Numerical Approximation for Integral Equations

A numerical algorithm, based on a decomposition technique, is presented for solving a class of nonlinear integral equations. The scheme is shown to be highly accurate, and only few terms are required to obtain accurate computable solutions. 1. Introduction. Adomian polynomial algorithm has been extensively used to solve linear and nonlinear problems arising in many interesting applications (see, e.g., [1, 2, 4, 5]). The algorithm (a decomposition method) assumes a series solution for the unknown quantity. It has been shown [3] that the series converges fast, and with only few terms, this series approximates the exact solution with a fairly reasonable error. In this note, we will adapt the algorithm and a modification version of the algorithm due to Wazwaz [7] to the solution of the nonlinear Volterra-Fredholm integral equations arising in the modeling of many applications [8]:


Introduction.
Adomian polynomial algorithm has been extensively used to solve linear and nonlinear problems arising in many interesting applications (see, e.g., [1,2,4,5]).The algorithm (a decomposition method) assumes a series solution for the unknown quantity.It has been shown [3] that the series converges fast, and with only few terms, this series approximates the exact solution with a fairly reasonable error.In this note, we will adapt the algorithm and a modification version of the algorithm due to Wazwaz [7] to the solution of the nonlinear Volterra-Fredholm integral equations arising in the modeling of many applications [8]: and analyze the solution.In (1.1), where K 1 (x, t) and K 2 (x, t) are referred to as the kernel, g 1 and g 2 are nonlinear functions of y, and f (x) a given function, g, K, and f are known functions, and λ 1 and λ 2 are parameters.The balance of this note is as follows.In Section 2, we describe the general algorithm as it applies to the solution of integral equations of the form (1.1).In Section 3, we adapt the algorithm to some problems.

Analysis.
In this section, we first describe the algorithm of the decomposition method as it applies to a general nonlinear equation of the form where N is a nonlinear operator on a Hilbert space H and f is a known element of H.We assume that for a given f , a unique solution u of (2.2) exists.
The standard decomposition method assumes a series solution for u given by and the nonlinear operator N can be decomposed into where the A n 's are Adomian's polynomials of y 0 ,...,y n given by If the series in (2.6) is convergent, then (2.6) holds upon setting Thus, one can recursively determine every term of the series ∞ n=0 y n .The convergence of this series has been established (see [2]).The two hypotheses necessary for proving convergence of the decomposition method as given in [2] are as follows.These hypotheses, for proving convergence, are generally satisfied in physical problems.
The modified Adomian method [7] may be roughly described as a reassignment of the initial approximants y 0 and y 1 .In particular, if f is split into two functions, say, f = f 1 + f 2 , then we may rewrite (2.7) as The choice of how to assign y 0 and y 1 is experimental, yet it leads to less computation and does accelerate the convergence.
We now describe the application of the decomposition method to an integral equation of the form (1.1).For the sake of simplicity, we will present the method for integral equations of the form (2.9) The adaption for the method to (1.1) is immediate.Assuming that g(y) is analytic (and thus satisfying Condition 2.2), we can write where A k are the specially generated Adomian polynomials which can be constructed by the following procedures.Assume that the Taylor expansion of g(y) around y 0 exists and is determined by g(y) = g y 0 + g (1) y 0 y − y 0 + 1 2! g (2) Substituting the difference y − y 0 from (2.3) into (2.11),we get g(y) = g y 0 + g (1) After expanding, this results in g(y) = g y 0 + g (1) Adomian polynomials are obtained by reordering and rearranging the terms given in (2.13).Indeed, to determine the Adomian polynomial, one needs to determine the order of each term in (2.13) which actually depends on both the subscripts and the exponents of the y n 's.To be more specific, we define the order of the component y m k to be mk, and y m k y n j to be mk + nj.Then the Adomian's polynomial A 0 depends upon y 0 with order 0, A 1 depends upon y 0 and y 1 with order 1, and so forth.Therefore, rearranging the terms in the expansion equation (2.13) according to the order, we will have A n as follows: (1) y 0 , A 2 = y 2 g (1) y 0 + y 2 1 2! g (2) y 0 , A 3 = y 3 g (1) y 0 + y 1 y 2 g (2) y 0 + y 3 1 3! g (3) If the series is convergent, then we can determine each term of the series ∞ n=0 y n recursively: The algorithm in (2.16) determines the y i 's and hence the solution y can be determined by (2.3).The decomposition method can be applied to solve problems in higher dimensions (see [2]).We will also apply the modified decomposition by writing f = f 1 + f 2 with appropriate choice for y 0 and y 1 .3. The solution of integral equations.In this section, we apply the algorithms described in Section 2 to some problems of integral equations.Most of the problems discussed were solved using Taylor-type method in [8].The decomposition method is an alternative method for solving these equations.Whenever appropriate, we will note the comparison.y(x) is approximated by using only five terms of decomposition polynomials: The exact solution of the integral equation is y(x) = x 2 − x.Comparing the approximate solution from the decomposition method with the exact solution of the integral equation at x = 0, 0.2, 0.4, 0.6, 0.8, and 1.0, we find the errors displayed in Table 3 Let y 0 (x) = e x , then   A comparison of the approximate solution from the decomposition method with the exact solution y(x) = e x of the integral equation at x = 0, 0.2, 0.4, 0.6, 0.8, and 1.0 yields the errors displayed in Table 3.2.
We can see also from Figure 3.2 that the approximation is very good.The solid curve, which represents the approximate solution almost coincides with the analytic solution (rhombuses curve).Problem 3.3.We apply the modified decomposition method  (3.9) A comparison of the approximate solution from the decomposition method with the exact solution y(x) = sin x of the integral equation at x = 0, 0.2, 0.4, 0.6, 0.8, and 1.0 yields the errors displayed in Table 3. 3.
We can see also from Figure 3.3 that the approximation is very good.
In this note, we presented decomposition method as an alternate method to other approximate methods to integral equations.In the above problems, the method yields accurate computable solutions with good approximation using only few terms.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning
y 0 , [1,2,6]etermined by(2.14),onecanrecurrentlydetermine the terms y n of the series from (2.7), and hence the solution y.It is easy to verify that when N(y) is g(y), formula (2.5) yields the same result as in(2.14).For a detailed description of the decomposition method, we refer the reader to[1,2,6].Substituting (2.3) and (2.10) into (1.1),we have

Table 3 . 1
.1.Figure3.1 shows the approximate and analytic solutions of the equation.The solid curve represents the approximate solution, while the rhombuses represent the analytic solution.It is obvious from the figure that the approximation is very good, although we used only four terms of decomposition polynomials.

Table 3
X Figure 3.3.Decomposition method versus analytic solution for y

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation