Heydari
In this paper, an efficient method is presented for solving two dimensional Fredholm and Volterra integral equations of the second kind. Chebyshev polynomials are applied to approximate a solution for these integral equations. This method transforms the integral equation to algebraic equations with unknown Chebyshev coefficients. The high accuracy of this method is verified through some numerical examples.
Mathematical subject classification: 65R20, 41A50, 41A55, 65M70.
Key words: Chebyshev polynomials, two dimensional integral equations, collocation method.
Two dimensional integral equations provide an important tool for modeling a numerous problems in engineering and science [2, 12]. These equations appear in electromagnetic and electrodynamic, elasticity and dynamic contact, heat and mass transfer, fluid mechanic, acoustic, chemical and electrochemical process, molecular physics, population, medicine and in many other fields [6, 7, 8, 14, 18, 20].
The Nystrom method  and collocation method [3, 21] are the most important approaches of the numerical solution of these integral equations. Recently, some new methods such as differential transform method are applied for solving two dimensional linear and nonlinear Volterra integral equations [1, 5, 19].
In this work, we will apply the Chebyshev polynomials for solving two dimensional integral equations of second kind. The use of the Chebyshev series for the numerical solution of linear integral equations has previously been discussed in  and references therein. The privilege of the method is simplicity and spectral accuracy [4, 11]. The examples confirm that the method is considerably fast and highly accurate as sometimes lead to exact solution. Also, this method lead to continuous solution covering all the domain.
The paper is organized in the following way. In Section 2 the famous Chebyshev polynomials and its application are introduced . In Section 3 the method of solution of the linear two dimensional integral equation of second kind isdescribed. In Section 4 we will try to generalize this method for nonlinearcases. In Section 5 some examples are chosen to show the ability and high accuracy of the method.
2 Chebyshev polynomials
Definition 1. If t = cosθ (0 <θ< π), the function
is a polynomial of t of degree n (n = 0, 1, 2, ...). Tn is called the Chebyshev polynomial of degree n . When θ increase from 0 to π , t decrease from 1 to -1. Then the interval [-1, 1] is domain of definition of Tn(t). It satisfies the orthogonality condition
Remark 1 (Chebyshev series expansion). Let be g(x) a function on [a, b]. For a given arbitrary natural number M, Chebyshev series expansion of g(x) have the form
Definition 2. Suppose f(x, t) be a continuous function on [-1, 1] × [-1, 1]. For a given natural number N, we set
and ‹ · , ·› denotes the inner product in function space L2([-1, 1] × [-1, 1]).
Remark 2. This paper discusses using Chebyshev polynomials of the first kind to approximate functions on finite interval, that is, on the interval [-1, 1].Practically, other polynomials, which are orthogonal on finite interval, can also be applied for approximating functions. But the partial sums of a first-kindChebyshev expansion of a continuous function in [-1, 1] cover faster than the partial sums of an expansion in any other orthogonal polynomials .
3 Solution of linear two dimensional integral equation
Consider the two dimensional linear Fredholm and Volterra integral equations as follows
where u(x, t) is an unknown scalar-valued function, f(x, t) and k(x, t, y, z) are continuous functions on [-1, 1]2 and [-1, 1]4 respectively. For the case which integration domain is [a, b] × [c, d] , we can use suitable change of variable to obtain these intervals.
At first, we consider two dimensional linear Fredholm integral equations are defined in (4). Function u(x, t) defined over [-1, 1] × [-1, 1] may be represented by Chebyshev series as
If the infinite series in (6) is truncated, then (6) can be written as
where N is any natural number. The method of collocation solves the (4) using the approximation (7) through the equations
for Gauss-Chebyshev-Lobatto as collocation points 
The interested reader can see more detail of collocation method in [3, 4, 11].
Similarly, function k(xr, ts, y, z) can be expressed as truncated Chebyshev series in the following form
for any natural number M. From (3) we have
By using Gauss-Chebyshev-Lobatto integration rule , for a given natural number n we have
where double prime denotes that the first and the last terms are halved. Now, by substituting (7) and (10) into (8) we obtain
But the Chebyshev polynomials are even for even degree and odd for odddegree. Hence,
So, from (13), (14), (15) and (16) we can obtain the system of linear equations,
where is computed by the following relation
Clearly, the obtained system contains (N + 1)2 equations in the same number as unknowns. It can be solved by Newton's iteration method to obtain the value of aij such that i, j = 0 ... N.
For the Volterra case, this method is valid. We just change (16) with the following equation
so that, if let
we will have
when respectively i + p = 1 , | i - p | = 1, i + p is even and i + p is odd.
4 Solution of nonlinear two dimensional integral equation
Consider the two dimensional nonlinear Fredholm and Volterra integral equations as follows
where k(x, t, y, z) is continuous on [-1, 1]4, and f(x, t) and F(u(y, z)) are continuous on [-1, 1]2 . Again, for the case which integration domain is [a, b] × [c, d] , we can use suitable change of variable to obtain this intervals.
Before solving the above equations we exchange them with following equations. The other cases can be approximated in this form using Taylor extension. It reduce the related computation effectively.
where p is a positive integer number and p > 2. Correspondingly the linear case, by using (7) and (24) and considering collocation points we have
Now, we replace (10) into above equation and if we let
This is a system of algebraic equations with (N + 1) unknowns and (N + 1) equations which can be solved by Newton's iteration method to obtain the value of aij such that i, j = 0 ... N.
In Volterra case, we let instead of νpq in (29)
where dij is linear or nonlinear combination of aij. Hence, we can calculate easily νpq and by (17) and (21).
Remark 3. In case F(u(x, t)) is strongly nonlinear, the Taylor series can be used to approximate F(u(x, t)) as a polynomial in u(x, t). Then the above method can be applied easily for general cases (25) and (26).
5 Numerical results
In this section, the illustrate examples are given to show efficiency the method proposed in Sections 3 and 4. All of the computations have been done using the Maple 12 with just 10 digits precision. In this study, our criterion of accuracy is the maximum absolute error in relevant intervals. In the other word, we investigate the value of infinity norm of error functions.
5.1 Linear examples
For the following cases, we let M = 6 and n = 15.
Example 1. Consider the following Fredholm integral equation
with exact solution
u(x, t) = x cos t - t.
By using (18) we obtain approximate solution
Example 2. Consider the following Fredholm integral equation
with exact solution u(x, t) = xe-t + t . By using (18) we have
Example 3. Consider the following Volterra integral equation
and exact solution is u(x, t) = xe-t . By applying (18) and considering (20) we obtain approximate solution as follows
Example 4. Consider the following Volterra integral equation 
with exact solution u(x, t) = x sin t. By applying (18) and considering (20) we obtain approximate solution as follows
In Table 1 we investigate the above examples and shows the maximum absolute error. The numerical results shows high accuracy even for small N.
Also, comparison between estimated absolute errors of Example 4 for N = 8 using presented method and differential transform method  are illustrated in Table 2. The results show more accuracy and smoother error function by described method.
5.2 Nonlinear examples
Example 5. Consider the following Fredholm integral equation
where x, t ∈ [0, 1] and exact solution is u(x, t) = x cos t. If we let M = 1, N = 6 and n = 15, by considering (29) we obtain approximate solution as follows
Also, the value of infinity norm of error function is 2.3 × 10-8.
Example 6. Consider the following Volterra integral equation 
In this case, we let M = 1, N = 2 and n = 15. By considering (30) and solving the obtained system we have
The values of aij lead to u(x, t) = x2 + t2 which is the exact solution.
Analytical solution of the two dimensional integral equations are usually difficult. In many cases, it is required to approximate solutions. In this work, the two dimensional linear and nonlinear integral equations of the second kind aresolved by using Chebyshev polynomials through collocation scheme. However this method only works when F is a power function as (27) and (28), we know other cases can be rewritten easily in this form using Taylor extension. The privilege of the method is simplicity and spectrally accuracy [4, 11]. The illustrative examples confirm the validity and efficiency of the method. This method can be extended for the system including such the equations. Also, development of the method can solve the two dimensional integro-differential equations.
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