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In this paper, we will use the successive approximation method for solving Fredholm integral equation of the second kind using Maple18. By means of this method, an algorithm is successfully established for solving the non-linear Fredholm integral equation of the second kind. Finally, several examples are presented to illustrate the application of the algorithm and results appear that this method is very effective and convenient to solve these equations.

The current research intends to the successive approximation method for solving nonlinear Fredholm integral equation of the second kind using Maple18.

Homotopy perturbation technique in [

Different types of analytical methods and numerical methods were used to solve the problem [

The main objective of this work is to use the successive approximations method in solving the nonlinear Fredholm integral equation of the second kind using Maple18.

The paper is arranged as follows: In Section 2, the successive approximations method. In Section 3, numerical examples are also considered to show the ability of the proposed method, and the conclusion is drawn in Section 4.

The nonlinear Fredholm integral equation of the second kind

u ( x ) = f ( x ) + λ ∫ a b K ( x , t ) F ( u ( t ) ) d t (1)

where u ( x ) is the unknown function to be determined, K ( x , t ) is the kernel, F ( u ( t ) ) is a nonlinear function of u ( t ) , and λ is a parameter. u 0 ( x ) = any selective real valued function,

u n + 1 ( x ) = f ( x ) + λ ∫ a b K ( x , t ) u n ( t ) d t , n ≥ 0. (2)

The question of convergence of u n ( x ) is justified by noting the following theorem

Theorem 1 see [

In this section, we solve some examples, and we can compare the numerical results with the exact solution.

Example 1. Consider the nonlinear Fredholm integral equation of the second kind

u ( x ) = cos ( x ) − π 2 48 + 1 12 ∫ 0 π t u 2 ( t ) d t , (3)

with the exact solution u ( x ) = cos ( x ) .

Example 2. Consider the nonlinear Fredholm integral equation of the second kind

u ( x ) = ln x + 143 144 + 1 36 ∫ 0 1 t u 2 ( t ) d t , (4)

with the exact solution u ( x ) = x + ln x .

x | E x a c t 1 = cos ( x ) | u = cos ( x ) − π 2 48 + 0.2049572390 | E r r o r = | E x a c t 1 − u | |
---|---|---|---|

0.1 | 0.9950042 | 0.9943446 | 0.0006595 |

0.2 | 0.9800666 | 0.9794071 | 0.0006595 |

0.3 | 0.9553365 | 0.9546770 | 0.0006595 |

0.4 | 0.9210610 | 0.9204015 | 0.0006595 |

0.5 | 0.8775826 | 0.8769230 | 0.0006595 |

0.6 | 0.8253356 | 0.8246761 | 0.0006595 |

0.7 | 0.7648422 | 0.7641827 | 0.0006595 |

0.8 | 0.6967067 | 0.6960472 | 0.0006595 |

0.9 | 0.6216100 | 0.6209504 | 0.0006595 |

1.0 | 0.5403023 | 0.5396428 | 0.0006595 |

E r r o r = | E x a c t 2 − u | | u = x + ln x + 1.34767 10000000 | E x a c t 2 = x + ln x | x |
---|---|---|---|

0.0000001 | −2.2025850 | −2.2025851 | 0.1 |

0.0000001 | −1.4094378 | −1.4094379 | 0.2 |

0.0000001 | −0.9039727 | −0.9039728 | 0.3 |

0.0000001 | −0.5162906 | −0.5162907 | 0.4 |

0.0000001 | −0.1931470 | −0.1931472 | 0.5 |

0.0000001 | 0.0891745 | 0.0891744 | 0.6 |

0.0000001 | 0.3433252 | 0.3433251 | 0.7 |

0.0000001 | 0.5768566 | 0.5768564 | 0.8 |

0.0000001 | 0.7946396 | 0.7946395 | 0.9 |

0.0000001 | 1.0000001 | 1.0000000 | 1.0 |

Example 3. Consider the nonlinear Fredholm integral equation of the second kind

u ( x ) = x e x − 1 288 ( 3 + e 2 ) x + 1 36 ∫ 0 1 x t u 2 ( t ) d t , (5)

with the exact solution u ( x ) = x e x .

Example 4. Consider the nonlinear Fredholm integral equation of the second kind

u ( x ) = e x + 1 144 ( 127 − e 2 ) + 1 36 ∫ 0 1 t ( u + u 2 ( t ) ) d t , (6)

with the exact solution u ( x ) = 1 + e x .

E r r o r = | E x a c t 3 − u | | u = x e x − ( 1 96 + 1 288 e x ) x + 0.03519196200 x | E x a c t 3 = x e x | x |
---|---|---|---|

0.0000881 | 0.1104290 | 0.1105171 | 0.1 |

0.0001762 | 0.2441043 | 0.2442806 | 0.2 |

0.0002643 | 0.4046933 | 0.4049576 | 0.3 |

0.0003525 | 0.5963774 | 0.5967299 | 0.4 |

0.0004406 | 0.8239201 | 0.8243606 | 0.5 |

0.0005287 | 1.0927426 | 1.0932713 | 0.6 |

0.0006168 | 1.4090101 | 1.4096269 | 0.7 |

0.0007049 | 1.7797278 | 1.7804327 | 0.8 |

0.0007930 | 2.2128498 | 2.2136428 | 0.9 |

0.0008811 | 2.7174007 | 2.7182818 | 1.0 |

E r r o r = | E x a c t 4 − u | | u = e x − 1 144 e 2 + 1.051185675 | E x a c t 4 = 1 + e x | x |
---|---|---|---|

0.0001272 | 2.1050437 | 2.1051709 | 0.1 |

0.0001272 | 2.2212755 | 2.2214028 | 0.2 |

0.0001272 | 2.3497316 | 2.3498588 | 0.3 |

0.0001272 | 2.4916975 | 2.4918247 | 0.4 |

0.0001272 | 2.6485941 | 2.6487213 | 0.5 |

0.0001272 | 2.8219916 | 2.8221188 | 0.6 |

0.0001272 | 3.0136255 | 3.0137527 | 0.7 |

0.0001272 | 3.2254137 | 3.2255409 | 0.8 |

0.0001272 | 3.4594759 | 3.4596031 | 0.9 |

0.0001272 | 3.7181546 | 3.7182818 | 1.0 |

Example 5. Consider the nonlinear Fredholm integral equation of the second kind

u ( x ) = sin ( x ) − π 2 64 + 1 48 ∫ 0 π t ( 1 + u 2 ( t ) ) d t , (7)

with the exact solution u ( x ) = sin ( x ) .

Example 6. Consider the nonlinear Fredholm integral equation of the second kind

u ( x ) = sin ( x ) + 1 − π 16 ( 1 + 5 π 12 ) + 1 48 ∫ 0 π t ( u + u 2 ( t ) ) d t (8)

E r r o r = | E x a c t 5 − u | | u = sin ( x ) − π 2 64 + 0.1543332664 | E x a c t 5 = sin ( x ) | x |
---|---|---|---|

0.0001207 | 0.0999541 | 0.0998334 | 0.1 |

0.0001207 | 0.1987900 | 0.1986693 | 0.2 |

0.0001207 | 0.2956409 | 0.2955202 | 0.3 |

0.0001207 | 0.3895390 | 0.3894183 | 0.4 |

0.0001207 | 0.4795462 | 0.4794255 | 0.5 |

0.0001207 | 0.5647632 | 0.5646425 | 0.6 |

0.0001207 | 0.6443384 | 0.6442177 | 0.7 |

0.0001207 | 0.7174768 | 0.7173561 | 0.8 |

0.0001207 | 0.7834476 | 0.7833269 | 0.9 |

0.0001207 | 0.8415917 | 0.8414710 | 1.0 |

E r r o r = | E x a c t 6 − u | | u = sin ( x ) + 0.9808838911 | E x a c t 6 = 1 + sin ( x ) | x |
---|---|---|---|

0.0191161 | 1.0807173 | 1.0998334 | 0.1 |

0.0191161 | 1.1795532 | 1.1986693 | 0.2 |

0.0191161 | 1.2764041 | 1.2955202 | 0.3 |

0.0191161 | 1.3703022 | 1.3894183 | 0.4 |

0.0191161 | 1.4603094 | 1.4794255 | 0.5 |

0.0191161 | 1.5455264 | 1.5646425 | 0.6 |

0.0191161 | 1.6251016 | 1.6442177 | 0.7 |

0.0191161 | 1.6982400 | 1.7173561 | 0.8 |

0.0191161 | 1.7642108 | 1.7833269 | 0.9 |

0.0191161 | 1.8223549 | 1.8414710 | 1.0 |

with the exact solution u ( x ) = 1 + sin ( x ) .

Example 7. Consider the nonlinear Fredholm integral equation of the second kind

u ( x ) = cos ( x ) + 7 6 − 5 π 2 144 + 1 36 ∫ 0 π t ( u + u 2 ( t ) ) d t , (9)

with the exact solution u ( x ) = 1 + cos (x)

E r r o r = | E x a c t 7 − u | | u = cos ( x ) + 1.345517155 − 5 π 2 144 | E x a c t 7 = 1 + cos ( x ) | x |
---|---|---|---|

0.0191161 | 1.0807173 | 1.0998334 | 0.1 |

0.0191161 | 1.1795532 | 1.1986693 | 0.2 |

0.0191161 | 1.2764041 | 1.2955202 | 0.3 |

0.0191161 | 1.3703022 | 1.3894183 | 0.4 |

0.0191161 | 1.4603094 | 1.4794255 | 0.5 |

0.0191161 | 1.5455264 | 1.5646425 | 0.6 |

0.0191161 | 1.6251016 | 1.6442177 | 0.7 |

0.0191161 | 1.6982400 | 1.7173561 | 0.8 |

0.0191161 | 1.7642108 | 1.7833269 | 0.9 |

0.0191161 | 1.8223549 | 1.8414710 | 1.0 |

In the paper, a successive approximations method is presented for solving the nonlinear Fredholm integral equation of the second kind using Maple18. The benefit of our method lies in the fact that, for some nonlinear problems, our method is still convergent as illustrated by figures and tables showing match the right accuracy, which shows the exact solution with the approximate solution is largely identical and noticeable Tables 1-7 represent the exact and numerical results of the examples in this article. Figures 1-7 readily show the comparison of exact solution and approximate solution. We can see from the figures that the approximate solution is very applicable to the exact solution and application is displayed through some examples. Numerical results show that the accuracy of the solutions obtained is good.

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University. The author, therefore, acknowledges with thanks to DSR technical and financial support.

The author declares no conflicts of interest regarding the publication of this paper.

Maturi, D.A. (2019) The Successive Approximation Method for Solving Nonlinear Fredholm Integral Equation of the Second Kind Using Maple. Advances in Pure Mathematics, 9, 832-843. https://doi.org/10.4236/apm.2019.910040