Life Science Journal 2013;10(5s) … · 2013-03-24 · HyperSingular Integral Equations and their...

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Modified Adomian decomposition method for solving a class of hypersingular integral equations of first kind

Y. Mahmoudi, M. Baghmisheh

Department of Mathematics, Tabriz branch, Islamic Azad University, Tabriz, Iran. Email: [email protected]

Abstract: In this paper, a new algorithm based on modified Adomian decomposition method is employed to obtain analytical solution of a class of hypersingular integral equations of the first kind. This method avoids the complex

function-theoretic, long computations of collocation polynomial-based methods and produces the exact solution. [Y. Mahmoudi, M. Baghmisheh. Modified Adomian decomposition method for solving a class of hypersingular integral equations of first kind. Life Sci J 2013;10(5s):292-297] (ISSN:1097-8135). http://www.lifesciencesite.com. 53

keywords: Hypersingular integral equations; Modified Adomian decomposition method. 1 Introduction

Hypersingular integral equation is considered as an important tool in applied Mathematics as it finds application in solving a large class of mixed boundary value problems arising in mathematical physics. Particularly, the crack problems in fracture mechanics or water wave scattering problems involving barriers, diffraction of electromagnetic waves and aerodynamics problems ([2,9,10,13]) could be reduced to hypersingular integral equations in single or disjoint multiple intervals.

A simple approximation method for solving a general hypersingular integral equation of the first kind where the kernel consists a hypersingular part and a regular part is introduced and developed in [12]. A method based on polynomial approximation

is used in [11] to produce the approximate solution of a class of singular integral equations of the second kind. Dutta and Banerjea ([6]) have solved a hypersingular integral equation in two intervals by using the solution of Cauchy type singular integral equations in two disjoint intervals. Gori et.al. ([7]) have constructed a quadrature rule based on the use of suitable refinable quasi-interpolatory operators, for the numerical evaluation of Hadamard finite-part integrals. Chen and Zhou ([3]) have developed an efficient method for solving hypersingular integral equation of the fist kind in reproducing kernel space in order to eliminate the singularity of the equation. In the present paper, we consider the following hypersingular integral equation of the second kind.

(1)

where the unknown function has square-root zero at the end-point, that is, with smooth. is bounded and belongs to , is a square-integrable function of and , and is smooth. The first integral in equation (1) is referred to as Hadamard finite part ([13])

(2)

A simple approximate method for solving (1), with its kernel consisting of a hypersingular part and a regular part is developed using second kind Chebyshev polynomials method by Mandal and Bera in [12]. A simple and efficient method for solving (1) in reproducing kernel spaces is developed by Chen and Zhou [3]. In this paper we use weighted modified Adomian decomposition method (WMADM) to obtain an analytical solution for equation (1). We apply the new method for the case that be a polynomial. 2 Adomian Decomposition Method

The decomposition procedure of Adomian was first proposed by the American mathematician, G. Adomian (1923-1996) and has been applied already to a wide class of stochastic and deterministic problems in science and engineering. It is based on the search for a solution in the form of a series and on decomposing the nonlinear operator into a series in which the terms are calculated recursively using Adomian polynomials [1].



In this method, the nonlinear function in the equation is decomposed into terms of special polynomials called Adomian's polynomials and then the terms of the solution which is regarded as a series, are determined recurrently. Consider the differential equation

(3) where represents a general nonlinear ordinary differential operator involving both linear and nonlinear parts and

is a given function. Equation (3) can be written as

(4) where is an easily invertible operator, which is usually taken as the highest-ordered derivative, is the reminder of the linear operator, and is the nonlinear term in . Applying the inverse operator to (4) yields

(5) and therefore

(6) where is the integration constant and satisfies . Based on the ADM, the solution of Eq. (3) is regarded as

(7) and the nonlinear term is decomposed as follows

(8) where the components are Adomian's polynomials which are calculated by the formula

(9)

Substituting (8) and (7) in (6) results

(10) Now according to the decomposition procedure of Adomian, we can obtain the components s by the following recurrent relation

(11)

The -term approximation of the solution is defined as and . As we know, the more terms added to the approximate solution, the more accurate it would be.

Convergence of Adomian decomposition scheme was established by many authors using fixed point theorem, see for example [4,5,8].

For inhomogeneous equations a simple strategy is used to increas the convergent rate. In performing the ADM, we can expand and then we use the following substitution in (10) to get the exact solution

(12)

where is computed with the initial conditions of the problem. The rate of convergence depends on choosing the functions and . Usually they have chosen in such a way that . This method is refereed as modified Adomian decomposition method (MADM).

3 Method of Solution In equation (1) it is assumed that

(13) where is a smooth function (see [13]). After substituting (13) in (1) we have

(14) Note that with Liebnites formula we have

Let us start with the following definition

(15)

It was proven in [12] that



(16) For example some first s are as follows

(17)

Let us rewrite (15) to the form

(18) For applying ADM to (15) suppose

(19) Substituting (19) to (15)

(20)

Now according to the decomposition procedure of ADM we can obtain the components as follows

(21)

In performing the MADM, we expand and obtain the components as follows

(22)

The rate of convergence of MADM depends on how to choose the functions and . Usually, they have been chosen in such a way that . The main problem in performing MADM is to choose and such that the rate of convergence to be increased. We apply the new method for the case that be a polynomial. We introduce a weighted MADM method which helps us in choosing and properly.

4 Illustrations Example 1: We assume the hypersingular integral equation (1) as follows



(23)

Applying in (23) we get

Here is a polynomial of degree two, then must be a polynomial of two. To start MADM we suppose

and as follows

(24)

Now applying MADM (22) to (24) we get

(25)

Note that for the first integral in we use (17), where as the second one is straightforward for evaluation. In the MADM, and should be chosen such that . Putting leads us to the following system of equations.

(26)

Where and , is the solution, that is (27)

and finally the solution of the integral equation (23) is given as

(28) which is the exact solution of the integral equation. Example 2: Another hypersingular integral equation is given by

(29)

Applying in (30) we get

(30) Here is a polynomial of degree three, then must be a polynomial of three. We suppose and as follows



(31)


(32)

Putting leads us to the following system of equations.

(33)

Where and , is the solution, that is (34)

and finally the solution of the integral equation ((29)) is given as

(35) which is the exact solution of the integral equation. Example 3: We assume the hypersingular integral equation ((1)) as follows

(37)

Applying in (Error! Reference source not found.) we get

(36) Here is a polynomial of degree two, then must be a polynomial of two. To start MADM we suppose

and as follows

(37)


(38)

Putting we get and , that is

(39)

and finally the solution of the integral equation (Error! Reference source not found.) is given as

(40)

which is the exact solution of the integral equation. 5 Conclusion

In this paper, a new scheme of modified Adomian decomposition method is introduced and used to solve a class of hypersingular integral equations of the second kind analytically. Unlike existing analytical and numerical schemes for solving the hypersingular integral equations, the proposed scheme is easy in use and avoids the complex

function-theoretic of analytical method and long computations of numerical methods. Acknowledgments The author would like to thank Tabriz Branch, Islamic Azad University for the financial support of this research, which is based on a research project contract.



Corresponding Author: Y. Mahmoudi Department of Mathematics, Tabriz branch, Islamic Azad University, Tabriz, Iran Email: [email protected] References

[1] Adomian G., Solving frontier problems of physics: the decomposition method, Kluwer Academic Publisher, Dordrecht, 1994.

[2] Chan Y., Fannjiang A.C., Paulino G.H., Integral equations with hyper singular kernels-theory and applications to fracture mechanics, J. Eng. Sci. 41 (2003) 683-720.

[3] Chen Z., Zhou Y.F., A new method for solving hypersingular integral equations of the first kind, Appl. Math. Lett. 24 (2011) 636-421.

[4] Cherruault Y., Convrgence of Adomian's method, Kybernetes, 18 (1989) 31-38.

[5] Chrysos M., Sanchez F., Cherruault Y., Improvement of convergence of Adomian's method using Padé approximants, Kybernetes, 31 (2002) 884-895.

[6] Dutta B., Banerjea S., Solution of a hypersingular integral equation in two disjoint

intervals, Appl. Math. Lett. 22 (2009) 1281-1285 .

[7] Gori L., Pellegrino E., Santi E., Numerical evaluation of certain hypersingular integrals using refinable operators, Math. Comput. Simul. 82 (2011) 132-143.

[8] Himoun N., Abbaoui K., Cherruault Y., New results of convergence of Adomian's method, Kybernetes, 10 (1999) 423-429.

[9] Kaya A.C., Erdogom F., On the solution of integral equation with strongly singular kernels, Quart. Appl. Math. 55 (1) (1987) 105-122.

[10] Lifanov I.K., Poltavskii L.N., Vainikko G.M., HyperSingular Integral Equations and their Applications, Chapman and Hall/ CRC, 2004.

[11] Mandal B.N., Bera G.H., Approximate solution for a class of singular integral equations of second kind, J. Comput. Appl. Math. 206 (2007) 189-195.

[12] Mandal B.N., Bera G.H., Approximate solution for a class of hypersingular integral equations, Appl. Math. Lett. 19 (2006) 1286-1290.

[13] Mandal B.N., Chakrabarti A., Applied singular integral equation, Science Publishers, 2011.

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