Medical, Pharma, Engineering, Science, Technology and Business

Department of Mathematics, Dhaka University, Dhaka, Bangladesh

- *Corresponding Author:
- Mohedul Hasan Md

Department of Mathematics

Dhaka University

Dhaka, Bangladesh

**Tel:**880-2-8628758

**E-mail:**[email protected]; [email protected]

**Received date:** February 24, 2016; **Accepted date:** March 17, 2017; **Published date:** March 24, 2017

**Citation: **Hasan M, Matin A (2017) Solving Nonlinear Integral Equations by using Adomian Decomposition Method. J Appl Computat Math 6:346. doi: 10.4172/2168-9679.1000346

**Copyright:** © 2017 Hasan M, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

In this paper, we propose a numerical method to solve the nonlinear integral equation of the second kind. We intend to approximate the solution of this equation by Adomian decomposition method using He’s polynomials. Several examples are given at the end of this paper with the exact solution is known. Also the error is estimated.

Nonlinear Fredholm integral equation; Adomian decomposition method; Adomian polynomials; Approximate solutions; OriginPro 8 and MATHEMATICA v9 softwares

Several scientific and engineering applications are usually described by integral equations. Integral equations arise in the potential theory more than any other field. Integral equations arise also in diffraction problems, conformal mapping, water waves, scattering in quantum mechanics, and population growth model. The electrostatic, electromagnetic scattering problems and propagation of acoustical and elastically waves are scientific fields where integral equations appear [1]. The Fredholm integral equation is of widespread use in many realms of engineering and applied mathematics [2].

Consider the general form non-linear Fredholm integral equation of the second kind

where *y(x)* is the unknown solution, a and b are real constants. The kernel *K(x,t)* and *f(x) *are known smooth functions on *R ^{2}* and

Consider the following non-linear Fredholm integral equation of the second kind of the form

(1)

We assume *G(y(t))* is a nonlinear function of *y(x)*. That means that the nonlinear Fredholm integral equation (1) contains the nonlinear function presented by *G(y(t))*. Assume that the solution of equation (1) can be written in the form

(2)

The comparisons of like powers of *p* give solutions of various orders and the best approximation is

The nonlinear term *G(y(t))* can be expressed in Adomian polynomials [3-5] as

(3)

where *H _{k}*’s are the so called Adomian polynomials which can be calculated by using the formula

(4)

Using (2), (3) and (4) into (1), we have

(5)

Equating the term with identical power of *p* in equation (5),

and so on.

and in general form we have

(6)

Using the recursive scheme (6), the *n*-term approximation series solution can be obtained as follows:

(7)

In this section, we will apply the Adomian decomposition method to compute a numerical solution for non-linear integral equation of the Fredholm type. Then we will compare between the results which we obtain by the numerical solution technique and the results of the exact solution. To illustrate this, we consider the following example:

**Example 1**

Consider the following nonlinear Fredholm integral equation of the second kind

(8)

where the exact solution of the equation is *y(x)=x*. In the following, we will compute Adomian polynomials for the nonlinear terms *y ^{2}(t)* that arises in nonlinear integral equation.

For *k*=0, equation (4) becomes

The Adomian polynomials for *G(y)=y ^{2}* are given by

By using the MATHEMATICA software, the next few terms, we have

and so on.

Applying the technique as stated above in equation (6), we have

In a similar manner, we stop the iteration at the tenth step. Therefore we can write

The table under shows the approximate solutions obtained by applying the Adomian Decomposition method giving to the value of *x*, which is in the interval [0-1] (**Table 1** and **Figure 1**).

Nodes (x) |
Exact solutions | Approximate solutions | Absolute error |
---|---|---|---|

0 | 0 | 0 | 0 |

0.10 | 0.100000 | 0.0999947 | 0.0000053 |

0.20 | 0.200000 | 0.1999890 | 0.0000110 |

0.30 | 0.300000 | 0.2999840 | 0.0000160 |

0.40 | 0.400000 | 0.3999790 | 0.0000210 |

0.50 | 0.500000 | 0.4999740 | 0.0000260 |

0.60 | 0.600000 | 0.5999680 | 0.0000320 |

0.70 | 0.700000 | 0.6999630 | 0.0000370 |

0.80 | 0.800000 | 0.7999580 | 0.0000420 |

0.90 | 0.900000 | 0.8999520 | 0.0000480 |

1.0 | 1.000000 | 0.9999470 | 0.0000530 |

**Table 1:** Numerical and exact solutions to the integral equation (8).

**Example 2**

Consider the following nonlinear Fredholm integral equation of the second kind

(9)

Applying above procedure, we have

In a similar manner, we stop the iteration at the tenth step. Therefore we can write

The exact solution of the equation is *3+x*. The table below shows the approximate solutions obtained by applying the Adomian decomposition method according to the value of *x*, which is confined between zero and one. We compared these results with the results which were obtained by the exact solution (**Table 2** and **Figure 2**).

Nodes (x) |
Exact solutions | Approximate solutions | Absolute error |
---|---|---|---|

0 | 3 | 3 | 0 |

0.10 | 3.100000 | 3.10349 | 0.00349 |

0.20 | 3.200000 | 3.20699 | 0.00699 |

0.30 | 3.300000 | 3.31048 | 0.01048 |

0.40 | 3.400000 | 3.41397 | 0.01397 |

0.50 | 3.500000 | 3.51747 | 0.01747 |

0.60 | 3.600000 | 3.62096 | 0.02096 |

0.70 | 3.700000 | 3.72445 | 0.02445 |

0.80 | 3.800000 | 3.82794 | 0.02794 |

0.90 | 3.900000 | 3.93144 | 0.03144 |

1.0 | 4.000000 | 4.03493 | 0.03493 |

**Table 2:** Numerical and exact solutions to the integral equation (9).

**Example 3**

Consider the following nonlinear Fredholm integral equation

(10)

The exact solution of the equation (10) is . In the following, we will calculate Adomian polynomials for the nonlinear terms *y ^{3}(t)* that arises in nonlinear integral equation.

For *k*=0, equation (4) becomes

The Adomian polynomials for *G(y)=y ^{3}* are given by

By using the MATHEMATICA v9 software, the next few terms, we have

and so on.

Applying the procedure as stated above in equation (6), we have

In a similar manner, we stop the iteration at the ninth step. Therefore we can write

The table under shows the approximate solutions obtained by applying the Adomian decomposition method according to the value of *x*, which is in the interval [0-1] (**Table 3** and **Figure 3**).

Nodes | Exact values | Approximate values | Absolute error |
---|---|---|---|

0.00 | 0.07542668890493687 | 0.07542668888687896 | 1.8057902×10^{-11} |

0.05 | 0.23093252624133365 | 0.23093252622349808 | 1.7835566×10^{-11} |

0.10 | 0.38075203836055493 | 0.3807520383433809 | 1.7174039×10^{-11} |

0.15 | 0.5211961716517719 | 0.5211961716356822 | 1.6089685×10^{-11} |

0.20 | 0.6488067254459994 | 0.6488067254313902 | 1.4609202×10^{-11} |

0.25 | 0.7604415043936764 | 0.7604415043809076 | 1.2768786×10^{-11} |

0.30 | 0.8533516897425216 | 0.8533516897319074 | 1.0614176×10^{-11} |

0.35 | 0.9252495243780194 | 0.9252495243698213 | 8.198108×10^{-12 } |

0.40 | 0.9743646449962113 | 0.9743646449906311 | 5.580202×10^{-12} |

0.45 | 0.9994876743237375 | 0.9994876743209126 | 2.824962×10^{-12} |

0.50 | 1 | 1 | 0 |

0.55 | 0.9758890068665379 | 0.9758890068693629 | 2.824962×10^{-12} |

0.60 | 0.9277483875940958 | 0.927748387599676 | 5.580202×10^{-12} |

0.65 | 0.8567635239987162 | 0.8567635240069144 | 8.198108×10^{-12} |

0.70 | 0.7646822990073733 | 0.7646822990179875 | 1.0614176×10^{-11} |

0.75 | 0.6537720579794186 | 0.6537720579921874 | 1.2768786×10^{-11} |

0.80 | 0.526763779138947 | 0.5267637791535562 | 1.4609202×10^{-11} |

0.85 | 0.38678482782732176 | 0.38678482784341145 | 1.6089685×10^{-11} |

0.90 | 0.23728195038933994 | 0.237281950406514 | 1.7174067×10^{-11} |

0.95 | 0.08193640383912822 | 0.08193640385696378 | 1.7835566×10^{-11} |

1.00 | -0.07542668890493687 | -0.07542668888687896 | 1.8057902×10^{-11} |

**Table 3:** Numerical and exact solutions to the integral equation (10).

This paper presents a technique to find the result of a nonlinear Fredholm integral equation by Adomian decomposition method (ADM). The estimated solutions obtained by the ADM are compared with exact solutions. It can be concluded that the ADM is effective and accuracy of the numerical results demonstrations that the proposed method is well suited for the solution of such kind problems.

- Wazwaz AM (2015) A First Course in Integral Equations. World Scientific.
- Jerri AA (1985) Introduction to Integral Equations with Applications. Marcel Dekker Inc, New York.
- Abbasbandy S, Shivanian E (2011) A new Analytical Technique to Solve Fredholm’s Integral Equations. Numerical Algorithms 56: 27-43.
- Ghorbani A (2009) Beyond Adomian polynomials: He polynomials. Chaos, Solitons and Fractals 39: 1486-1492.
- Ghorbani A, Saberi-Nadjafi J (2007) He's homotopy perturbation method for calculating adomian polynomials. International Journal of Nonlinear Sciences and Numerical Simulation 8: 229-232.

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