Abstract
We are concerning with two analytical methods; the classical method of successive approximations (Picard method) [14] which consists the construction of a sequence of functions such that the limit of this sequence of functions in the sense of uniform convergence is the solution of a quadratic integral equation, and Adomian method which gives the solution as a series see ([1-6], [12] and [13]). The existence and uniqueness of the solution and the convergence will be discussed for each method. Mathematical subject classification: Primary: 39B82; Secondary: 44B20, 46C05.
quadratic integral equation; Picard method; Adomian method; continuous unique solution; convergence analysis; error analysis
Picard and Adomian methods for quadratic integral equation
El-Sayed A.M.A.I; Hashem H.H.G.II; Ziada E.A.A.II
IFaculty of Science, Alexandria University, Alexandria, Egypt
IIFaculty of Engineering, Mansoura University, Mansoura, Egypt E-mails: amasayed@hotmail.com / hendhghashem@yahoo.com / eng_emanziada@yahoo.com
ABSTRACT
We are concerning with two analytical methods; the classical method of successive approximations (Picard method) [14] which consists the construction of a sequence of functions such that the limit of this sequence of functions in the sense of uniform convergence is the solution of a quadratic integral equation, and Adomian method which gives the solution as a series see ([1-6], [12] and [13]). The existence and uniqueness of the solution and the convergence will be discussed for each method.
Mathematical subject classification: Primary: 39B82; Secondary: 44B20, 46C05.
Key words: quadratic integral equation, Picard method, Adomian method, continuous unique solution, convergence analysis, error analysis.
1 Introduction
Quadratic integral equations (QIEs) are often applicable in the theory of radiative transfer, kinetic theory of gases, in the theory of neutron transport and in the traffic theory. The quadratic integral equations can be very often encountered in many applications.
The quadratic integral equations have been studied in several papers andmonographs (see for examples [7-11] and [16-22]).
The Picard-Lindelof theorem for the initial value problem
was proved in [14] and the solution can be shown as the limit of constructed sequence.
The existence of continuous solution of the nonlinear quadratic integral equation
was proved in [18] by using Tychonov fixed point theorem where f and g satisfy Carathéodory condition.
In this work, we will prove the existence and the uniqueness of continuous solution for (1) by using the principle of contraction mapping. Also, we are concerning with the two methods; Picard method and Adomian method.
2 Main Theorem
Now, equation (1) will be investigated under the assumptions:
is continuous on I where I = [0,1];
are continuous and there exist positive constants M1 and M2 such that |g(t,x)| < M1 and |f(t,x)| < M2 on D;
(iii) f,g satisfy Lipschitz condition with Lipschitz constants L1 and L2 such that,
Let C = C(I) be the space of all real valued functions which are continuous on I.
Define the operator F as
Theorem 1. Let the assumptions (i)-(iii) be satisfied. If h = (L1M2+L2M1) < 1, then the nonlinear quadratic integral equation (1) has a unique positive solution x∈C.
Proof. It is clear that the operator F maps C into C.
Now define a subset S of C as
Then the operator F maps S into S, since for x ∈ S
Moreover it is easy to see that S is a closed subset of C. In order to show thatF is a contraction we compute
Since
3 Method of successive approximations (Picard method)
Applying Picard method to the quadratic integral equation (1), the solution is constructed by the sequence
All the functions xn(t) are continuous functions and xn can be written as a sum of successive differences:
This means that convergence of the sequence xnis equivalent to convergence of the infinite series Σ(xj-xj-1) and the solution will be,
i.e. if the infinite series Σ(xj-xj-1) converges, then the sequence xn(t) will converge to x(t). To prove the uniform convergence of {xn(t)}, we shall consider the associated series
From (2) for n = 1, we get
and
Now, we shall obtain an estimate for xn(t)-xn-1(t), n > 2
using assumptions (ii) and (iii), we get
Putting n = 2, then using (3) we get
Repeating this technique, we obtain the general estimate for the terms of the series:
Since (L1M2+L2M1) < 1, then the uniform convergence of
is proved and so the sequence {xn(t)} is uniformly convergent.
Since f(t,x) and g(t,x) are continuous in x, then
thus, the existence of a solution is proved.
To prove the uniqueness, let y(t) be a continuous solution of (1). Then
and
using assumptions (ii) and (iii), we get
But
and using (4) then we get
Hence
When g(t,x) = 1, then M1 = 1 and L1 = 0 and we obtain the original Picard theorem [14] and [15].
Corollary 1. Let the assumptions of Theorem 1 (with g(t,x) = 1) be satisfied. If L2 < 1, then the integral equation
has a unique continuous solution.
4 Adomian Decomposition Method (ADM)
The Adomian decomposition method (ADM) is a nonnumerical method forsolving a wide variety of functional equations and usually gets the solution in a series form.
Since the beginning of the 1980s, Adomian ([1-6] and [12-13]) has presented and developed a so-called decomposition method for solving algebraic, differential, integro-differential, differential-delay, and partial differential equations. The solution is found as an infinite series which converges rapidly to accurate solutions. The method has many advantages over the classical techniques, mainly, it makes unnecessary the linearization, perturbation and other restrictive methods and assumptions which may change the problem being solved, sometimes seriously. In recent decades, there has been a great deal of interest in the Adomian decomposition method. The method was successfully applied to a large amount of applications in applied sciences. For more details about the method and its application, see ([1-6] and [12-13]).
In this section, we shall study Adomian decomposition method (ADM) for the quadratic integral equation (1).
The solution algorithm of the quadratic integral equation (1) using ADM is,
where Aiand Bi are Adomian polynomials of the nonlinear terms g(t,x) and f(s,x) respectively, which have the form
and the solution will be,
4.1 Convergence analysis
Theorem 2. Let the solution of the QIE (1) exists. If |x1(t)| < l,l is a positive constant, then the series solution (9) of the QIE (1) using ADM converges.
Proof. Define the sequence {Sp} such that,
is the sequence of partial sums from the series solution , and we have
Let Sp and Sq be two arbitrary partial sums with p > q. Now, we are going to prove that {Sp} is a Cauchy sequence in this Banach space E.
Let p = q+1 then,
From the triangle inequality we have,
Now 0 < h < 1, and p > q implies that (1-hp-q) < 1. Consequently,
converges.
□
4.2 Error analysis
Theorem 3. The maximum absolute truncation error of the series solution (9) to the problem (1) is estimated to be,
Proof. From Theorem 2 we have,
so, the maximum absolute truncation error in the interval I is,
5 Numerical Examples
In this section, we shall study some numerical examples and applying Picard and ADM methods, then comparing the results.
Example 1. Consider the following nonlinear QIE,
and has the exact solution x(t) = t2.
Applying Picard method to equation (10), we get
and the solution will be,
Applying ADM to equation (10), we get
where Ai are Adomian polynomials of the nonlinear term x2, and the solution will be,
Table 1 shows a comparison between the absolute error of Picard (when n = 5) and ADM solutions (when q = 5).
Example 2. Consider the following nonlinear QIE,
and has the exact solution x(t) = t3.
Applying Picard method to equation (11), we get
and the solution will be,
Applying ADM to equation (11), we get
where Ai and Bi are Adomian polynomials of the nonlinear terms x2 and x3 respectively, and the solution will be,
Table 2 shows a comparison between the absolute error of Picard (when n = 3) and ADM solutions (when q = 3).
Example 3. Consider the following nonlinear QIE [9],
Applying Picard method to equation (12), we get
and the solution will be,
Applying ADM to equation (12), we get
where Ai are Adomian polynomials of the nonlinear term cos ()and the solution will be,
Table 3 shows a caparison between ADM and Picard solutions.
Example 4. Consider the following nonlinear QIE [8],
Applying Picard method to equation (13), we get
and the solution will be,
Applying ADM to equation (13), we get
where Ai are Adomian polynomials of the nonlinear term ln(1+s|x(s)|) and the solution will be,
Table 4 shows a caparison between ADM and Picard solutions.
6 Conclusion
We used two analytical methods to solve QIEs; Picard method and ADM, from the results in the tables we see that Picard method gives more accurate solution than ADM.
Received: 07/VI/09
Accepted: 02/XII/09.
#CAM-118/09
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Publication Dates
-
Publication in this collection
22 Nov 2010 -
Date of issue
2010
History
-
Accepted
02 Dec 2009 -
Received
07 June 2009