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Modified homotopy perturbation method for solving two-dimensional fredholm integral equations.

1. Introduction

The HPM was first introduced by Ji-Huan He (see [2]-[8]). This method has been favorably applied to various types of linear and nonlinear problems.

For example, the authors of [9] has presented an extension of the HPM for solving Fredholm integral equations. In [10] this method has been applied to nonlinear Fredholm integral equations and in [11] to the system of linear Fredholm integral equations. In [12] the HPM has been applied for solving nonlinear (algebraic) equations. The HPM has been applied for solving partial differential equations in [13], the convergence of this method for solving partial differential equations has been investigated in [14].

In [15] the HPM has been developed for solving linear and nonlinear fractional diffusion-wave equations. In [16] a class of hyperchoatic systems has been solved by the multistage homotopy perturbation method. In [17] this method has been applied to fractional BBM Burgers equations. In [18], the HPM has been applied for non-linear system of second-order boundary value problems and in [19] it has been also applied for the coupled Schrodinger KDV equation. In [20] this method has been developed for solving system of Volterra integral equations of the second kind. In [21], the HPM has been also applied to quadratic Riccati differential equation of fractional order.

Subject of presented paper is to apply a modification of the HPM for solving the two-dimensional linear Fredholm integral equations of the second kind of the form:

[phi](x, t) = f(x, t) + [[integral].sup.d.sub.c][[integral].sup.b.sub.a]K(x, t, y, z)[phi](y, z)dydz, x[member of][a, b], t[member of][c, d] (1.1)

where the kernel K(x, t, y, z) and f(x, t) are given continuous functions and [phi](x, t) is solution to be obtained.

Here without lose of generality, we assume that the kernel function has a degenerate form:

K(x, t, y, z) = [n.summation over (i=1)][m.summation over (i=1)][v.sub.i](x, t)[w.sub.i](y, z). (1.2)

Note that non-degenerate differentiable kernels may be converted to the degenerate form, by using Taylor series expansion.

The rest of this paper is organized as follows: In section 2, we give some results on the existence and uniqueness of the solution of 2DLFIE. In section 3, we introduce a modification of HPM. We present some examples and numerical results in section 4 and finally a conclusion is given in section 5.

2. Some Results on the Existence and Uniqueness of Solution

In this section, we give some theoretical results about existence and uniqueness of solution of (1.1). To this end, we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

which is an integral operator and therefore (1.1) can be written in the operator form

[phi] - A[phi] = f (2.2)

Theorem 2.1. Let K :[a, b] x [c, d] x [a, b] x [c, d] [right arrow] R be continuous, then the operator A defined by (2.1) is bounded with the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

Proof. See [22].

Now we recall the following theorem from [1].

Theorem 2.2. Let A is a bounded operator on C([a, b] x [c, d]) with [parallel]A[parallel] < 1 and I denotes the identity operator. Then I - A has a bounded inverse on C([a, b] x [c, d]) that is given by the Neumann series

[(I - A).sup.-1] = [[infinity].summation over (k=0)][A.sup.k]

and satisfies

[parallel][(I - A).sup.-1][parallel] [less than or equal to] 1/1 - [parallel]A[parallel].

Therefore theorem 2.2 ensure that the following condition is a sufficient (not necessary) condition for existence and uniqueness of the solution of (2.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3. Modified Homotopy Perturbation Method (MHPM)

In this section, we apply the modified perturbation method to (1.1) with (1.2). To this end, we define a new convex homotopy perturbation as (see [9])

H(u, p, m) = (1 - p)F(u) + pL(u) + [p.sup.2](1 - p) [[n.summation over (i=1)][m.sub.i][v.sub.i](x, t) = 0 (3.1)

where m = [[m.sub.1], ..., [m.sub.n]] and [m.sub.i], i = 1, ..., n are called accelerating parameters and

F(u) = u(x, t) - f(x, t),

L(u) = u(x, t) - f(x, t) - [[integral].sup.d.sub.c][[integral].sup.b.sub.a] v(x, t)w(y, z)u(y, z)dydz = 0 (3.2)

MHPM uses the homotopy parameter p as an expanding parameter to obtain

u = [u.sub.0] + [pu.sub.1] + [p.sup.2][u.sub.2] + ... (3.3)

when p [right arrow] 1, (3.3) becomes the approximate solution of Eq. (1.1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.4)

Now we consider two cases for solving (1.1).

For the first case, we assume that K(x, t, y, z) = v(x, t)w(y, z), then from (1.1) and (1.2) we obtain

[phi](x, t) = f(x, t) + [[integral].sup.d.sub.c][[integral].sup.b.sub.a]v(x, t)w(y, z)[phi](y, z)dydz, a [less than or equal to] x [is less than or equal to] b, c [less than or equal to] t [less than or equal to] d (3.5)

and from (3.1)

H(u, p, m) = (1 - p)F(u) + pL(u) + [p.sup.2](1 - p) [mv(x, t)] = 0 (3.6)

Substituting from (3.2) into (3.6) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

u(x, t) - f(x, t) - pv(x, t)[[integral].sup.d.sub.c][[integral].sup.b.sub.a]w(y, z)u(y, z)dydz + [p.sup.2]mv(x, t) - [p.sup.3]mv(x, t) (3.7)

by substituting from (3.3) into (3.7) and equating coefficients of the identical powers of p, we obtain

[p.sup.0]: [u.sub.0] - f = 0 [??] [u.sub.0](x, t) = f(x, t)

[p.sup.1]: [u.sub.1](x, t) = v(x, t) [[integral].sup.d.sub.c][[integral].sup.b.sub.a]w(y, z)f(y, z)dydz

[p.sup.2]: [u.sub.2](x, t) = v(x, t) [[integral].sup.d.sub.c][[integral].sup.b.sub.a]w(y, z)[u.sub.1](y, z)dydz - mv(x, t)

[p.sup.3]: [u.sub.3](x, t) = v(x, t) [[integral].sup.d.sub.c][[integral].sup.b.sub.a]w(y, z)[u.sub.1](y, z)dydz + mv(x, t)

and

[u.sub.r+1](x, t) = v(x, t) [[integral].sup.d.sub.c][[integral].sup.b.sub.a]w(y, z)[u.sub.r](y, z)dydz r = 3, 4,....

Now we find m in such a way that u3 = 0, to this end we have

[u.sub.1](x, t) = [alpha]v(x, t) with [alpha] = [[integral].sup.d.sub.c][[integral].sup.b.sub.a] w(y, z)f(y, z)dydz

and so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[u.sub.3](x, t) = [([alpha][beta] - m)[beta] + m] v(x, t)

so that setting ([alpha][beta] - m)[beta] + m = 0, implies

m = [alpha][[beta].sup.2]/[beta] - 1 (3.8)

provided that [beta] [not equal to] = 1.

But [u.sub.3] = 0 implies that [u.sub.4] = [u.sub.5] = ... = 0, therefore u(x, t) = [u.sub.0](x, t) + [u.sub.1](x, t) + [u.sub.2](x, t)

For the second case we consider the general form (1.2) for the kernel of Eq. (1.1). In this case we consider the convex homotopy (3.1) and proceed as the first case to get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

whit

[[beta].sub.ij] = [[integral].sup.d.sub.c][[integral].sup.b.sub.a][w.sub.i](y, z)[v.sub.j](y, z)dydz, i = 1, ..., n, j = 1, ..., n

also

[u.sub.3](x, t) = [n.summation over (i=1)][[n.summation over (j=1)]([n.summation over (k=1)][[alpha].sub.k][[beta].sub.jk] - [m.sub.j])[[beta].sub.ij] + [m.sub.i]][v.sub.i](x, t)

and finally

[u.sub.r+1](x, t) = [n.sumation over (i=1)][v.sub.i](x, t) [[integral].sup.d.sub.c] [[integral].sup.b.sub.a][w.sub.i](y, z)[u.sub.r](y, z)dydz, r = 3, 4, ...

Consequently setting u3 = 0 implies

[m.sub.i] + [n.summation over (j=1)]([n,summation over (k=1)][[alpha].sub.k][[beta].sub.jk] - [m.sub.j])[[beta].sub.ij] = 0, i = 1, ..., n

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.9)

which is a linear system for the unknowns [m.sub.i]: i = 1, ..., n. By solving this system we obtain unknowns [m.sub.i], i = 1, ..., n, hence [u.sub.2](x, t) will be known and we obtain

[phi](x, t) [equivalent] u(x, t) = [u.sub.0](x, t) + [u.sub.1](x, t) + [u.sub.2](x, t).

4. Examples

In this section, we give some examples to clarify accuracy of the presented method. All computations have been done by programming in Maple software.

Example 4.1. ([23]) As first example we consider the following two-dimensional linear fredholm integral equation

[phi](x, t) = [xe.sup.-t] + 4xsint - 7/3 + [[integral].sup.1.sub.-1] [[integral].sup.1.sub.-1](xsint + [ye.sup.z])[phi](y, z)dydz, x, t[member of][-1, 1] (4.1)

with the exact solution [phi](x, t) = [xe.sup.-t] - 1.

By MHPM we obtain the following system for the unknowns [m.sub.1], [m.sub.2]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

that implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and we obtain the approximate solution as

u(x, t) = [xe.sup.-t] - (0.1e - 18)xsint - 1.0000000000000000001

that is indeed the exact solution because the error function is

e(x, t) = [phi](x, t) - u(x, t) = 0.1(xsint + 1) x [10.sup.-18].

To compare, we report the results of [23] in Table 1, including the values of error function in some points.

Example 4.2. We consider the second integral equation of the form (Example.3 of [23])

[phi](x, t) = xcost-[1/6]sin(1)(sin(1) + 3) + [[integral].sup.1.sub.0][[integral].sup.1.sub.0](ysinz + 1)[phi](y, z)dydz, x, t[member of][0, 1] (4.2)

with the exact solution [phi](x, t) = xcost.

Similar to example 1 we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

that implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

therefore

u(x, t) = xcost + 0.20000000000000000000e - 19

with the error function e(x, t)- = 0.2 x [10.sup.-19]. In Table 2 we compare the absolute errors of presented method and the method of [23] in some arbitrary points

Example 4.3. We consider in this example an integral equation of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.3)

with the exact solution [phi](x, t) = xsin(x - t) + t.

By the same way of previous examples we obtain

u(x,t) = xsin(x -1) + 0.99999999999999999940t - 0.19000000000000000000e - 16

with the error e(x, t) = 0.6 x [10.sup.-18]t + 0.19 x [10.sup.-16].

Example 4.4. In this example we consider the two dimensional integral equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.4)

with exact solution u(x, t) = [xe.sup.x-t] + xt. And by MHPM we obtain u(x, t) = [xe.sup.x-t] + xt - (0.49787068367863942979e - 18)x - (0.29872241020718365787e - 18)t

with error

e(x, t) = 0.4979 x [10.sup.-18]x + 0.2987 x [10.sup.-18]t.

5. Conclusion

The modified homotopy perturbation method has been applied for solving the two dimensional linear Fredholm integral equations of the second kind. As examples show, the presented method reduces the computational difficulties of other methods and all the calculations can be done simple. Therefore this method be recommended for solving problems in applied science and engineering.

References

[1] R. Kress, Linear Integral Equations. Springer-Verlag, 1999.

[2] J. H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng., 173, pp. 257-262, 1999.

[3] J. H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, International Journal ofNon-Linear Mechanics, 35(1), pp. 37-43, 2000.

[4] J. H. He, The homotopy perturbation method for nonlinear oscilatots, Appl. Math. Comput., 151(1), pp. 287-292, 2004.

[5] J. H. He, Comparison of homotopy perturbation method and homotopy analysis method, Appl. Math. Comput., 156, pp. 527-539, 2004.

[6] J. H. He, Asymtotology by homotopy perturbation method, Appl. Math. Comput., 156(3), pp. 591-596, 2004.

[7] J. H. He, Homotopy perturbation method for solving boundary problems, Physical Letters, A 350(1-2), pp. 827-833, 2006.

[8] J. H. He. Recent development of the homotopy perturbation method, Topological Methods in Nonlinear Analysis, 31(2), pp. 205-209, 2008.

[9] A. Golbabai, B. Keramati, Modified homotopy perturbation method for solving Fredholm integral equations, Chaos, Solitons and Fractals, 37, pp. 1528-1537, 2008.

[10] M. Javidi, A. Golbabai, Modified homotopy perturbation method for solving nonlinear Fredholm integral equations, Chaos, Solitons and Fractals, 4(3), pp. 1408-1412, 2009.

[11] M. Javidi, Modified homotopy perturbation method for solving system of linear Fredholm integral equations, Math. Comput. Model., in press (doi:10.1016/j.mcm.2009.02.003), 2009.

[12] M. Javidi, A. Golbabai, A third-order Newton type method for nonlinear equations based on modified homotopy perturbation method, Appl. Math. Comput., 191(1), pp. 199-205, 2007.

[13] J. Biazar, H. Ghazvini, Homotopy perturbation method for solving partial differential equations, Computers Math. with applications, 56, pp. 453-458, 2008.

[14] J. Biazar, H. Ghazvini, Convergence of the homotopy perturbation method for partial differential equations, Nonlinear Analysis: Real World Applications, 10(5) (2009) 2633-2640.

[15] H. Jafari, S. Seifi, Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation, Commun. Non. Sci. Num. Simul., 14(5), pp. 20062012, 2009.

[16] Y. Yu, H. Li, Application of the multistage homotopy-perturbation method to solve a class of hyperchaotic systems, Chaos, Solitons and Fractals, In Press, 2009.

[17] L. Song, H. Zhang, Solving the fractional BBM Burgers equation using the homotopy analysis method, Chaos, Solitons and Fractals, 40(4), pp. 1616-1622, 2009.

[18] A. Saadatmandi, M. Dehghan, A. Eftekhari, Application of He's homotopy perturbation method for non-linear system of second-order boundary value problems, Nonlinear Analysis: Real World Applications, 10(3), pp. 1912-1922, 2009.

[19] S. Kugukarslan, Homotopy perturbation method for coupled Schrodinger KdV equation, Nonlinear Analysis: Real World Applications, 10(4), pp. 2264-2271, 2009.

[20] J. Biazar, H. Ghazvini, He's homotopy perturbation method for solving system of Volterra integral equations of the second kind, Chaos, Solitons and Fractals, 39(2), pp. 770-777, 2009.

[21] Z. Odibat, S. Momani, Modified homotopy perturbation method:application to quadratic Riccati differential equation of fractional order, Chaos, Solitons and Fractals, 36(1), pp. 167-174, 2008.

[22] M. Y. Rahimi, S. Shahmorad, F. Talati, A. Tari, An operational method for the Numerical solution of two dimensional linear Fredholm integral equations with an error estimation, Bulletin of Iranian Mathematical Society, In press.

[23] A. Tari, S. Shahmorad, A computational method for solving two dimensional linear Fredholm integral equations of the second kind, ANZIANJ., 49, pp. 543-549, 2008.

A. Tari

Department of Mathematics, Shahed University, Tehran--Iran.

E-mail: tari@shahed.ac.ir, at4932@gmail.com.
Table 1: Numerical results of example 1.

(x, t) Abs.Er.MHPM Abs.Er.[23]

(0.25, 0.25) 0.1062e--18 0.8407e--12
(0.50, 0.50) 0.1240e--18 0.2003e--11
(0.75, 0.75) 0.1511e--18 0.5596e--11
(1,1) 0.1841e--18 0.7111e--10

Table 2: Numerical results of example 3 of [23].

(x, t) Abs.Er.MHPM Abs.Er.[23]

(0.50, 0.50) 0.2000e--19 0.7607e--16
(0.50, 1) 0.2000e--19 0.8804e--14
(1, 0.5) 0.2000e--19 0.1522e--15
(1,1) 0.2000e--19 0.1761e--13
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Author:Tari, A.
Publication:International Journal of Computational and Applied Mathematics
Date:Sep 1, 2010
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